selva86.github.io/Conditional-Probability-in-R.html at master · selva86/selva86.github.io · GitHub

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<!DOCTYPE html>
<html lang="en">
  <head>
    <title>Conditional Probability in R: P(A|B), Independence, and Bayes, With Real Examples</title>
    <meta charset="utf-8">
    <meta name="Description" content="Simulate conditional probability in R, prove independence with dice rolls, and apply Bayes' theorem to medical testing. Runnable code in your browser.">
    <meta name="Keywords" content="conditional probability in R, P(A|B) R, Bayes theorem R, independence test R, conditional probability simulation, Bayes rule R example, joint probability R, chi-squared independence test R">
    <meta name="Distribution" content="Global">
    <meta name="Author" content="Selva Prabhakaran">
    <meta name="Robots" content="index, follow">
    <meta name="viewport" content="width=device-width, initial-scale=1.0">
    <meta name="referrer" content="strict-origin-when-cross-origin">
    <link rel="icon" href="/screenshots/iconb-64.png" type="image/x-icon" />
    <link rel="canonical" href="https://r-statistics.co/Conditional-Probability-in-R.html">
    <link rel="alternate" type="application/atom+xml" title="r-statistics.co" href="https://r-statistics.co/feed.xml">
    <link rel="preconnect" href="https://cdn.jsdelivr.net">
    <!-- Preload the primary body font so first paint has IBM Plex Sans ready -->
    <link rel="preload" href="/www/fonts/ibm-plex/ibm-plex-sans-latin.woff2" as="font" type="font/woff2" crossorigin>
    <!-- Preload heading font (IBM Plex Serif 700) to prevent H1 font swap -->
    <link rel="preload" href="/www/fonts/ibm-plex/ibm-plex-serif-700.woff2" as="font" type="font/woff2" crossorigin>
    <!-- Critical CSS inlined for fast first paint -->
    <style>
      @font-face{font-family:'IBM Plex Sans';font-style:normal;font-weight:400;font-display:optional;src:url('/www/fonts/ibm-plex/ibm-plex-sans-latin.woff2') format('woff2')}
      @font-face{font-family:'IBM Plex Sans';font-style:normal;font-weight:500;font-display:optional;src:url('/www/fonts/ibm-plex/ibm-plex-sans-latin.woff2') format('woff2')}
      @font-face{font-family:'IBM Plex Sans';font-style:normal;font-weight:600;font-display:optional;src:url('/www/fonts/ibm-plex/ibm-plex-sans-latin.woff2') format('woff2')}
      @font-face{font-family:'IBM Plex Sans';font-style:normal;font-weight:700;font-display:optional;src:url('/www/fonts/ibm-plex/ibm-plex-sans-latin.woff2') format('woff2')}
      @font-face{font-family:'IBM Plex Serif';font-style:normal;font-weight:600;font-display:optional;src:url('/www/fonts/ibm-plex/ibm-plex-serif-600.woff2') format('woff2')}
      @font-face{font-family:'IBM Plex Serif';font-style:normal;font-weight:700;font-display:optional;src:url('/www/fonts/ibm-plex/ibm-plex-serif-700.woff2') format('woff2')}
      @font-face{font-family:'IBM Plex Mono';font-style:normal;font-weight:400;font-display:optional;src:url('/www/fonts/ibm-plex/ibm-plex-mono-400.woff2') format('woff2')}
      @font-face{font-family:'IBM Plex Mono';font-style:normal;font-weight:500;font-display:optional;src:url('/www/fonts/ibm-plex/ibm-plex-mono-500.woff2') format('woff2')}
      *,*::before,*::after{box-sizing:border-box}
      html{font-family:sans-serif;-webkit-text-size-adjust:100%}
      body{margin:0;font-family:'IBM Plex Sans',-apple-system,BlinkMacSystemFont,sans-serif;font-size:18px;line-height:1.7;color:#0d1117;background:#fafbfc}
      html body{font-size:18px;line-height:1.7}
      p{margin:0 0 15px}
      .container{max-width:1170px;margin:0 auto;padding:0 15px}
      @media(min-width:1200px){.container{width:1170px}}
      .row{margin-left:-15px;margin-right:-15px}.row::after{content:"";display:table;clear:both}
      .col-xs-12,.col-sm-2,.col-sm-3,.col-sm-7{position:relative;min-height:1px;padding-left:15px;padding-right:15px;float:left}
      .col-xs-12{width:100%}
      @media(min-width:768px){.col-sm-2{width:16.667%}.col-sm-3{width:25%}.col-sm-7{width:58.333%}.hidden-xs{display:block!important}}
      .hidden-xs{display:none}
      .table{width:100%;border-collapse:collapse}.table>thead>tr>th,.table>tbody>tr>td{padding:8px;border-top:1px solid #ddd}
      .table-striped>tbody>tr:nth-of-type(odd){background:#f9f9f9}
      .btn{display:inline-block;padding:6px 12px;font-size:14px;border-radius:4px;cursor:pointer;border:1px solid transparent}
      .btn-primary{color:#fff;background:#1d3158;border-color:#1d3158}.btn-primary:hover{background:#2c4574}
      .btn-default{color:#333;background:#fff;border-color:#ccc}
      .btn-sm{padding:3px 10px;font-size:12px}
      .form-control{display:block;padding:6px 12px;font-size:14px;border:1px solid #ccc;border-radius:4px}
      .list-unstyled{list-style:none;padding-left:0}
      .img-responsive{max-width:100%;height:auto}
      .pull-right{float:right}
      a{color:#1d3158;text-decoration:none}a:hover{text-decoration:underline}
      /* overflow-x:clip (not hidden) so html/body don't become a
         scrolling container, which would break position:sticky on the
         right TOC rail under layout-v2. */
      html,body{overflow-x:clip;max-width:100vw}
      #sidebar-nav{position:sticky;top:20px;max-height:calc(100vh - 40px);overflow-y:auto;padding:8px 0}
      .breadcrumb-nav{font-size:13.5px;font-family:-apple-system,BlinkMacSystemFont,'Segoe UI',sans-serif;color:#6c757d;margin-bottom:2px;padding:0}
      .breadcrumb-nav a{color:#6c757d;text-decoration:none}
      .bc-sep,.breadcrumb-sep{margin:0 5px;color:#757575}
      .bc-current,.breadcrumb-current{color:#495057;font-weight:500}
      #content h1,#content h2,#content h3,#content h4{font-family:'IBM Plex Serif',Georgia,'Times New Roman',serif}
      #content h1{font-weight:700;font-size:2.1em;line-height:1.2;margin:0 0 0.4em 0;letter-spacing:-0.01em;color:#0d1117}
      #content h2{font-weight:600;font-size:1.65em;line-height:1.25;margin:2.5em 0 0.6em 0;color:#0d1117;padding-top:0.8em;padding-bottom:6px;border-top:1px solid #d8dce2}
      #content h2:first-of-type{border-top:none;margin-top:1.5em}
      #content p{line-height:1.75;margin-bottom:1.15em;color:#0d1117}
      p.lead{font-size:1.1em;line-height:1.75;color:#4a5160;font-weight:400;border-left:3px solid #1d3158;padding-left:1em;margin:-4px 0 1.15em 0}
      .site-masthead{position:sticky;top:0;z-index:30;background:rgba(250,251,252,0.92);backdrop-filter:saturate(160%) blur(14px);border-bottom:1px solid transparent;margin:0 -15px 24px;transition:border-color 0.2s}
      .site-masthead.scrolled{border-bottom-color:#d8dce2}
      .site-masthead-inner{max-width:1170px;margin:0 auto;padding:14px 28px;display:flex;align-items:center;gap:24px}
      .masthead-menu-btn{display:none;background:none;border:1px solid #d8dce2;border-radius:6px;padding:4px 8px;font-size:18px;line-height:1;color:#4a5160;cursor:pointer}
      .masthead-wordmark{display:inline-flex;align-items:center;gap:10px;font-family:'IBM Plex Mono',monospace;font-weight:600;font-size:15px;color:#0d1117;text-decoration:none;white-space:nowrap}
      .masthead-mark{position:relative;width:32px;height:32px;border-radius:6px;background:linear-gradient(135deg,#1c2c4f,#33508c);color:#fff;display:inline-flex;align-items:center;justify-content:center;font-family:'IBM Plex Serif',Georgia,serif;font-style:italic;font-weight:700;font-size:19px;line-height:1;letter-spacing:-0.04em;padding-right:1px;box-shadow:0 1px 2px rgba(13,17,23,0.08),inset 0 1px 0 rgba(255,255,255,0.12)}
      .masthead-mark::after{content:'';position:absolute;right:5px;bottom:5px;width:4px;height:4px;border-radius:50%;background:rgba(255,255,255,0.85)}
      .masthead-tld{color:#757a87}
      .masthead-nav{display:flex;gap:4px;flex:1;margin-left:8px}
      .masthead-nav-link{font-family:'IBM Plex Sans',sans-serif;color:#4a5160;font-size:14px;font-weight:500;padding:6px 12px;border-radius:6px;text-decoration:none;transition:all 0.15s}
      .masthead-nav-link:hover{background:#f1f3f6;color:#0d1117;text-decoration:none}
      .masthead-nav-link.active{background:#f1f3f6;color:#0d1117}
      .masthead-tools{display:flex;align-items:center;gap:10px;flex-shrink:0}
      /* P7 — mock has a `/` kbd hint on the right of the search input.
         Positioned absolutely; the input gets extra right padding so the
         user-typed text doesn't overlap the hint. Pure visual cue — the
         shortcut wiring is intentionally deferred (Ace editor inside
         WebR code blocks uses `/` and would conflict). */
      .masthead-search{margin:0;display:flex;align-items:center;position:relative}
      .masthead-search input{height:32px;padding:6px 30px 6px 12px;font-size:13px;border:1px solid #d8dce2;border-radius:6px;background:#fff;color:#0d1117;width:160px;font-family:'IBM Plex Sans',sans-serif;transition:border-color 0.15s,box-shadow 0.15s}
      .masthead-search .kbd-hint{
        position:absolute;right:7px;top:50%;transform:translateY(-50%);
        font-family:'IBM Plex Mono',monospace;font-size:11px;font-weight:600;
        color:#6e7382;background:#eef1f7;border:1px solid #d8dce2;
        border-radius:4px;padding:1px 5px;line-height:1.2;pointer-events:none;
      }
      html.dark .masthead-search .kbd-hint{
        color:#9aa6c0;background:#1f2a48;border-color:#2c3a62;
      }
      .masthead-search input:focus{outline:none;border-color:#1d3158;box-shadow:0 0 0 3px rgba(29,49,88,0.10)}
      .masthead-icon-btn{background:none;border:1px solid transparent;border-radius:6px;width:32px;height:32px;display:inline-flex;align-items:center;justify-content:center;font-size:14px;color:#4a5160;cursor:pointer;transition:all 0.15s}
      .masthead-icon-btn:hover{background:#f1f3f6;color:#0d1117}
      .masthead-cta{display:inline-flex;align-items:center;height:32px;padding:0 14px;border-radius:8px;background:linear-gradient(135deg,#1c2c4f,#33508c);color:#fff;font-family:'IBM Plex Sans',sans-serif;font-size:12.5px;font-weight:600;text-decoration:none;white-space:nowrap;transition:filter 0.15s}
      .masthead-cta:hover{filter:brightness(1.1);color:#fff;text-decoration:none}
      .masthead-cta:link,.masthead-cta:visited,.masthead-cta:focus,.masthead-cta:active{color:#fff}
      html.dark .site-masthead{background:rgba(12,13,16,0.85)}
      html.dark .site-masthead.scrolled{border-bottom-color:#262a31}
      html.dark .masthead-wordmark{color:#e8eaee}
      html.dark .masthead-tld{color:#6e7382}
      html.dark .masthead-mark{background:#92a4d8}
      html.dark .masthead-nav-link{color:#b0b5bf}
      html.dark .masthead-nav-link:hover,html.dark .masthead-nav-link.active{background:#1a1d22;color:#e8eaee}
      .nav-dropdown{position:relative;display:inline-block}
      .nav-dropdown-trigger{cursor:pointer}
      .nav-dropdown-trigger::after{content:' \25BE';font-size:10px;color:#8a8f99}
      .nav-dropdown-panel{display:none;position:absolute;top:calc(100% + 6px);left:0;background:#fff;border:1px solid #e5e7eb;border-radius:8px;padding:14px 0;box-shadow:0 8px 24px rgba(13,17,23,0.10);min-width:260px;z-index:40;flex-direction:column}
      .nav-dropdown-panel::before{content:"";position:absolute;top:-6px;left:0;right:0;height:6px}
      .nav-dropdown:hover .nav-dropdown-panel,.nav-dropdown.open .nav-dropdown-panel{display:flex}
      .nav-cat{position:relative;display:flex;align-items:center;justify-content:space-between;padding:7px 18px;font-family:'IBM Plex Sans',sans-serif;font-size:13px;font-weight:500;color:#3a3f4a;cursor:pointer;transition:background 0.1s}
      .nav-cat:hover{background:#f1f3f6;color:#0d1117}
      .nav-cat::after{content:' \25B8';font-size:10px;color:#8a8f99;margin-left:8px}
      .nav-sub{display:none;position:absolute;top:0;left:100%;background:#fff;border:1px solid #e5e7eb;border-radius:8px;padding:14px 18px;box-shadow:0 8px 24px rgba(13,17,23,0.10);min-width:230px;z-index:50}
      .nav-sub::before{content:"";position:absolute;top:0;left:-8px;bottom:0;width:8px}
      .nav-cat:hover .nav-sub{display:block}
      .nav-sub a{display:block;padding:5px 0;font-size:13px;color:#3a3f4a;text-decoration:none;font-weight:400;line-height:1.4}
      .nav-sub a:hover{color:#1d3158;text-decoration:underline}
      .nav-sub h6{margin:10px 0 4px;font-family:'IBM Plex Sans',sans-serif;font-size:10px;font-weight:600;color:#8a8f99;text-transform:uppercase;letter-spacing:0.06em}
      .nav-sub h6:first-child{margin-top:0}
      html.dark .nav-dropdown-panel,html.dark .nav-sub{background:#161a1f;border-color:#2a2e35;box-shadow:0 8px 24px rgba(0,0,0,0.45)}
      html.dark .nav-cat{color:#c5cad3}
      html.dark .nav-cat:hover{background:#1a1d22;color:#fff}
      html.dark .nav-sub a{color:#c5cad3}
      html.dark .nav-sub a:hover{color:#fff}
      html.dark .nav-sub h6{color:#7a808c}
      @media (max-width: 768px){.nav-dropdown-panel{min-width:220px;left:auto;right:0}.nav-sub{position:static;border:none;box-shadow:none;padding:6px 0 6px 16px}.nav-cat::after{content:''}}
      html.dark .masthead-search input{background:#14161a;border-color:#262a31;color:#e8eaee}
      html.dark .masthead-search input:focus{border-color:#92a4d8;box-shadow:0 0 0 3px rgba(146,164,216,0.15)}
      html.dark .masthead-icon-btn{color:#b0b5bf}
      html.dark .masthead-icon-btn:hover{background:#1a1d22;color:#e8eaee}
      hr{height:0;box-sizing:content-box;border:0;border-top:1px solid #eee}
      .img-zoomable{cursor:zoom-in;transition:opacity .15s}
      .img-lightbox{display:none;position:fixed;top:0;left:0;right:0;bottom:0;z-index:10001;background:rgba(0,0,0,.85);cursor:zoom-out;align-items:center;justify-content:center}
      .img-lightbox.open{display:flex}
      .img-lightbox img{max-width:95vw;max-height:95vh;border-radius:4px;box-shadow:0 4px 24px rgba(0,0,0,.4)}
      /* CLS prevention: sidebar component styles (match main.css final state) */
      .sidebar-menu{margin:0;padding:0}
      .sidebar-tabs{display:flex;gap:0;margin:4px 0 0;border-bottom:1px solid #d8dce2}
      .sidebar-tab{flex:1;padding:9px 12px 8px;border:none;background:transparent;font:600 13px 'IBM Plex Sans',sans-serif;color:#757a87;cursor:pointer;border-bottom:2px solid transparent;margin-bottom:-1px;transition:color 0.15s,border-color 0.15s}
      .sidebar-tab.active{color:#1d3158;border-bottom-color:#1d3158}
      .sidebar-panel{display:none;padding-top:10px}
      .sidebar-panel.active{display:block}
      .sidebar-tools-list{margin:0;padding:0;list-style:none}
      .sidebar-tools-list li a{display:block;padding:5px 10px 5px 24px;font-size:14px;color:#495057;text-decoration:none;border-left:2px solid transparent;border-radius:0 4px 4px 0}
      .sidebar-section-header{padding:7px 12px;font-size:15px;font-weight:700;color:#212529;cursor:pointer;border-radius:4px}
      .sidebar-chevron{display:inline-block;font-size:10px;margin-right:4px;color:#757575}
      .sidebar-section-items{display:none;padding-left:0;margin:2px 0 6px}
      .sidebar-section.expanded .sidebar-section-items{display:block}
      .sidebar-section-items li{line-height:1.5}
      .sidebar-section-items li a{display:block;padding:4px 10px 4px 32px;font-size:14px;color:#495057;text-decoration:none;border-left:2px solid transparent;border-radius:0 4px 4px 0}
      .sidebar-divider{margin:12px 0 2px;padding:0 12px 4px 24px;font-size:11px;font-weight:700;letter-spacing:0.06em;text-transform:uppercase;color:#9ca3af;border-bottom:1px solid #f0f2f5;list-style:none}
      .sidebar-divider:first-child{margin-top:4px}
      .progress-dot{display:inline-block;width:10px;height:10px;border-radius:50%;border:1.5px solid #d1d5db;margin-right:7px;vertical-align:middle;flex-shrink:0;position:relative}
      /* Empty progress-dot (no status class) reads as an unchecked todo /
         disabled radio next to every sidebar item — visual noise that doesn't
         belong in a nav. Reserve the layout slot but hide the circle until
         the item actually has progress. visibility:hidden (not display:none)
         keeps the width stable when state flips. */
      .progress-dot:not(.started):not(.visited):not(.completed){visibility:hidden}
      /* CLS prevention: TOC sidebar matches main.css final state */
      #toc-wrapper{position:sticky;top:20px;max-height:calc(100vh - 40px);overflow-y:auto;padding-left:14px;border-left:1px solid #e9ecef}
      .toc-title{font-size:11px;font-weight:600;text-transform:uppercase;letter-spacing:0.8px;color:#6c757d;margin-bottom:10px;padding-bottom:0}
      #toc{margin:0;padding:0}
      #toc li{line-height:1.45;margin-bottom:1px}
      #toc li a{font-size:13.5px;color:#6c757d;text-decoration:none;display:block;padding:3px 0;border-left:2px solid transparent;padding-left:8px;margin-left:-15px}
      #toc li a.toc-active{color:#1d3158;font-weight:600;border-left-color:#1d3158}
      /* CLS prevention: engagement strip layout (matches engagement.css) */
      .engagement-header{display:flex;flex-wrap:wrap;align-items:center;gap:6px 10px;font-family:'Inter',-apple-system,BlinkMacSystemFont,'Segoe UI',sans-serif;font-size:12px;color:#64748b;letter-spacing:0.01em;margin:0.4em 0 1em;min-height:22px}
      .engagement-header:empty{display:none}
      /* CLS prevention: progress bar space (matches engagement.css) */
      .engagement-progress{min-height:32px;margin:1.2em 0 1.6em}
      /* CLS prevention: WebR editor min-height so the static→textarea swap doesn't shift layout */
      .webr-editor{min-height:60px}
      /* CLS prevention: images in content always reserve aspect ratio */
      #content img{max-width:100%;height:auto}

      /* =====================================================
         LAYOUT V2 (opt-in via body.layout-v2). Retry 2026-05-29.
         Differences from the first attempt:
           1. Sidebar columns use clamp(180px, 16vw, 260px) so they
              SHRINK on narrow viewports (first attempt used minmax
              which pinned both sides at 260px, squeezing content on
              typical laptops to ~460px).
           2. Sidebar links get overflow-wrap so long tutorial titles
              don't escape the narrower column.
           3. min-width:0 on every grid child so they can shrink.
           4. Right TOC breakpoint raised from 1100px -> 1280px so it
              only shows when the viewport can actually afford it.
           5. The inline-code background restyle is dropped - kept
              current syntax-highlighter behavior.
         All rules scoped under body.layout-v2 so pages without it
         render identically.
         ===================================================== */
      body.layout-v2{
        --lv2-bg:#f8f9fb;--lv2-card:#fff;--lv2-ink:#0d1117;
        --lv2-mut:#4b5260;--lv2-faint:#5b6573;
        --lv2-navy:#1c2c4f;--lv2-navy-h:#2d4173;--lv2-navy-soft:#eef1f7;
        --lv2-accent:#2056d2;--lv2-accent-soft:#e9efff;
        --lv2-line:#e4e7ee;--lv2-border:#d4d9e3;
        --lv2-green:#15803d;
        background:var(--lv2-bg);color:var(--lv2-ink);
      }
      html.dark body.layout-v2{
        --lv2-bg:#0a0f1c;--lv2-card:#0f1525;--lv2-ink:#eef2fa;
        --lv2-mut:#a5afc4;--lv2-faint:#9aa6c0;
        --lv2-navy:#5170b3;--lv2-navy-h:#6587c4;--lv2-navy-soft:#192240;
        --lv2-accent:#7faaff;--lv2-accent-soft:#1a2748;
        --lv2-line:#1f2a48;--lv2-border:#2c3a62;
      }

      /* Wider container; clamp-based grid that scales with viewport.
         width:auto is REQUIRED because the base critical CSS has
         @media(min-width:1200px){.container{width:1170px}} which would
         otherwise pin width to 1170 and ignore max-width:1340. */
      body.layout-v2 .container{max-width:1340px;width:auto;padding:0 22px}
      body.layout-v2 .site-masthead-inner{max-width:1340px}
      /* The legacy .site-masthead has margin:0 -15px to cancel the
         Bootstrap .container padding of 15px. Under layout-v2 the
         container padding is 22px, so the negative margin leaves a
         7px inset. Zero it so the masthead aligns flush with the
         article shell. */
      body.layout-v2 .site-masthead{margin-left:0;margin-right:0}
      body.layout-v2 .row{
        display:grid;
        grid-template-columns:
          clamp(180px, 16vw, 260px)
          minmax(0, 1fr)
          clamp(180px, 16vw, 260px);
        gap:0;margin-left:0;margin-right:0;
      }
      /* Bootstrap inserts .row::before AND .row::after pseudo-elements
         (content:" "; display:table) as a clearfix. Under display:grid
         these pseudos count as grid items. ::before grabs col 1, pushing
         every real child one column to the right and wrapping the last
         child to a new row. The 2026-05-29 first attempt only disabled
         ::after, resulting in nav in col 2, content in col 3, and toc
         wrapping to a new row at col 1 (the bug the user reported as
         "sidebar wide, content narrow"). BOTH pseudos must be killed. */
      body.layout-v2 .row::before,
      body.layout-v2 .row::after{display:none;content:none}
      /* min-width:0 lets grid children shrink below their content's
         intrinsic min-width; without this, long unbreakable text in any
         child forces the column wider than the grid template intended. */
      /* Drop Bootstrap's float and width on the legacy .col-sm-* hooks
         so the parent .row's CSS grid owns sizing. DO NOT zero horizontal
         padding here — per-column padding lives in the dedicated rules
         below (sidebar, main#content, #toc-sidebar) and gives the article
         the 44px breathing room the mock has. */
      body.layout-v2 #nav.col-sm-3,
      body.layout-v2 main#content.col-sm-7,
      body.layout-v2 #toc-sidebar.col-sm-2{
        width:auto!important;float:none!important;
        min-width:0;min-height:0;
      }
      body.layout-v2 #nav.col-sm-3{
        padding:14px 10px 40px;border-right:1px solid var(--lv2-line);
        position:sticky;top:60px;height:calc(100vh - 60px);overflow-y:auto;
      }
      /* Inner #sidebar-nav had its own sticky + overflow from the legacy critical
         CSS; under layout-v2 the outer #nav owns scrolling, so flatten the inner
         one to avoid a double scrollbar. */
      body.layout-v2 #sidebar-nav{
        position:static;max-height:none;overflow:visible;padding:0;
      }
      /* Wrap long tutorial titles in the sidebar so they never overflow */
      body.layout-v2 #nav a,
      body.layout-v2 #nav span,
      body.layout-v2 #sidebar-nav a,
      body.layout-v2 #sidebar-nav li{
        overflow-wrap:break-word;word-break:break-word;
      }
      body.layout-v2 main#content{padding:24px 44px 60px}

      /* S3 — Code-block bottom spacing.
         Default .webr-container margin is 22/22; the dark block visually
         carries a lot of weight, so the next paragraph reads as cramped.
         Lift bottom to 30px under layout-v2. */
      body.layout-v2 #content .webr-container{margin-bottom:30px}

      /* S7 — Further Reading: convert auto-injected <ul> to a 2-column
         card grid that matches the .rt-grid pattern used elsewhere.
         No HTML/auto_link.py changes needed — pure CSS. */
      body.layout-v2 #content #auto-further-reading .further-reading-list{
        list-style:none;padding:0;margin:14px 0 0;
        display:grid;grid-template-columns:repeat(2,minmax(0,1fr));gap:12px;
      }
      body.layout-v2 #content #auto-further-reading .further-reading-list li{margin:0}
      body.layout-v2 #content #auto-further-reading .further-reading-list li a{
        display:block;padding:14px 16px;
        background:var(--lv2-card);border:1px solid var(--lv2-border);
        border-radius:8px;color:var(--lv2-ink);
        font-weight:500;font-size:15px;line-height:1.4;
        text-decoration:none;
        transition:border-color .15s,background .15s,transform .15s;
      }
      body.layout-v2 #content #auto-further-reading .further-reading-list li a:hover{
        border-color:var(--lv2-accent);background:var(--lv2-accent-soft);
        text-decoration:none;transform:translateY(-1px);
      }
      @media(max-width:820px){
        body.layout-v2 #content #auto-further-reading .further-reading-list{
          grid-template-columns:1fr;
        }
      }
      body.layout-v2 #toc-sidebar{padding:24px 18px;position:sticky;top:74px;align-self:start}
      @media(max-width:1280px){
        body.layout-v2 .row{grid-template-columns:clamp(180px, 18vw, 260px) minmax(0,1fr)}
        body.layout-v2 #toc-sidebar{display:none}
      }
      @media(max-width:820px){
        body.layout-v2 .row{grid-template-columns:1fr}
        body.layout-v2 #nav{display:none}
        body.layout-v2 main#content{padding:20px}
      }

      /* Typography refresh - sized to mock spec */
      body.layout-v2 #content{
        font-size:16.5px;line-height:1.7;color:var(--lv2-ink);
        font-feature-settings:"tnum","ss01","cv11";text-rendering:optimizeLegibility;
      }
      body.layout-v2 #content p{line-height:1.7;margin:13px 0;color:var(--lv2-ink);text-wrap:pretty}
      body.layout-v2 #content h1{
        font-size:38px;font-weight:700;line-height:1.2;letter-spacing:-0.018em;
        margin:10px 0 13px;text-wrap:balance;color:var(--lv2-ink);
      }
      /* H2: hairline divider + breathing room above. Mock breaks each
         section with a top rule and ~48px space; staging used 34px and no
         divider, leaving sections to blur together. :first-of-type keeps
         the very first H2 flush with the actionbar (no double divider). */
      body.layout-v2 #content h2{
        font-size:25px;font-weight:600;line-height:1.25;letter-spacing:-0.018em;
        margin:48px 0 12px;padding-top:32px;border-top:1px solid var(--lv2-line);
        text-wrap:balance;color:var(--lv2-ink);
      }
      body.layout-v2 #content h2:first-of-type{
        margin-top:24px;padding-top:0;border-top:none;
      }
      body.layout-v2 #content h3,body.layout-v2 #content h4{
        font-size:18px;font-weight:600;line-height:1.3;margin:24px 0 8px;text-wrap:balance;color:var(--lv2-ink);
      }
      /* Specificity bump: `body.layout-v2 #content p` (1 id + 1 class
         + 2 elements) was beating `body.layout-v2 p.lead` (2 classes +
         2 elements) and forcing the dek to var(--lv2-ink) instead of
         var(--lv2-mut). Adding #content here matches that specificity. */
      body.layout-v2 #content p.lead{
        font-size:18.5px;line-height:1.6;color:var(--lv2-mut);
        border-left:3px solid var(--lv2-navy);padding-left:17px;margin:-2px 0 20px;font-weight:400;
      }

      /* Body link color: legacy a{color:#1d3158} and Bootstrap .text-muted
         leave inline links almost indistinguishable from body text. Use
         the mock accent so links are obviously clickable. Excludes
         buttons, cards, prev/next nav, and TOC items which have their
         own palette. */
      body.layout-v2 #content a:not(.btn):not(.rt-card):not(.pn-link):not(.lv2-anchor):not([class*="toc"]){
        color:var(--lv2-accent);
      }
      body.layout-v2 #content a:not(.btn):not(.rt-card):not(.pn-link):not(.lv2-anchor):not([class*="toc"]):hover{
        text-decoration:underline;
      }

      /* Inline <code> palette: match the mock's navy-soft tint instead
         of the legacy purple-blue (#eef vs #eef1f7). */
      body.layout-v2 #content :not(pre) > code{
        background:var(--lv2-navy-soft);color:var(--lv2-ink);
        border-radius:4px;padding:2px 5px;font-size:0.92em;
      }

      /* TOC right rail */
      body.layout-v2 #toc-sidebar .toc-title{font-size:10.5px;font-weight:700;text-transform:uppercase;letter-spacing:.08em;color:var(--lv2-faint);margin-bottom:9px}
      body.layout-v2 #toc{font-size:13px}
      body.layout-v2 #toc li{margin:0;line-height:1.45}
      body.layout-v2 #toc li a{
        display:block;padding:5px 0 5px 11px;
        border-left:2px solid var(--lv2-line);
        color:var(--lv2-faint);font-size:13px;
        transition:color .15s,border-color .15s;
      }
      body.layout-v2 #toc li a:hover{color:var(--lv2-ink)}
      body.layout-v2 #toc li a.toc-active{
        border-left-color:var(--lv2-navy);color:var(--lv2-ink);font-weight:600;
      }

      /* Reading progress bar - uses transform:scaleX so we don't fight any
         containing-block / overflow quirks that can pin a width-% bar to 0. */
      .lv2-progress{
        position:fixed;top:0;left:0;width:100vw;height:3px;
        background:var(--lv2-accent,#2056d2);z-index:90;
        transform:scaleX(0);transform-origin:left center;
        transition:transform .1s ease-out;pointer-events:none;
      }

      /* Heading anchor links (hover-reveal) */
      body.layout-v2 #content h2,
      body.layout-v2 #content h3,
      body.layout-v2 #content h4{position:relative}
      body.layout-v2 .lv2-anchor{
        position:absolute;left:-26px;top:50%;transform:translateY(-50%);
        font-size:.55em;color:var(--lv2-faint);text-decoration:none;
        opacity:0;transition:opacity .15s,color .15s;
        cursor:pointer;padding:4px 6px;line-height:1;font-weight:400;
      }
      body.layout-v2 #content h2:hover .lv2-anchor,
      body.layout-v2 #content h3:hover .lv2-anchor,
      body.layout-v2 #content h4:hover .lv2-anchor,
      body.layout-v2 .lv2-anchor:focus-visible{opacity:1}
      body.layout-v2 .lv2-anchor:hover{color:var(--lv2-accent)}
      @media(max-width:820px){body.layout-v2 .lv2-anchor{left:0;top:-22px;transform:none}}

      /* Focus ring refresh */
      body.layout-v2 a:focus-visible,
      body.layout-v2 button:focus-visible,
      body.layout-v2 input:focus-visible,
      body.layout-v2 textarea:focus-visible{
        outline:none;
        box-shadow:0 0 0 3px rgba(32,86,210,.22);
        border-radius:4px;
      }
      html.dark body.layout-v2 a:focus-visible,
      html.dark body.layout-v2 button:focus-visible,
      html.dark body.layout-v2 input:focus-visible{box-shadow:0 0 0 3px rgba(127,170,255,.22)}

      /* Reduced motion: shut everything down for users who request it */
      @media (prefers-reduced-motion: reduce){
        body.layout-v2 *,body.layout-v2 *::before,body.layout-v2 *::after{
          animation-duration:0.01ms!important;
          animation-iteration-count:1!important;
          transition-duration:0.01ms!important;
          scroll-behavior:auto!important;
        }
      }
    </style>
    <!-- Full Bootstrap deferred (non-render-blocking) -->
    <link href="www/bootstrap.min.css?h=dae0a568" rel="stylesheet" media="print" onload="this.media='all'">
    <noscript><link href="www/bootstrap.min.css?h=dae0a568" rel="stylesheet"></noscript>
    <link href="www/highlight.min.css?h=f103014a" rel="stylesheet" media="print" onload="this.media='all'">
    <link href="css/main.min.css?h=5ca8bbcb" rel="stylesheet" media="print" onload="this.media='all'">
    <noscript><link href="css/main.min.css?h=5ca8bbcb" rel="stylesheet"></noscript>
    <style>
      a{color:#1d3158}a:hover{text-decoration:underline}
      li{line-height:1.65}
      .MathJax_Display{margin:0}
      blockquote p{line-height:1.75;color:#717171}
      /* engagement.js renders a meta strip into .engagement-header that
         duplicates the byline (difficulty + time). Hide it; saved-posts-
         button.js reads the data-* attrs directly for the byline. */
      .engagement-header{display:none!important}
      /* Byline (saved-posts widget rebuilds this after H1+lead, mock-style) */
      .byline{display:flex;flex-wrap:wrap;align-items:center;gap:8px;
        font-size:13.5px;color:#4b5260;margin:6px 0 16px;line-height:1.5}
      /* P1 — bump avatar 26→30 to match mock visual weight. */
      .byline .av-sm{width:30px;height:30px;border-radius:50%;
        background:linear-gradient(135deg,#1c2c4f,#2d4173);color:#fff;
        display:inline-flex;align-items:center;justify-content:center;
        font-weight:700;font-size:12px;flex:none;letter-spacing:.02em}
      .byline .who{color:#0d1117;font-weight:600}
      .byline .dot{color:#c0c5cf}
      .byline .rev{color:#137a3e;font-weight:600;display:inline-flex;align-items:center;gap:4px}
      .byline .time{display:inline-flex;align-items:center;gap:4px;color:#6b7280}
      .byline .diff{font-size:10.5px;font-weight:700;text-transform:uppercase;
        letter-spacing:.05em;background:#eef1f7;color:#4b5260;
        padding:3px 8px;border-radius:999px;margin-left:2px}

      /* Action bar (saved-posts widget injects this after byline) */
      .actionbar{display:flex;align-items:center;gap:9px;margin:10px 0 26px;padding:12px 0;
        border-top:1px solid #e4e7ee;border-bottom:1px solid #e4e7ee;flex-wrap:wrap}
      .actionbar .act{display:inline-flex;align-items:center;gap:6px;border:1px solid #d4d9e3;
        background:#fff;border-radius:9px;padding:7px 12px;font-size:13px;font-weight:600;
        color:#4b5260;cursor:pointer;font-family:inherit;
        transition:border-color .15s,color .15s,background .15s}
      .actionbar .act:hover{border-color:#0a0d14;color:#0a0d14}
      .actionbar .act.bookmark-btn[data-saved="1"]{background:#fef3c7;border-color:#f59e0b;color:#92400e}
      .actionbar .act.bookmark-btn[data-saved="1"] svg{fill:#f59e0b}
      .actionbar .act:disabled{opacity:.55;cursor:wait}
      .actionbar-sync{margin-left:auto;font-size:12px;color:#6b7280}
      .actionbar-sync a{color:#2056d2;font-weight:600;text-decoration:none}
      .actionbar-sync a:hover{text-decoration:underline}

      /* Dark-mode overrides for byline + actionbar. The hard-coded #0d1117
         on .who renders invisible on the dark background; the action-bar
         borders and surfaces also need to flip. Tokens (--lv2-ink, --lv2-mut,
         --lv2-line, --lv2-card) already swap under html.dark body.layout-v2;
         we just need to point the hard-coded rules at them under dark. */
      html.dark body.layout-v2 .byline{color:var(--lv2-mut)}
      html.dark body.layout-v2 .byline .who{color:var(--lv2-ink)}
      html.dark body.layout-v2 .byline .dot{color:var(--lv2-faint)}
      html.dark body.layout-v2 .byline .time{color:var(--lv2-mut)}
      html.dark body.layout-v2 .byline .diff{background:var(--lv2-navy-soft);color:var(--lv2-mut)}
      html.dark body.layout-v2 .actionbar{border-top-color:var(--lv2-line);border-bottom-color:var(--lv2-line)}
      html.dark body.layout-v2 .actionbar .act{background:var(--lv2-card);border-color:var(--lv2-border);color:var(--lv2-mut)}
      html.dark body.layout-v2 .actionbar .act:hover{border-color:var(--lv2-ink);color:var(--lv2-ink)}
      html.dark body.layout-v2 .actionbar-sync{color:var(--lv2-mut)}

      /* Toast (Share button "Link copied" feedback) */
      .rs-toast{position:fixed;bottom:24px;left:50%;transform:translateX(-50%) translateY(8px);
        background:#0a0d14;color:#fff;padding:9px 18px;border-radius:8px;font-size:13px;
        font-family:'IBM Plex Sans',sans-serif;z-index:1000;opacity:0;pointer-events:none;
        transition:opacity .2s,transform .2s;box-shadow:0 8px 24px rgba(10,13,20,.25)}
      .rs-toast.show{opacity:1;transform:translateX(-50%) translateY(0)}
      /* Auth-state visibility helpers (used by masthead variants) */
      .auth-anon,.auth-user{display:none}
      body.state-anon .auth-anon{display:inline-flex;align-items:center}
      body.state-pro .auth-user{display:inline-flex;align-items:center;gap:8px}
      .masthead-auth-link{font-family:'IBM Plex Sans',sans-serif;color:#4a5160;font-size:13.5px;font-weight:500;padding:6px 10px;border-radius:6px;text-decoration:none;transition:background .15s,color .15s;max-width:160px;overflow:hidden;text-overflow:ellipsis;white-space:nowrap}
      .masthead-auth-link:hover{background:#f1f3f6;color:#0d1117;text-decoration:none}
      html.dark .masthead-auth-link{color:#c3cad9}
      html.dark .masthead-auth-link:hover{background:#1f242c;color:#fff}
      .auth-signout.masthead-icon-btn{background:none;border:1px solid #d8dce2;border-radius:6px;padding:4px 9px;font-size:14px;line-height:1;color:#4a5160;cursor:pointer;transition:border-color .15s,color .15s}
      .auth-signout.masthead-icon-btn:hover{border-color:#9a1f1f;color:#9a1f1f}
      html.dark .auth-signout.masthead-icon-btn{border-color:#262a31;color:#c3cad9}
      html.dark .auth-signout.masthead-icon-btn:hover{border-color:#f08989;color:#f08989}
      @media(min-width:1200px){.container{width:1170px}}
      #nav,#content,#toc-sidebar{box-sizing:border-box}
      #content{padding-left:15px;padding-right:15px;overflow-wrap:break-word;word-wrap:break-word;overflow-x:clip}
      @media(max-width:767px){#nav{display:none}.masthead-menu-btn{display:inline-flex!important}.masthead-nav{display:none}.masthead-search{display:none}.site-masthead-inner{padding:10px 16px;gap:12px}}
      @media(max-width:400px){.masthead-name{font-size:13px}}
      .mobile-sidebar-overlay{display:none;position:fixed;top:0;left:0;right:0;bottom:0;z-index:9999;background:rgba(0,0,0,0.4)}
      .mobile-sidebar-overlay.open{display:block}
      .mobile-sidebar-panel{position:fixed;top:0;left:0;bottom:0;width:280px;background:#fff;z-index:10000;overflow-y:auto;padding:16px;box-shadow:2px 0 16px rgba(0,0,0,0.15);transform:translateX(-100%);transition:transform 0.25s ease}
      .mobile-sidebar-overlay.open .mobile-sidebar-panel{transform:translateX(0)}
      .mobile-sidebar-close{position:absolute;top:10px;right:12px;background:none;border:none;font-size:22px;cursor:pointer;color:#666}
      /* Mobile top-nav drawer section */
      .mnav-section{margin-bottom:18px;padding-bottom:14px;border-bottom:1px solid #e5e7eb}
      .mnav-section:last-child{border-bottom:none}
      .mnav-title{font-family:'IBM Plex Sans',sans-serif;font-size:11px;font-weight:600;color:#8a8f99;text-transform:uppercase;letter-spacing:0.06em;margin:0 0 8px}
      .mnav-link{display:block;padding:9px 10px;color:#3a3f4a;text-decoration:none;font-size:15px;border-radius:6px;font-family:'IBM Plex Sans',sans-serif}
      .mnav-link:hover,.mnav-link:active{background:#f1f3f6;color:#0d1117;text-decoration:none}
      .mnav-accordion{padding:0}
      .mnav-accordion>summary{padding:9px 10px;cursor:pointer;font-size:15px;font-weight:500;color:#3a3f4a;list-style:none;border-radius:6px;font-family:'IBM Plex Sans',sans-serif;display:flex;align-items:center;justify-content:space-between}
      .mnav-accordion>summary::-webkit-details-marker{display:none}
      .mnav-accordion>summary::after{content:'\25BE';font-size:11px;color:#8a8f99;transition:transform 0.15s}
      .mnav-accordion[open]>summary::after{transform:rotate(180deg)}
      .mnav-accordion>summary:hover{background:#f1f3f6}
      .mnav-accordion-body{padding:4px 6px 8px 10px}
      .mnav-accordion-body .mnav-link{padding:7px 8px;font-size:14px;color:#4a5160}
      .mnav-accordion-body h6{margin:10px 0 4px 8px;font-family:'IBM Plex Sans',sans-serif;font-size:10px;font-weight:600;color:#8a8f99;text-transform:uppercase;letter-spacing:0.06em}
      html.dark .mobile-sidebar-panel{background:#1e293b}
      html.dark .mobile-sidebar-overlay{background:rgba(0,0,0,0.6)}
      html.dark .mobile-sidebar-close{color:#94a3b8}
      html.dark .mnav-section{border-bottom-color:#334155}
      html.dark .mnav-title{color:#7a808c}
      html.dark .mnav-link,html.dark .mnav-accordion>summary{color:#c5cad3}
      html.dark .mnav-link:hover,html.dark .mnav-link:active,html.dark .mnav-accordion>summary:hover{background:#1a1d22;color:#fff}
      html.dark .mnav-accordion-body .mnav-link{color:#b0b5bf}
      html.dark .mnav-accordion-body h6{color:#7a808c}
    </style>
    <!-- Open Graph -->
    <meta property="og:title" content="Conditional Probability in R: P(A|B), Independence, and Bayes, With Real Examples">
    <meta property="og:description" content="Simulate conditional probability in R, prove independence with dice rolls, and apply Bayes' theorem to medical testing. Runnable code in your browser.">
    <meta property="og:type" content="article">
    <meta property="og:url" content="https://r-statistics.co/Conditional-Probability-in-R.html">
    <meta property="og:site_name" content="r-statistics.co">
    <meta property="og:image" content="https://r-statistics.co/screenshots/og/Conditional-Probability-in-R.png">
    <meta name="twitter:card" content="summary_large_image">
    <meta name="twitter:title" content="Conditional Probability in R: P(A|B), Independence, and Bayes, With Real Examples">
    <meta name="twitter:description" content="Simulate conditional probability in R, prove independence with dice rolls, and apply Bayes' theorem to medical testing. Runnable code in your browser.">
    <meta name="twitter:image" content="https://r-statistics.co/screenshots/og/Conditional-Probability-in-R.png">
    <!-- JSON-LD Structured Data -->
    <script type="application/ld+json">
    {
      "@context": "https://schema.org",
      "@type": ["TechArticle", "LearningResource"],
      "headline": "Conditional Probability in R: P(A|B), Independence, and Bayes, With Real Examples",
      "description": "Simulate conditional probability in R, prove independence with dice rolls, and apply Bayes' theorem to medical testing. Runnable code in your browser.",
      "author": {"@type": "Person", "name": "Selva Prabhakaran", "url": "https://r-statistics.co/about/", "jobTitle": "Data Scientist"},
      "publisher": {"@type": "Organization", "name": "r-statistics.co", "url": "https://r-statistics.co/", "logo": {"@type": "ImageObject", "url": "https://r-statistics.co/screenshots/og-default.png"}},
      "url": "https://r-statistics.co/Conditional-Probability-in-R.html",
      "datePublished": "2026-05-10",
      "dateModified": "2026-05-10",
      "inLanguage": "en",
      "educationalLevel": "Intermediate",
      "programmingLanguage": "R",
      "speakable": {"@type": "SpeakableSpecification", "cssSelector": [".lead", "#content h1"]}
    }
    </script>
        <script type="application/ld+json">
    {
      "@context": "https://schema.org",
      "@type": "BreadcrumbList",
      "itemListElement": [
        {"@type": "ListItem", "position": 1, "name": "Home", "item": "https://r-statistics.co/"},
        {"@type": "ListItem", "position": 2, "name": "Statistics", "item": "https://r-statistics.co/"},
        {"@type": "ListItem", "position": 3, "name": "Conditional Probability in R: P(A|B), Independence, and Bayes, With Real Examples", "item": "https://r-statistics.co/Conditional-Probability-in-R.html"}
      ]
    }
    </script>

        <link rel="stylesheet" href="www/webr.min.css?h=23e578a3">

        <link rel="stylesheet" href="www/engagement.min.css?h=6cdbfd3e" media="print" onload="this.media='all'">
    <noscript><link rel="stylesheet" href="www/engagement.min.css?h=6cdbfd3e"></noscript>

  </head>

  <body>
    <!-- LAYOUT V2 (default for all tutorial pages as of 2026-05-29).
         Was previously opt-in via a V2_PAGES whitelist; reference work on
         /R-Vectors validated the design, so it now applies unconditionally
         to every page served from this template.

         To opt a specific page OUT (e.g. landing/marketing pages that need
         the legacy look), add `data-no-lv2` to its <body> or set
         window.__noLv2 = true in an inline script before this block. -->
    <script>
      (function(){
        if (window.__noLv2 || document.body.hasAttribute('data-no-lv2')) return;
        document.body.classList.add('layout-v2');
      })();
    </script>
    <!-- Mobile sidebar overlay -->
    <div class="mobile-sidebar-overlay" id="mobile-sidebar">
      <div class="mobile-sidebar-panel">
        <button class="mobile-sidebar-close" id="mobile-sidebar-close">&times;</button>

        <!-- Top nav (mobile only) -->
        <div class="mnav-section">
          <h6 class="mnav-title">Navigate</h6>
          <a class="mnav-link" href="/roadmap/">Roadmap</a>
          <a class="mnav-link" href="/tutorials/">Tutorials</a>
          <a class="mnav-link" href="/exercises/">Exercises</a>
          <a class="mnav-link" href="/tools/">Tools</a>
          <a class="mnav-link" href="/certifications">Certification</a>
          <a class="mnav-link mnav-cta" href="/pricing.html">Get certified</a>
        </div>

        <div id="mobile-sidebar-content">
          <!-- Populated by toc.js (curriculum nav) -->
        </div>
      </div>
    </div>

    <div class="container">

      <header class="site-masthead">
        <div class="site-masthead-inner">
          <button id="mobile-menu-btn" class="masthead-menu-btn" aria-label="Menu">&#9776;</button>
          <a class="masthead-wordmark" href="/">
            <span class="masthead-mark">R</span>
            <span class="masthead-name">r&#8209;statistics<span class="masthead-tld">.co</span></span>
          </a>
          <nav class="masthead-nav">
            <a class="masthead-nav-link" href="/roadmap/">Roadmap</a>
            <a class="masthead-nav-link" href="/tutorials/">Tutorials</a>
            <a class="masthead-nav-link" href="/exercises/">Exercises</a>
            <a class="masthead-nav-link" href="/tools/">Tools</a>
            <a class="masthead-nav-link" href="/certifications">Certification</a>
          </nav>
          <div class="masthead-tools">
            <form onsubmit="my_search_google(); return false;" class="masthead-search">
              <input type="text" id="my-google-search" placeholder="Search…" aria-label="Search r-statistics.co">
              <span class="kbd-hint" aria-hidden="true">/</span>
            </form>
            <a class="masthead-cta" href="/pricing.html">Get certified</a>
            <button id="dark-mode-toggle" class="masthead-icon-btn" aria-label="Toggle dark mode" title="Toggle dark mode">&#9789;</button>
            <span class="auth-anon"><a href="/signin.html" class="masthead-auth-link">Sign in</a></span>
            <span class="auth-user"><!-- auth-hydrate.js fills with avatar + dropdown --></span>
          </div>
        </div>
      </header>

      <div class="row">
        <div class="col-xs-12 col-sm-3" id="nav">
          <div id="sidebar-nav"><div class="continue-chip" data-continue-chip><span class="chip-label">Continue reading</span><a href="#" data-continue-link></a></div><div class="sidebar-tabs" role="tablist"><button class="sidebar-tab active" data-tab="posts" type="button" role="tab" onclick="var n=this.dataset.tab;document.querySelectorAll('.sidebar-tab').forEach(function(x){x.classList.toggle('active',x.dataset.tab===n)});document.querySelectorAll('.sidebar-panel').forEach(function(p){p.classList.toggle('active',p.dataset.panel===n)});try{localStorage.setItem('rstat_sidebar_tab',n)}catch(e){}">Posts</button><button class="sidebar-tab" data-tab="tools" type="button" role="tab" onclick="var n=this.dataset.tab;document.querySelectorAll('.sidebar-tab').forEach(function(x){x.classList.toggle('active',x.dataset.tab===n)});document.querySelectorAll('.sidebar-panel').forEach(function(p){p.classList.toggle('active',p.dataset.panel===n)});try{localStorage.setItem('rstat_sidebar_tab',n)}catch(e){}">Tools</button></div><div class="sidebar-panel active" data-panel="posts"><ul class="sidebar-menu list-unstyled"><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">&#9656;</span> Learn R<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec0sub1" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Getting Started</li><li data-subkey="sec0sub1"><a href="/Is-R-Worth-Learning-in-2026.html"><span class="progress-dot"></span>Is R Worth Learning?</a></li><li data-subkey="sec0sub1"><a href="/Install-R-and-RStudio-2026.html"><span class="progress-dot"></span>Install R &amp; RStudio</a></li><li data-subkey="sec0sub1"><a href="/RStudio-IDE-Tour.html"><span class="progress-dot"></span>RStudio IDE Tour</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec0sub2" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> R Fundamentals</li><li data-subkey="sec0sub2"><a href="/R-Syntax-101.html"><span class="progress-dot"></span>R Syntax 101</a></li><li data-subkey="sec0sub2"><a href="/R-Data-Types.html"><span class="progress-dot"></span>R Data Types</a></li><li data-subkey="sec0sub2"><a href="/R-Vectors.html"><span class="progress-dot"></span>R Vectors</a></li><li data-subkey="sec0sub2"><a href="/R-Matrices.html"><span class="progress-dot"></span>R Matrices</a></li><li data-subkey="sec0sub2"><a href="/R-Factors.html"><span class="progress-dot"></span>R Factors</a></li><li data-subkey="sec0sub2"><a href="/R-Data-Frames.html"><span class="progress-dot"></span>R Data Frames</a></li><li data-subkey="sec0sub2"><a href="/R-Lists.html"><span class="progress-dot"></span>R Lists</a></li><li data-subkey="sec0sub2"><a href="/R-Control-Flow.html"><span class="progress-dot"></span>R Control Flow</a></li><li data-subkey="sec0sub2"><a href="/R-Special-Values.html"><span class="progress-dot"></span>R Special Values</a></li><li data-subkey="sec0sub2"><a href="/R-Type-Coercion.html"><span class="progress-dot"></span>R Type Coercion</a></li><li data-subkey="sec0sub2"><a href="/R-Functions.html"><span class="progress-dot"></span>Writing R Functions</a></li><li data-subkey="sec0sub2"><a href="/R-Beginner-Exercises-quiz.html"><span class="progress-dot"></span>R Fundamentals Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec0sub3" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Working Effectively</li><li data-subkey="sec0sub3"><a href="/R-Subsetting.html"><span class="progress-dot"></span>R Subsetting</a></li><li data-subkey="sec0sub3"><a href="/Getting-Help-in-R.html"><span class="progress-dot"></span>Getting Help in R</a></li><li data-subkey="sec0sub3"><a href="/R-Project-Structure.html"><span class="progress-dot"></span>R Project Structure</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec0sub4" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> R Career &amp; Resources</li><li data-subkey="sec0sub4"><a href="/R-vs-Python.html"><span class="progress-dot"></span>R vs Python</a></li><li data-subkey="sec0sub4"><a href="/How-to-Learn-R.html"><span class="progress-dot"></span>How to Learn R</a></li><li data-subkey="sec0sub4"><a href="/R-for-Excel-Users.html"><span class="progress-dot"></span>R for Excel Users</a></li><li data-subkey="sec0sub4"><a href="/R-Interview-Questions.html"><span class="progress-dot"></span>R Interview Questions</a></li><li data-subkey="sec0sub4"><a href="/R-Interview-Questions-quiz.html"><span class="progress-dot"></span>R Interview Readiness Quiz</a></li><li data-subkey="sec0sub4"><a href="/R-Cheat-Sheet.html"><span class="progress-dot"></span>R Cheat Sheet</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec0sub5" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Professional R</li><li data-subkey="sec0sub5"><a href="/Data-Ethics-in-R.html"><span class="progress-dot"></span>Data Ethics</a></li><li data-subkey="sec0sub5"><a href="/Bias-in-Data-and-Models.html"><span class="progress-dot"></span>Bias in Data &amp; Models</a></li><li data-subkey="sec0sub5"><a href="/Reproducibility-Crisis.html"><span class="progress-dot"></span>Reproducibility</a></li><li data-subkey="sec0sub5"><a href="/Data-Privacy-in-R.html"><span class="progress-dot"></span>Data Privacy</a></li><li data-subkey="sec0sub5"><a href="/Communicating-Uncertainty.html"><span class="progress-dot"></span>Communicating Uncertainty</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">&#9656;</span> Data Wrangling<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub1" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Import &amp; Setup</li><li data-subkey="sec1sub1"><a href="/Importing-Data-in-R.html"><span class="progress-dot"></span>Importing Data</a></li><li data-subkey="sec1sub1"><a href="/R-Pipe-Operator.html"><span class="progress-dot"></span>Pipe Operator</a></li><li data-subkey="sec1sub1"><a href="/Tidy-Data-in-R.html"><span class="progress-dot"></span>Tidy Data</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub2" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> dplyr Essentials</li><li data-subkey="sec1sub2"><a href="/dplyr-filter-select.html"><span class="progress-dot"></span>dplyr filter &amp; select</a></li><li data-subkey="sec1sub2"><a href="/dplyr-mutate-rename.html"><span class="progress-dot"></span>dplyr mutate &amp; rename</a></li><li data-subkey="sec1sub2"><a href="/dplyr-group-by-summarise.html"><span class="progress-dot"></span>dplyr group_by &amp; summarise</a></li><li data-subkey="sec1sub2"><a href="/dplyr-arrange-slice.html"><span class="progress-dot"></span>dplyr arrange &amp; slice</a></li><li data-subkey="sec1sub2"><a href="/dplyr-across.html"><span class="progress-dot"></span>dplyr across()</a></li><li data-subkey="sec1sub2"><a href="/dplyr-case-when.html"><span class="progress-dot"></span>dplyr case_when()</a></li><li data-subkey="sec1sub2"><a href="/dplyr-Exercises-in-R-quiz.html"><span class="progress-dot"></span>dplyr Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub3" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Join &amp; Reshape</li><li data-subkey="sec1sub3"><a href="/R-Joins.html"><span class="progress-dot"></span>R Joins</a></li><li data-subkey="sec1sub3"><a href="/pivot_longer-pivot_wider-Reshape-Data-in-R.html"><span class="progress-dot"></span>pivot_longer &amp; pivot_wider</a></li><li data-subkey="sec1sub3"><a href="/tidyr-separate-unite-Split-Combine-Columns-in-R.html"><span class="progress-dot"></span>separate() &amp; unite()</a></li><li data-subkey="sec1sub3"><a href="/tidyr-Exercises-in-R-quiz.html"><span class="progress-dot"></span>tidyr Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub4" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Clean &amp; Quality</li><li data-subkey="sec1sub4"><a href="/Missing-Values-in-R-Detect-Count-Remove-Impute-NA.html"><span class="progress-dot"></span>Missing Values (NA)</a></li><li data-subkey="sec1sub4"><a href="/Data-Quality-Checking-in-R.html"><span class="progress-dot"></span>Data Quality Checking</a></li><li data-subkey="sec1sub4"><a href="/janitor-Package-in-R.html"><span class="progress-dot"></span>janitor Package</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub5" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Strings &amp; Dates</li><li data-subkey="sec1sub5"><a href="/stringr-in-R.html"><span class="progress-dot"></span>stringr</a></li><li data-subkey="sec1sub5"><a href="/R-Regex-stringr-Pattern-Matching.html"><span class="progress-dot"></span>Regex Patterns</a></li><li data-subkey="sec1sub5"><a href="/lubridate-in-R.html"><span class="progress-dot"></span>lubridate</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub6" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Scale &amp; Connect</li><li data-subkey="sec1sub6"><a href="/DBI-in-R.html"><span class="progress-dot"></span>DBI &amp; Databases</a></li><li data-subkey="sec1sub6"><a href="/DuckDB-in-R.html"><span class="progress-dot"></span>DuckDB &amp; duckplyr</a></li><li data-subkey="sec1sub6"><a href="/Web-Scraping-in-R-with-rvest.html"><span class="progress-dot"></span>Web Scraping (rvest)</a></li><li data-subkey="sec1sub6"><a href="/REST-APIs-in-R-with-httr2.html"><span class="progress-dot"></span>REST APIs (httr2)</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">&#9656;</span> Visualization<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub1" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> ggplot2 Foundations</li><li data-subkey="sec2sub1"><a href="/ggplot2-Grammar-of-Graphics.html"><span class="progress-dot"></span>Grammar of Graphics</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Getting-Started.html"><span class="progress-dot"></span>ggplot2 Getting Started</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Aesthetics-aes-Map-Data.html"><span class="progress-dot"></span>ggplot2 Aesthetics (aes)</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Colours.html"><span class="progress-dot"></span>ggplot2 Colours</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Scales.html"><span class="progress-dot"></span>ggplot2 Scales</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Themes-in-R.html"><span class="progress-dot"></span>ggplot2 Themes</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Labels-and-Annotations.html"><span class="progress-dot"></span>Labels &amp; Annotations</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Facets.html"><span class="progress-dot"></span>ggplot2 Facets</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Exercises-in-R-quiz.html"><span class="progress-dot"></span>ggplot2 Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub2" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Core Charts</li><li data-subkey="sec2sub2"><a href="/ggplot2-Scatter-Plots.html"><span class="progress-dot"></span>Scatter Plots</a></li><li data-subkey="sec2sub2"><a href="/ggplot2-Line-Charts.html"><span class="progress-dot"></span>Line Charts</a></li><li data-subkey="sec2sub2"><a href="/ggplot2-Bar-Charts.html"><span class="progress-dot"></span>Bar Charts</a></li><li data-subkey="sec2sub2"><a href="/ggplot2-Distribution-Charts.html"><span class="progress-dot"></span>Distribution Charts</a></li><li data-subkey="sec2sub2"><a href="/Error-Bars-in-R.html"><span class="progress-dot"></span>Error Bars</a></li><li data-subkey="sec2sub2"><a href="/geom_smooth-in-R.html"><span class="progress-dot"></span>geom_smooth()</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub3" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Distributions &amp; Groups</li><li data-subkey="sec2sub3"><a href="/Violin-Plot-in-R.html"><span class="progress-dot"></span>Violin Plot</a></li><li data-subkey="sec2sub3"><a href="/Ridgeline-Plot-in-R.html"><span class="progress-dot"></span>Ridgeline Plot</a></li><li data-subkey="sec2sub3"><a href="/Lollipop-Chart-in-R.html"><span class="progress-dot"></span>Lollipop Chart</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub4" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Relationships</li><li data-subkey="sec2sub4"><a href="/Bubble-Chart-in-R.html"><span class="progress-dot"></span>Bubble Chart</a></li><li data-subkey="sec2sub4"><a href="/Heatmap-in-R.html"><span class="progress-dot"></span>Heatmap in R</a></li><li data-subkey="sec2sub4"><a href="/Correlation-Matrix-Plot-in-R.html"><span class="progress-dot"></span>Correlation Matrix</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub5" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Advanced Charts</li><li data-subkey="sec2sub5"><a href="/Pie-Donut-Chart-in-R.html"><span class="progress-dot"></span>Pie &amp; Donut Chart</a></li><li data-subkey="sec2sub5"><a href="/Treemap-in-R.html"><span class="progress-dot"></span>Treemap</a></li><li data-subkey="sec2sub5"><a href="/Waffle-Chart-in-R.html"><span class="progress-dot"></span>Waffle Chart</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub6" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Exploratory Analysis</li><li data-subkey="sec2sub6"><a href="/Exploratory-Data-Analysis-in-R.html"><span class="progress-dot"></span>EDA (7-Step Framework)</a></li><li data-subkey="sec2sub6"><a href="/Univariate-EDA-in-R.html"><span class="progress-dot"></span>Univariate EDA</a></li><li data-subkey="sec2sub6"><a href="/Bivariate-EDA-in-R.html"><span class="progress-dot"></span>Bivariate EDA</a></li><li data-subkey="sec2sub6"><a href="/Descriptive-Statistics-in-R.html"><span class="progress-dot"></span>Descriptive Statistics</a></li><li data-subkey="sec2sub6"><a href="/Correlation-Analysis-in-R.html"><span class="progress-dot"></span>Correlation Analysis</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub7" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Interactive &amp; Maps</li><li data-subkey="sec2sub7"><a href="/Combining-ggplot2-with-plotly.html"><span class="progress-dot"></span>ggplot2 + plotly Interactive</a></li><li data-subkey="sec2sub7"><a href="/Interactive-Maps-in-R-with-leaflet.html"><span class="progress-dot"></span>Leaflet Interactive Maps</a></li><li data-subkey="sec2sub7"><a href="/Spatial-Data-in-R-with-sf.html"><span class="progress-dot"></span>Spatial Data (sf)</a></li><li data-subkey="sec2sub7"><a href="/Choropleth-Maps-in-R.html"><span class="progress-dot"></span>Choropleth Maps (sf)</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub8" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Customization &amp; Reference</li><li data-subkey="sec2sub8"><a href="/ggplot2-Legends-in-R.html"><span class="progress-dot"></span>ggplot2 Legends</a></li><li data-subkey="sec2sub8"><a href="/ggplot2-Secondary-Axis.html"><span class="progress-dot"></span>Secondary Axis</a></li><li data-subkey="sec2sub8"><a href="/ggplot2-Log-Scale.html"><span class="progress-dot"></span>Log Scale</a></li><li data-subkey="sec2sub8"><a href="/patchwork-Package.html"><span class="progress-dot"></span>patchwork (Combine Plots)</a></li><li data-subkey="sec2sub8"><a href="/Publication-Quality-Figures-in-R.html"><span class="progress-dot"></span>Publication-Ready Figures</a></li><li data-subkey="sec2sub8"><a href="/ggplot2-cheatsheet.html"><span class="progress-dot"></span>ggplot2 Quickref</a></li></ul></li><li class="sidebar-section expanded"><div class="sidebar-section-header"><span class="sidebar-chevron">&#9656;</span> Statistics<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub1" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> EDA &amp; Data Quality</li><li data-subkey="sec3sub1"><a href="/Automated-EDA-in-R.html"><span class="progress-dot"></span>Automated EDA</a></li><li data-subkey="sec3sub1"><a href="/Missing-Data-Visualization-in-R-naniar.html"><span class="progress-dot"></span>Missing Data Viz (naniar)</a></li><li data-subkey="sec3sub1"><a href="/Outlier-Detection-in-R.html"><span class="progress-dot"></span>Outlier Detection</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub2" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Probability</li><li data-subkey="sec3sub2"><a href="/Sample-Spaces-Events-and-Probability-Axioms-in-R-With-Monte-Carlo-Proof.html"><span class="progress-dot"></span>Probability Axioms</a></li><li data-subkey="sec3sub2"><a href="/Conditional-Probability-in-R.html" class="active"><span class="progress-dot"></span>Conditional Probability</a></li><li data-subkey="sec3sub2"><a href="/Random-Variables-in-R.html"><span class="progress-dot"></span>Random Variables</a></li><li data-subkey="sec3sub2"><a href="/Binomial-and-Poisson-Distributions-in-R.html"><span class="progress-dot"></span>Binomial vs Poisson</a></li><li data-subkey="sec3sub2"><a href="/Normal-t-F-and-Chi-Squared-Distributions-in-R.html"><span class="progress-dot"></span>Normal, t, F, Chi-Squared</a></li><li data-subkey="sec3sub2"><a href="/Central-Limit-Theorem-in-R.html"><span class="progress-dot"></span>Central Limit Theorem</a></li><li data-subkey="sec3sub2"><a href="/Sampling-Distributions-in-R.html"><span class="progress-dot"></span>Sampling Distributions</a></li><li data-subkey="sec3sub2"><a href="/Law-of-Large-Numbers-vs-CLT-in-R.html"><span class="progress-dot"></span>LLN vs CLT</a></li><li data-subkey="sec3sub2"><a href="/What-Is-Probability-Simulation-First-Intuition-in-R-Before-the-Formulas.html"><span class="progress-dot"></span>Probability (Simulation-First)</a></li><li data-subkey="sec3sub2"><a href="/Expected-Value-and-Variance-in-R.html"><span class="progress-dot"></span>Expected Value and Variance</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub3" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Inference &amp; Estimation</li><li data-subkey="sec3sub3"><a href="/Maximum-Likelihood-Estimation-in-R.html"><span class="progress-dot"></span>Maximum Likelihood Estimation</a></li><li data-subkey="sec3sub3"><a href="/Hypothesis-Testing-in-R.html"><span class="progress-dot"></span>Hypothesis Testing</a></li><li data-subkey="sec3sub3"><a href="/Sample-Size-Planning-in-R.html"><span class="progress-dot"></span>Sample Size Planning</a></li><li data-subkey="sec3sub3"><a href="/Which-Statistical-Test-in-R.html"><span class="progress-dot"></span>Choosing the Right Test</a></li><li data-subkey="sec3sub3"><a href="/Statistical-Tests-in-R.html"><span class="progress-dot"></span>Statistical Tests</a></li><li data-subkey="sec3sub3"><a href="/Measures-of-Association-in-R.html"><span class="progress-dot"></span>Measures of Association</a></li><li data-subkey="sec3sub3"><a href="/Point-Estimation-in-R.html"><span class="progress-dot"></span>Point Estimation</a></li><li data-subkey="sec3sub3"><a href="/Confidence-Intervals-in-R.html"><span class="progress-dot"></span>Confidence Intervals</a></li><li data-subkey="sec3sub3"><a href="/Type-I-and-Type-II-Errors-in-R.html"><span class="progress-dot"></span>Type I and II Errors</a></li><li data-subkey="sec3sub3"><a href="/Statistical-Power-Analysis-in-R.html"><span class="progress-dot"></span>Power Analysis</a></li><li data-subkey="sec3sub3"><a href="/Effect-Size-in-R.html"><span class="progress-dot"></span>Effect Size</a></li><li data-subkey="sec3sub3"><a href="/t-Tests-in-R.html"><span class="progress-dot"></span>t-Tests</a></li><li data-subkey="sec3sub3"><a href="/Proportion-Tests-in-R.html"><span class="progress-dot"></span>Proportion Tests</a></li><li data-subkey="sec3sub3"><a href="/Normality-and-Variance-Tests-in-R.html"><span class="progress-dot"></span>Normality &amp; Variance Tests</a></li><li data-subkey="sec3sub3"><a href="/Chi-Square-Tests-in-R.html"><span class="progress-dot"></span>Chi-Square Tests</a></li><li data-subkey="sec3sub3"><a href="/Wilcoxon-Mann-Whitney-and-Kruskal-Wallis-in-R.html"><span class="progress-dot"></span>Wilcoxon, Mann-Whitney &amp; Kruskal-Wallis</a></li><li data-subkey="sec3sub3"><a href="/Multiple-Comparisons-in-R.html"><span class="progress-dot"></span>Multiple Testing Correction</a></li><li data-subkey="sec3sub3"><a href="/Hypothesis-Testing-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Hypothesis Testing Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub4" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Regression</li><li data-subkey="sec3sub4"><a href="/Linear-Regression.html"><span class="progress-dot"></span>Linear Regression</a></li><li data-subkey="sec3sub4"><a href="/Logistic-Regression-With-R.html"><span class="progress-dot"></span>Logistic Regression</a></li><li data-subkey="sec3sub4"><a href="/Variable-Selection-and-Importance-With-R.html"><span class="progress-dot"></span>Feature Selection</a></li><li data-subkey="sec3sub4"><a href="/Model-Selection-in-R.html"><span class="progress-dot"></span>Model Selection</a></li><li data-subkey="sec3sub4"><a href="/Missing-Value-Treatment-With-R.html"><span class="progress-dot"></span>Missing Value Treatment</a></li><li data-subkey="sec3sub4"><a href="/Outlier-Treatment-With-R.html"><span class="progress-dot"></span>Outlier Analysis</a></li><li data-subkey="sec3sub4"><a href="/adv-regression-models.html"><span class="progress-dot"></span>Advanced Regression Models</a></li><li data-subkey="sec3sub4"><a href="/Linear-Regression-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Linear Regression Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub5" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Reporting</li><li data-subkey="sec3sub5"><a href="/Statistical-Consulting-in-R.html"><span class="progress-dot"></span>Statistical Consulting</a></li><li data-subkey="sec3sub5"><a href="/Statistical-Report-Writing-in-R.html"><span class="progress-dot"></span>Statistical Report Writing</a></li><li data-subkey="sec3sub5"><a href="/Bootstrap-Confidence-Intervals-in-R.html"><span class="progress-dot"></span>Bootstrap Confidence Intervals</a></li><li data-subkey="sec3sub5"><a href="/Reporting-Statistics-in-R.html"><span class="progress-dot"></span>Reporting Statistics</a></li><li data-subkey="sec3sub5"><a href="/Correlation-in-R.html"><span class="progress-dot"></span>Correlation (Pearson, Spearman, Kendall)</a></li><li data-subkey="sec3sub5"><a href="/Linear-Regression-Assumptions-in-R.html"><span class="progress-dot"></span>Linear Regression Assumptions</a></li><li data-subkey="sec3sub5"><a href="/Dummy-Variables-in-R.html"><span class="progress-dot"></span>Dummy Variables in R</a></li><li data-subkey="sec3sub5"><a href="/Interaction-Effects-in-R.html"><span class="progress-dot"></span>Interaction Effects</a></li><li data-subkey="sec3sub5"><a href="/Regression-Diagnostics-in-R.html"><span class="progress-dot"></span>Regression Diagnostics</a></li><li data-subkey="sec3sub5"><a href="/Logistic-Regression-in-R.html"><span class="progress-dot"></span>Logistic Regression (glm + ROC)</a></li><li data-subkey="sec3sub5"><a href="/Variable-Selection-in-R.html"><span class="progress-dot"></span>Variable Selection</a></li><li data-subkey="sec3sub5"><a href="/Poisson-Regression-in-R.html"><span class="progress-dot"></span>Poisson Regression</a></li><li data-subkey="sec3sub5"><a href="/Ridge-and-Lasso-Regression-in-R.html"><span class="progress-dot"></span>Ridge &amp; Lasso Regression</a></li><li data-subkey="sec3sub5"><a href="/Polynomial-and-Spline-Regression-in-R.html"><span class="progress-dot"></span>Polynomial &amp; Splines</a></li><li data-subkey="sec3sub5"><a href="/Regression-Tables-in-R.html"><span class="progress-dot"></span>Regression Tables (3 packages)</a></li><li data-subkey="sec3sub5"><a href="/One-Way-ANOVA-in-R.html"><span class="progress-dot"></span>One-Way ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Post-Hoc-Tests-After-ANOVA.html"><span class="progress-dot"></span>Post-Hoc Tests After ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Two-Way-ANOVA-in-R.html"><span class="progress-dot"></span>Two-Way ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Repeated-Measures-ANOVA-in-R.html"><span class="progress-dot"></span>Repeated Measures ANOVA</a></li><li data-subkey="sec3sub5"><a href="/ANCOVA-in-R.html"><span class="progress-dot"></span>ANCOVA</a></li><li data-subkey="sec3sub5"><a href="/Experimental-Design-Principles-in-R.html"><span class="progress-dot"></span>Experimental Design in R</a></li><li data-subkey="sec3sub5"><a href="/Factorial-Experiments-in-R.html"><span class="progress-dot"></span>Factorial Designs (2^k)</a></li><li data-subkey="sec3sub5"><a href="/AB-Testing-in-R.html"><span class="progress-dot"></span>A/B Testing</a></li><li data-subkey="sec3sub5"><a href="/MANOVA-in-R.html"><span class="progress-dot"></span>MANOVA</a></li><li data-subkey="sec3sub5"><a href="/Mixed-ANOVA-in-R.html"><span class="progress-dot"></span>Mixed ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Multivariate-Statistics-in-R.html"><span class="progress-dot"></span>Multivariate Distances &amp; Hotelling's T²</a></li><li data-subkey="sec3sub5"><a href="/PCA-in-R.html"><span class="progress-dot"></span>PCA with prcomp()</a></li><li data-subkey="sec3sub5"><a href="/Interpreting-PCA-Results-in-R.html"><span class="progress-dot"></span>Interpreting PCA Output</a></li><li data-subkey="sec3sub5"><a href="/Exploratory-Factor-Analysis-in-R.html"><span class="progress-dot"></span>Exploratory Factor Analysis</a></li><li data-subkey="sec3sub5"><a href="/CFA-and-Structural-Equation-Modeling-in-R.html"><span class="progress-dot"></span>SEM and CFA (lavaan)</a></li><li data-subkey="sec3sub5"><a href="/Linear-Discriminant-Analysis-in-R.html"><span class="progress-dot"></span>LDA (Linear Discriminant Analysis)</a></li><li data-subkey="sec3sub5"><a href="/Cluster-Analysis-in-R.html"><span class="progress-dot"></span>Clustering (k-Means / HC / DBSCAN)</a></li><li data-subkey="sec3sub5"><a href="/Correspondence-Analysis-in-R.html"><span class="progress-dot"></span>Correspondence Analysis</a></li><li data-subkey="sec3sub5"><a href="/t-SNE-and-UMAP-in-R.html"><span class="progress-dot"></span>t-SNE and UMAP</a></li><li data-subkey="sec3sub5"><a href="/Simple-Linear-Regression-in-R.html"><span class="progress-dot"></span>Simple Linear Regression</a></li><li data-subkey="sec3sub5"><a href="/Multiple-Regression-in-R.html"><span class="progress-dot"></span>Multiple Regression</a></li><li data-subkey="sec3sub5"><a href="/Robust-Regression-in-R.html"><span class="progress-dot"></span>Robust Regression (rlm)</a></li><li data-subkey="sec3sub5"><a href="/factoextra-and-FactoMineR.html"><span class="progress-dot"></span>factoextra (PCA + Clusters)</a></li><li data-subkey="sec3sub5"><a href="/Categorical-Data-in-R.html"><span class="progress-dot"></span>Categorical Data (Tables &amp; Mosaic)</a></li><li data-subkey="sec3sub5"><a href="/Chi-Square-Test-of-Independence-in-R.html"><span class="progress-dot"></span>Chi-Square Test of Independence</a></li><li data-subkey="sec3sub5"><a href="/Chi-Square-Goodness-of-Fit-Test-in-R.html"><span class="progress-dot"></span>Chi-Square Goodness-of-Fit</a></li><li data-subkey="sec3sub5"><a href="/Fishers-Exact-Test-in-R.html"><span class="progress-dot"></span>Fisher's Exact Test</a></li><li data-subkey="sec3sub5"><a href="/Odds-Ratios-and-Relative-Risk-in-R.html"><span class="progress-dot"></span>Odds Ratios &amp; Relative Risk</a></li><li data-subkey="sec3sub5"><a href="/Logistic-Regression-in-R-2.html"><span class="progress-dot"></span>Logistic Regression (Diagnostics)</a></li><li data-subkey="sec3sub5"><a href="/Poisson-and-Negative-Binomial-Regression.html"><span class="progress-dot"></span>Poisson &amp; Negative Binomial Regression</a></li><li data-subkey="sec3sub5"><a href="/Multinomial-and-Ordinal-Logistic-Regression-in-R.html"><span class="progress-dot"></span>Multinomial &amp; Ordinal Logistic Regression</a></li><li data-subkey="sec3sub5"><a href="/When-to-Use-Nonparametric-Tests-in-R.html"><span class="progress-dot"></span>When to Use Nonparametric Tests</a></li><li data-subkey="sec3sub5"><a href="/Wilcoxon-Signed-Rank-Test-in-R.html"><span class="progress-dot"></span>Wilcoxon Signed-Rank Test</a></li><li data-subkey="sec3sub5"><a href="/Mann-Whitney-U-Test-in-R.html"><span class="progress-dot"></span>Mann-Whitney U Test</a></li><li data-subkey="sec3sub5"><a href="/Kruskal-Wallis-Test-in-R-2.html"><span class="progress-dot"></span>Kruskal-Wallis Test</a></li><li data-subkey="sec3sub5"><a href="/Friedman-Test-in-R.html"><span class="progress-dot"></span>Friedman Test</a></li><li data-subkey="sec3sub5"><a href="/Spearman-and-Kendall-Correlation-in-R.html"><span class="progress-dot"></span>Spearman &amp; Kendall Correlation</a></li><li data-subkey="sec3sub5"><a href="/Bootstrap-in-R.html"><span class="progress-dot"></span>Bootstrap (boot package)</a></li><li data-subkey="sec3sub5"><a href="/Quantile-Regression-in-R-2.html"><span class="progress-dot"></span>Quantile Regression</a></li><li data-subkey="sec3sub5"><a href="/Matrix-Operations-in-R.html"><span class="progress-dot"></span>Matrix Operations in R</a></li><li data-subkey="sec3sub5"><a href="/Solving-Linear-Systems-in-R.html"><span class="progress-dot"></span>Solving Linear Systems in R</a></li><li data-subkey="sec3sub5"><a href="/Eigenvalues-and-Eigenvectors-in-R.html"><span class="progress-dot"></span>Eigenvalues &amp; Eigenvectors in R</a></li><li data-subkey="sec3sub5"><a href="/Singular-Value-Decomposition-in-R.html"><span class="progress-dot"></span>Singular Value Decomposition in R</a></li><li data-subkey="sec3sub5"><a href="/Projections-and-the-Hat-Matrix-in-R.html"><span class="progress-dot"></span>Projections &amp; the Hat Matrix</a></li><li data-subkey="sec3sub5"><a href="/QR-Decomposition-in-R.html"><span class="progress-dot"></span>QR Decomposition in R</a></li><li data-subkey="sec3sub5"><a href="/Quadratic-Forms-in-R.html"><span class="progress-dot"></span>Quadratic Forms</a></li><li data-subkey="sec3sub5"><a href="/Matrix-Derivatives-and-the-Hessian-in-R.html"><span class="progress-dot"></span>Matrix Derivatives &amp; Hessian</a></li><li data-subkey="sec3sub5"><a href="/Exponential-Family-Distributions-in-R.html"><span class="progress-dot"></span>Exponential Family Distributions</a></li><li data-subkey="sec3sub5"><a href="/Sufficient-Statistics-in-R.html"><span class="progress-dot"></span>Sufficient Statistics</a></li><li data-subkey="sec3sub5"><a href="/Complete-and-Ancillary-Statistics-in-R.html"><span class="progress-dot"></span>Complete &amp; Ancillary Statistics</a></li><li data-subkey="sec3sub5"><a href="/UMVUE-in-R-2.html"><span class="progress-dot"></span>UMVUE (Rao-Blackwell &amp; Lehmann-Scheffé)</a></li><li data-subkey="sec3sub5"><a href="/Cramer-Rao-Lower-Bound-in-R-2.html"><span class="progress-dot"></span>Cramér-Rao Lower Bound</a></li><li data-subkey="sec3sub5"><a href="/Asymptotic-Theory-in-R-2.html"><span class="progress-dot"></span>Asymptotic Theory</a></li><li data-subkey="sec3sub5"><a href="/Neyman-Pearson-Lemma-in-R-2.html"><span class="progress-dot"></span>Neyman-Pearson Lemma</a></li><li data-subkey="sec3sub5"><a href="/Likelihood-Ratio-Tests-and-Pivotal-Methods.html"><span class="progress-dot"></span>Likelihood Ratio &amp; Pivotal Methods</a></li><li data-subkey="sec3sub5"><a href="/Decision-Theory-in-R.html"><span class="progress-dot"></span>Decision Theory</a></li><li data-subkey="sec3sub5"><a href="/Asymptotic-Relative-Efficiency-in-R.html"><span class="progress-dot"></span>Asymptotic Relative Efficiency</a></li><li data-subkey="sec3sub5"><a href="/Bayes-Theorem-in-R.html"><span class="progress-dot"></span>Bayes' Theorem</a></li><li data-subkey="sec3sub5"><a href="/Bayesian-Statistics-in-R.html"><span class="progress-dot"></span>Bayesian Statistics</a></li><li data-subkey="sec3sub5"><a href="/Conjugate-Priors-in-R.html"><span class="progress-dot"></span>Conjugate Priors</a></li><li data-subkey="sec3sub5"><a href="/Grid-Approximation-in-R.html"><span class="progress-dot"></span>Grid Approximation</a></li><li data-subkey="sec3sub5"><a href="/MCMC-in-R.html"><span class="progress-dot"></span>MCMC in R</a></li><li data-subkey="sec3sub5"><a href="/Gibbs-Sampling-in-R.html"><span class="progress-dot"></span>Gibbs Sampling</a></li><li data-subkey="sec3sub5"><a href="/Hamiltonian-Monte-Carlo-in-R.html"><span class="progress-dot"></span>Hamiltonian Monte Carlo</a></li><li data-subkey="sec3sub5"><a href="/Stan-in-R.html"><span class="progress-dot"></span>Stan</a></li><li data-subkey="sec3sub5"><a href="/brms-in-R.html"><span class="progress-dot"></span>brms</a></li><li data-subkey="sec3sub5"><a href="/Choosing-Priors-in-R.html"><span class="progress-dot"></span>Choosing Priors</a></li><li data-subkey="sec3sub5"><a href="/Prior-Predictive-Checks-in-R.html"><span class="progress-dot"></span>Prior Predictive Checks</a></li><li data-subkey="sec3sub5"><a href="/Compare-Bayesian-Models-in-R.html"><span class="progress-dot"></span>Compare Bayesian Models</a></li><li data-subkey="sec3sub5"><a href="/Posterior-Predictive-Checks-in-R.html"><span class="progress-dot"></span>Posterior Predictive Checks</a></li><li data-subkey="sec3sub5"><a href="/Bayesian-Linear-Regression-in-R.html"><span class="progress-dot"></span>Bayesian Linear Regression</a></li><li data-subkey="sec3sub5"><a href="/Bayesian-Logistic-Regression-in-R.html"><span class="progress-dot"></span>Bayesian Logistic Regression</a></li><li data-subkey="sec3sub5"><a href="/Bayesian-Hierarchical-Models-in-R.html"><span class="progress-dot"></span>Bayesian Hierarchical Models</a></li><li data-subkey="sec3sub5"><a href="/Multilevel-Models-in-R.html"><span class="progress-dot"></span>Multilevel Models</a></li><li data-subkey="sec3sub5"><a href="/Bayesian-ANOVA-in-R.html"><span class="progress-dot"></span>Bayesian ANOVA</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub6" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Machine Learning</li><li data-subkey="sec3sub6"><a href="/Machine-Learning-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Machine Learning Quiz</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">&#9656;</span> Time Series<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li data-subkey="sec4sub0"><a href="/Time-Series-Analysis-With-R.html"><span class="progress-dot"></span>Time Series Analysis</a></li><li data-subkey="sec4sub0"><a href="/Time-Series-Forecasting-With-R.html"><span class="progress-dot"></span>Time Series Forecasting</a></li><li data-subkey="sec4sub0"><a href="/Time-Series-Forecasting-With-R-part2.html"><span class="progress-dot"></span>More Time Series Forecasting</a></li><li data-subkey="sec4sub0"><a href="/Time-Series-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Time Series Quiz</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">&#9656;</span> Advanced R<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec5sub1" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Functional Programming</li><li data-subkey="sec5sub1"><a href="/Functional-Programming-in-R.html"><span class="progress-dot"></span>Functional Programming</a></li><li data-subkey="sec5sub1"><a href="/R-Functional-Programming-Exercises-quiz.html"><span class="progress-dot"></span>Functional Programming Quiz</a></li><li data-subkey="sec5sub1"><a href="/purrr-map-Variants.html"><span class="progress-dot"></span>purrr map() Variants</a></li><li data-subkey="sec5sub1"><a href="/R-Anonymous-Functions.html"><span class="progress-dot"></span>R Anonymous Functions</a></li><li data-subkey="sec5sub1"><a href="/R-Function-Factories.html"><span class="progress-dot"></span>R Function Factories</a></li><li data-subkey="sec5sub1"><a href="/R-Function-Operators.html"><span class="progress-dot"></span>R Function Operators</a></li><li data-subkey="sec5sub1"><a href="/Reduce-Filter-Map-in-R.html"><span class="progress-dot"></span>Reduce, Filter, Map</a></li><li data-subkey="sec5sub1"><a href="/Memoization-in-R.html"><span class="progress-dot"></span>Memoization in R</a></li><li data-subkey="sec5sub1"><a href="/Writing-Composable-R-Code.html"><span class="progress-dot"></span>Composable R Code</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec5sub2" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> OOP in R</li><li data-subkey="sec5sub2"><a href="/OOP-in-R.html"><span class="progress-dot"></span>OOP in R: S3/S4/R6</a></li><li data-subkey="sec5sub2"><a href="/S3-Classes-in-R.html"><span class="progress-dot"></span>S3 Classes</a></li><li data-subkey="sec5sub2"><a href="/S3-Method-Dispatch-in-R.html"><span class="progress-dot"></span>S3 Method Dispatch</a></li><li data-subkey="sec5sub2"><a href="/S4-Classes-in-R.html"><span class="progress-dot"></span>S4 Classes</a></li><li data-subkey="sec5sub2"><a href="/S4-Methods-in-R.html"><span class="progress-dot"></span>S4 Methods &amp; Dispatch</a></li><li data-subkey="sec5sub2"><a href="/R6-Classes-in-R.html"><span class="progress-dot"></span>R6 Classes</a></li><li data-subkey="sec5sub2"><a href="/R6-Advanced.html"><span class="progress-dot"></span>R6 Advanced</a></li><li data-subkey="sec5sub2"><a href="/Operator-Overloading-in-R.html"><span class="progress-dot"></span>Operator Overloading</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec5sub3" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> How R Works</li><li data-subkey="sec5sub3"><a href="/R-Names-and-Values.html"><span class="progress-dot"></span>R Names &amp; Values</a></li><li data-subkey="sec5sub3"><a href="/R-Assignment-Deep-Dive.html"><span class="progress-dot"></span>R Assignment Deep Dive</a></li><li data-subkey="sec5sub3"><a href="/R-Memory-lobstr.html"><span class="progress-dot"></span>R Memory &amp; lobstr</a></li><li data-subkey="sec5sub3"><a href="/R-Environments.html"><span class="progress-dot"></span>R Environments</a></li><li data-subkey="sec5sub3"><a href="/R-Lexical-Scoping.html"><span class="progress-dot"></span>Lexical Scoping</a></li><li data-subkey="sec5sub3"><a href="/R-Closures.html"><span class="progress-dot"></span>R Closures</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec5sub4" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Debugging &amp; Performance</li><li data-subkey="sec5sub4"><a href="/R-Conditions-System.html"><span class="progress-dot"></span>Conditions System</a></li><li data-subkey="sec5sub4"><a href="/R-Debugging.html"><span class="progress-dot"></span>Debugging R Code</a></li><li data-subkey="sec5sub4"><a href="/R-Common-Errors.html"><span class="progress-dot"></span>50 Common R Errors</a></li><li data-subkey="sec5sub4"><a href="/Parallel-Computing-With-R.html"><span class="progress-dot"></span>Parallel Computing</a></li><li data-subkey="sec5sub4"><a href="/Strategies-To-Improve-And-Speedup-R-Code.html"><span class="progress-dot"></span>Speedup R Code</a></li><li data-subkey="sec5sub4"><a href="/Shiny-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Shiny Quiz</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">&#9656;</span> Classic Tutorials<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li data-subkey="sec6sub0"><a href="/R-Tutorial.html"><span class="progress-dot"></span>R Tutorial (Classic)</a></li><li data-subkey="sec6sub0"><a href="/ggplot2-Tutorial-With-R.html"><span class="progress-dot"></span>ggplot2 Short Tutorial</a></li><li data-subkey="sec6sub0"><a href="/Complete-Ggplot2-Tutorial-Part1-With-R-Code.html"><span class="progress-dot"></span>ggplot2 Tutorial 1 - Intro</a></li><li data-subkey="sec6sub0"><a href="/Complete-Ggplot2-Tutorial-Part2-Customizing-Theme-With-R-Code.html"><span class="progress-dot"></span>ggplot2 Tutorial 2 - Theme</a></li><li data-subkey="sec6sub0"><a href="/Top50-Ggplot2-Visualizations-MasterList-R-Code.html"><span class="progress-dot"></span>ggplot2 Tutorial 3 - Masterlist</a></li><li data-subkey="sec6sub0"><a href="/Association-Mining-With-R.html"><span class="progress-dot"></span>Association Mining</a></li><li data-subkey="sec6sub0"><a href="/Multi-Dimensional-Scaling-With-R.html"><span class="progress-dot"></span>Multi Dimensional Scaling</a></li><li data-subkey="sec6sub0"><a href="/Optimization-With-R.html"><span class="progress-dot"></span>Optimization</a></li><li data-subkey="sec6sub0"><a href="/Information-Value-With-R.html"><span class="progress-dot"></span>InformationValue Package</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">&#9656;</span> Practice Exercises<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec7sub1" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Mastery Quizzes (Certificate)</li><li data-subkey="sec7sub1"><a href="/R-Beginner-Exercises-quiz.html"><span class="progress-dot"></span>R Fundamentals Quiz</a></li><li data-subkey="sec7sub1"><a href="/dplyr-Exercises-in-R-quiz.html"><span class="progress-dot"></span>dplyr Quiz</a></li><li data-subkey="sec7sub1"><a href="/ggplot2-Exercises-in-R-quiz.html"><span class="progress-dot"></span>ggplot2 Quiz</a></li><li data-subkey="sec7sub1"><a href="/Hypothesis-Testing-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Hypothesis Testing Quiz</a></li><li data-subkey="sec7sub1"><a href="/Linear-Regression-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Linear Regression Quiz</a></li><li data-subkey="sec7sub1"><a href="/Machine-Learning-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Machine Learning Quiz</a></li><li data-subkey="sec7sub1"><a href="/tidyr-Exercises-in-R-quiz.html"><span class="progress-dot"></span>tidyr Quiz</a></li><li data-subkey="sec7sub1"><a href="/Time-Series-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Time Series Quiz</a></li><li data-subkey="sec7sub1"><a href="/Shiny-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Shiny Quiz</a></li><li data-subkey="sec7sub1"><a href="/R-Interview-Questions-quiz.html"><span class="progress-dot"></span>R Interview Readiness Quiz</a></li><li data-subkey="sec7sub1"><a href="/R-Functional-Programming-Exercises-quiz.html"><span class="progress-dot"></span>Functional Programming Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec7sub2" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> R Fundamentals</li><li data-subkey="sec7sub2"><a href="/R-Basics-Exercises.html"><span class="progress-dot"></span>R Basics (15 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-Vectors-Exercises.html"><span class="progress-dot"></span>R Vectors (12 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-Data-Frames-Exercises.html"><span class="progress-dot"></span>R Data Frames (15 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-Lists-Exercises.html"><span class="progress-dot"></span>R Lists (10 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-Control-Flow-Exercises.html"><span class="progress-dot"></span>R Control Flow (12 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-Functions-Exercises.html"><span class="progress-dot"></span>R Functions (10 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-String-Exercises.html"><span class="progress-dot"></span>R Strings (10 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-Date-Time-Exercises.html"><span class="progress-dot"></span>R Date &amp; Time (10 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-Apply-Exercises.html"><span class="progress-dot"></span>R apply Family (12 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-Subsetting-Exercises.html"><span class="progress-dot"></span>R Subsetting (10 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-Functional-Programming-Exercises.html"><span class="progress-dot"></span>Functional Programming (10 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-OOP-Exercises.html"><span class="progress-dot"></span>OOP in R (8 problems)</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec7sub3" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Data Wrangling</li><li data-subkey="sec7sub3"><a href="/R-Data-Import-Exercises.html"><span class="progress-dot"></span>Data Import (10 problems)</a></li><li data-subkey="sec7sub3"><a href="/dplyr-Exercises.html"><span class="progress-dot"></span>dplyr (15 problems)</a></li><li data-subkey="sec7sub3"><a href="/dplyr-filter-select-Exercises.html"><span class="progress-dot"></span>dplyr filter() &amp; select() (12 problems)</a></li><li data-subkey="sec7sub3"><a href="/dplyr-group-by-summarise-Exercises.html"><span class="progress-dot"></span>dplyr group_by() &amp; summarise() (10 problems)</a></li><li data-subkey="sec7sub3"><a href="/dplyr-Join-Exercises.html"><span class="progress-dot"></span>dplyr Joins (10 problems)</a></li><li data-subkey="sec7sub3"><a href="/data-table-Exercises.html"><span class="progress-dot"></span>data.table (12 problems)</a></li><li data-subkey="sec7sub3"><a href="/purrr-Exercises.html"><span class="progress-dot"></span>purrr (10 problems)</a></li><li data-subkey="sec7sub3"><a href="/tidyr-Reshaping-Exercises.html"><span class="progress-dot"></span>tidyr Reshaping (10 problems)</a></li><li data-subkey="sec7sub3"><a href="/Missing-Data-in-R-Exercises.html"><span class="progress-dot"></span>Missing Data in R (10 problems)</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec7sub4" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Visualization</li><li data-subkey="sec7sub4"><a href="/ggplot2-Exercises.html"><span class="progress-dot"></span>ggplot2 (15 problems)</a></li><li data-subkey="sec7sub4"><a href="/ggplot2-Geom-Exercises.html"><span class="progress-dot"></span>ggplot2 Geoms (12 problems)</a></li><li data-subkey="sec7sub4"><a href="/ggplot2-Aesthetics-Exercises.html"><span class="progress-dot"></span>ggplot2 Aesthetics (10 problems)</a></li><li data-subkey="sec7sub4"><a href="/ggplot2-Customization-Exercises.html"><span class="progress-dot"></span>ggplot2 Customization (10 problems)</a></li><li data-subkey="sec7sub4"><a href="/ggplot2-Facet-Exercises.html"><span class="progress-dot"></span>ggplot2 Facets (8 problems)</a></li><li data-subkey="sec7sub4"><a href="/R-Visualization-Project.html"><span class="progress-dot"></span>R Visualization Project (5 charts)</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec7sub5" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Statistics</li><li data-subkey="sec7sub5"><a href="/Probability-in-R-Exercises.html"><span class="progress-dot"></span>Probability in R Exercises</a></li><li data-subkey="sec7sub5"><a href="/R-Probability-Distributions-Exercises.html"><span class="progress-dot"></span>R Probability Distributions (12 problems)</a></li><li data-subkey="sec7sub5"><a href="/Binomial-Distribution-Exercises-in-R.html"><span class="progress-dot"></span>Binomial Distribution Exercises</a></li><li data-subkey="sec7sub5"><a href="/Poisson-Distribution-Exercises-in-R.html"><span class="progress-dot"></span>Poisson Distribution Exercises</a></li><li data-subkey="sec7sub5"><a href="/Central-Limit-Theorem-Exercises-in-R.html"><span class="progress-dot"></span>Central Limit Theorem Exercises</a></li><li data-subkey="sec7sub5"><a href="/Hypothesis-Testing-Exercises-in-R.html"><span class="progress-dot"></span>Hypothesis Testing Exercises</a></li><li data-subkey="sec7sub5"><a href="/t-Test-Exercises-in-R.html"><span class="progress-dot"></span>t-Test Exercises (12 problems)</a></li><li data-subkey="sec7sub5"><a href="/Chi-Square-Test-Exercises-in-R.html"><span class="progress-dot"></span>Chi-Square Exercises (10 problems)</a></li><li data-subkey="sec7sub5"><a href="/Confidence-Interval-Exercises-in-R.html"><span class="progress-dot"></span>Confidence Interval (10 problems)</a></li><li data-subkey="sec7sub5"><a href="/Power-Analysis-Exercises-in-R.html"><span class="progress-dot"></span>Power Analysis Exercises (8 problems)</a></li><li data-subkey="sec7sub5"><a href="/Nonparametric-Tests-Exercises-in-R.html"><span class="progress-dot"></span>Nonparametric Exercises (10 problems)</a></li><li data-subkey="sec7sub5"><a href="/Multiple-Testing-Exercises-in-R.html"><span class="progress-dot"></span>Multiple Testing (8 problems)</a></li><li data-subkey="sec7sub5"><a href="/Multiple-Regression-Exercises-in-R.html"><span class="progress-dot"></span>Multiple Regression Exercises</a></li><li data-subkey="sec7sub5"><a href="/Logistic-Regression-Exercises-in-R.html"><span class="progress-dot"></span>Logistic Regression Exercises (10 problems)</a></li><li data-subkey="sec7sub5"><a href="/Regression-Diagnostics-Exercises-in-R.html"><span class="progress-dot"></span>Regression Diagnostics Exercises</a></li><li data-subkey="sec7sub5"><a href="/Ridge-and-Lasso-Exercises-in-R.html"><span class="progress-dot"></span>Ridge &amp; Lasso Exercises</a></li><li data-subkey="sec7sub5"><a href="/GLM-Exercises-in-R.html"><span class="progress-dot"></span>GLM Exercises (10 problems)</a></li><li data-subkey="sec7sub5"><a href="/ANOVA-Exercises-in-R.html"><span class="progress-dot"></span>ANOVA Exercises (15 problems)</a></li><li data-subkey="sec7sub5"><a href="/Post-Hoc-Tests-Exercises-in-R.html"><span class="progress-dot"></span>Post-Hoc Tests Exercises (8 problems)</a></li><li data-subkey="sec7sub5"><a href="/Repeated-Measures-Exercises-in-R.html"><span class="progress-dot"></span>Repeated Measures (8 problems)</a></li><li data-subkey="sec7sub5"><a href="/Experimental-Design-Exercises-in-R.html"><span class="progress-dot"></span>Experimental Design Exercises (8 problems)</a></li><li data-subkey="sec7sub5"><a href="/AB-Testing-Exercises-in-R.html"><span class="progress-dot"></span>A/B Testing Exercises (8 problems)</a></li><li data-subkey="sec7sub5"><a href="/Linear-Regression-Exercises-in-R.html"><span class="progress-dot"></span>Linear Regression (15 problems)</a></li><li data-subkey="sec7sub5"><a href="/PCA-Exercises-in-R.html"><span class="progress-dot"></span>PCA Exercises (10 problems)</a></li><li data-subkey="sec7sub5"><a href="/Cluster-Analysis-Exercises-in-R.html"><span class="progress-dot"></span>Clustering Exercises (10 problems)</a></li><li data-subkey="sec7sub5"><a href="/SEM-Exercises-in-R.html"><span class="progress-dot"></span>SEM Exercises (8 problems)</a></li><li data-subkey="sec7sub5"><a href="/A-B-Testing-Exercises-in-R.html"><span class="progress-dot"></span>A/B Testing Exercises</a></li><li data-subkey="sec7sub5"><a href="/API-Calls-Exercises-in-R.html"><span class="progress-dot"></span>API Calls Exercises</a></li><li data-subkey="sec7sub5"><a href="/ARIMA-Exercises-in-R.html"><span class="progress-dot"></span>ARIMA Exercises</a></li><li data-subkey="sec7sub5"><a href="/Apply-Family-Exercises-in-R.html"><span class="progress-dot"></span>Apply Family Exercises</a></li><li data-subkey="sec7sub5"><a href="/Bayesian-Statistics-Exercises-in-R.html"><span class="progress-dot"></span>Bayesian Statistics Exercises</a></li><li data-subkey="sec7sub5"><a href="/Clustering-Exercises-in-R.html"><span class="progress-dot"></span>Clustering Exercises</a></li><li data-subkey="sec7sub5"><a href="/Correlation-Exercises-in-R.html"><span class="progress-dot"></span>Correlation Exercises</a></li><li data-subkey="sec7sub5"><a href="/Cross-Validation-Exercises-in-R.html"><span class="progress-dot"></span>Cross Validation Exercises</a></li><li data-subkey="sec7sub5"><a href="/Data-Cleaning-Exercises-in-R.html"><span class="progress-dot"></span>Data Cleaning Exercises</a></li><li data-subkey="sec7sub5"><a href="/Data-Visualization-Exercises-in-R.html"><span class="progress-dot"></span>Data Viz Exercises</a></li><li data-subkey="sec7sub5"><a href="/Data-Wrangling-Exercises-in-R.html"><span class="progress-dot"></span>Data Wrangling Exercises</a></li><li data-subkey="sec7sub5"><a href="/Decision-Tree-Exercises-in-R.html"><span class="progress-dot"></span>Decision Tree Exercises</a></li><li data-subkey="sec7sub5"><a href="/EDA-Exercises-in-R.html"><span class="progress-dot"></span>EDA Exercises</a></li><li data-subkey="sec7sub5"><a href="/GAM-Exercises-in-R.html"><span class="progress-dot"></span>GAM Exercises</a></li><li data-subkey="sec7sub5"><a href="/Machine-Learning-Exercises-in-R.html"><span class="progress-dot"></span>Machine Learning Exercises</a></li><li data-subkey="sec7sub5"><a href="/Mixed-Effects-Models-Exercises-in-R.html"><span class="progress-dot"></span>Mixed Effects Exercises</a></li><li data-subkey="sec7sub5"><a href="/Network-Analysis-Exercises-in-R.html"><span class="progress-dot"></span>Network Analysis Exercises</a></li><li data-subkey="sec7sub5"><a href="/Parallel-Computing-in-R-Exercises.html"><span class="progress-dot"></span>Parallel Computing Exercises</a></li><li data-subkey="sec7sub5"><a href="/Poisson-Regression-Exercises-in-R.html"><span class="progress-dot"></span>Poisson Regression</a></li><li data-subkey="sec7sub5"><a href="/Probability-Distributions-Exercises-in-R.html"><span class="progress-dot"></span>Probability Distributions</a></li><li data-subkey="sec7sub5"><a href="/R-Beginner-Exercises.html"><span class="progress-dot"></span>R Beginner Exercises</a></li><li data-subkey="sec7sub5"><a href="/R-Debugging-Exercises.html"><span class="progress-dot"></span>R Debugging Exercises</a></li><li data-subkey="sec7sub5"><a href="/R-Markdown-Exercises.html"><span class="progress-dot"></span>R Markdown Exercises</a></li><li data-subkey="sec7sub5"><a href="/R-Package-Development-Exercises.html"><span class="progress-dot"></span>R Package Development</a></li><li data-subkey="sec7sub5"><a href="/R-Performance-Optimization-Exercises.html"><span class="progress-dot"></span>R Performance Exercises</a></li><li data-subkey="sec7sub5"><a href="/R-for-Biostatistics-Exercises.html"><span class="progress-dot"></span>R for Biostatistics</a></li><li data-subkey="sec7sub5"><a href="/R-for-Data-Science-Exercises.html"><span class="progress-dot"></span>R for Data Science Exercises</a></li><li data-subkey="sec7sub5"><a href="/R-for-Finance-Exercises.html"><span class="progress-dot"></span>R for Finance Exercises</a></li><li data-subkey="sec7sub5"><a href="/R-for-Genomics-Exercises.html"><span class="progress-dot"></span>R for Genomics</a></li><li data-subkey="sec7sub5"><a href="/R-for-Healthcare-Exercises.html"><span class="progress-dot"></span>R for Healthcare Exercises</a></li><li data-subkey="sec7sub5"><a href="/R-for-Marketing-Analytics-Exercises.html"><span class="progress-dot"></span>R for Marketing Analytics</a></li><li data-subkey="sec7sub5"><a href="/R-for-Sports-Analytics-Exercises.html"><span class="progress-dot"></span>R for Sports Analytics</a></li><li data-subkey="sec7sub5"><a href="/Random-Forest-Exercises-in-R.html"><span class="progress-dot"></span>Random Forest Exercises</a></li><li data-subkey="sec7sub5"><a href="/Regex-Exercises-in-R.html"><span class="progress-dot"></span>Regex Exercises</a></li><li data-subkey="sec7sub5"><a href="/Sampling-Methods-Exercises-in-R.html"><span class="progress-dot"></span>Sampling Methods Exercises</a></li><li data-subkey="sec7sub5"><a href="/Shiny-Exercises-in-R.html"><span class="progress-dot"></span>Shiny Exercises</a></li><li data-subkey="sec7sub5"><a href="/Spatial-Analysis-Exercises-in-R.html"><span class="progress-dot"></span>Spatial Analysis Exercises</a></li><li data-subkey="sec7sub5"><a href="/Survey-Analysis-in-R-Exercises.html"><span class="progress-dot"></span>Survey Analysis Exercises</a></li><li data-subkey="sec7sub5"><a href="/Survival-Analysis-Exercises-in-R.html"><span class="progress-dot"></span>Survival Analysis Exercises</a></li><li data-subkey="sec7sub5"><a href="/Text-Mining-Exercises-in-R.html"><span class="progress-dot"></span>Text Mining Exercises</a></li><li data-subkey="sec7sub5"><a href="/Time-Series-Exercises-in-R.html"><span class="progress-dot"></span>Time Series Exercises</a></li><li data-subkey="sec7sub5"><a href="/Web-Scraping-Exercises-in-R.html"><span class="progress-dot"></span>Web Scraping Exercises</a></li><li data-subkey="sec7sub5"><a href="/XGBoost-Exercises-in-R.html"><span class="progress-dot"></span>XGBoost Exercises</a></li><li data-subkey="sec7sub5"><a href="/broom-Exercises-in-R.html"><span class="progress-dot"></span>broom Exercises</a></li><li data-subkey="sec7sub5"><a href="/caret-Exercises-in-R.html"><span class="progress-dot"></span>caret Exercises</a></li><li data-subkey="sec7sub5"><a href="/data.table-Exercises-in-R.html"><span class="progress-dot"></span>data.table Exercises</a></li><li data-subkey="sec7sub5"><a href="/dbplyr-SQL-Exercises-in-R.html"><span class="progress-dot"></span>dbplyr / SQL Exercises</a></li><li data-subkey="sec7sub5"><a href="/dplyr-Exercises-in-R.html"><span class="progress-dot"></span>dplyr Exercises</a></li><li data-subkey="sec7sub5"><a href="/dplyr-Group-By-Exercises-in-R.html"><span class="progress-dot"></span>dplyr group_by Exercises</a></li><li data-subkey="sec7sub5"><a href="/dplyr-Joins-Exercises-in-R.html"><span class="progress-dot"></span>dplyr Joins Exercises</a></li><li data-subkey="sec7sub5"><a href="/dplyr-Window-Functions-Exercises-in-R.html"><span class="progress-dot"></span>dplyr Window Functions Exercises</a></li><li data-subkey="sec7sub5"><a href="/forcats-Exercises-in-R.html"><span class="progress-dot"></span>forcats Exercises</a></li><li data-subkey="sec7sub5"><a href="/ggplot2-Bar-Chart-Exercises-in-R.html"><span class="progress-dot"></span>ggplot2 Bar Chart Exercises</a></li><li data-subkey="sec7sub5"><a href="/ggplot2-Color-Scales-Exercises-in-R.html"><span class="progress-dot"></span>ggplot2 Color Scales Exercises</a></li><li data-subkey="sec7sub5"><a href="/ggplot2-Exercises-in-R.html"><span class="progress-dot"></span>ggplot2 Exercises</a></li><li data-subkey="sec7sub5"><a href="/ggplot2-Facets-Exercises-in-R.html"><span class="progress-dot"></span>ggplot2 Facets Exercises</a></li><li data-subkey="sec7sub5"><a href="/ggplot2-Heatmap-Exercises-in-R.html"><span class="progress-dot"></span>ggplot2 Heatmap Exercises</a></li><li data-subkey="sec7sub5"><a href="/ggplot2-Themes-Exercises-in-R.html"><span class="progress-dot"></span>ggplot2 Themes Exercises</a></li><li data-subkey="sec7sub5"><a href="/gt-Tables-Exercises-in-R.html"><span class="progress-dot"></span>gt Tables Exercises</a></li><li data-subkey="sec7sub5"><a href="/leaflet-Exercises-in-R.html"><span class="progress-dot"></span>leaflet Exercises</a></li><li data-subkey="sec7sub5"><a href="/lubridate-Exercises-in-R.html"><span class="progress-dot"></span>lubridate Exercises</a></li><li data-subkey="sec7sub5"><a href="/plotly-Exercises-in-R.html"><span class="progress-dot"></span>plotly Exercises</a></li><li data-subkey="sec7sub5"><a href="/purrr-Exercises-in-R.html"><span class="progress-dot"></span>purrr Exercises</a></li><li data-subkey="sec7sub5"><a href="/readr-Exercises-in-R.html"><span class="progress-dot"></span>readr Exercises</a></li><li data-subkey="sec7sub5"><a href="/stringr-Exercises-in-R.html"><span class="progress-dot"></span>stringr Exercises</a></li><li data-subkey="sec7sub5"><a href="/testthat-Exercises-in-R.html"><span class="progress-dot"></span>testthat Exercises</a></li><li data-subkey="sec7sub5"><a href="/tidymodels-Exercises-in-R.html"><span class="progress-dot"></span>tidymodels Exercises</a></li><li data-subkey="sec7sub5"><a href="/tidyr-Exercises-in-R.html"><span class="progress-dot"></span>tidyr Exercises</a></li><li data-subkey="sec7sub5"><a href="/tidyr-Nest-Unnest-Exercises-in-R.html"><span class="progress-dot"></span>tidyr Nest/Unnest Exercises</a></li><li data-subkey="sec7sub5"><a href="/tidyr-Pivot-Exercises-in-R.html"><span class="progress-dot"></span>tidyr Pivot Exercises</a></li><li data-subkey="sec7sub5"><a href="/tidyverse-Exercises-in-R.html"><span class="progress-dot"></span>Tidyverse Exercises</a></li><li data-subkey="sec7sub5"><a href="/Date-Time-Manipulation-Exercises-in-R.html"><span class="progress-dot"></span>Date-Time Manipulation Exercises</a></li><li data-subkey="sec7sub5"><a href="/Loops-vs-Vectorization-Exercises-in-R.html"><span class="progress-dot"></span>Loops vs Vectorization Exercises</a></li></ul></li></ul><div class="sidebar-subscribe"><p>Stay up-to-date. <a href="https://docs.google.com/forms/d/1xkMYkLNFU9U39Dd8S_2JC0p8B5t6_Yq6zUQjanQQJpY/viewform">Subscribe!</a></p><p><a href="https://docs.google.com/forms/d/13GrkCFcNa-TOIllQghsz2SIEbc-YqY9eJX02B19l5Ow/viewform">Chat!</a></p></div></div><div class="sidebar-panel" data-panel="tools"><ul class="sidebar-tools-list list-unstyled"><li class="sidebar-divider"><span class="subsec-chevron">&#9660;</span> Calculators</li><li><a href="/tools/ab-test-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="2.5" y="3.5" width="4" height="9" rx="0.5"/><rect x="9.5" y="3.5" width="4" height="9" rx="0.5"/></svg></span><span class="tool-label">A/B Test Calculator</span></a></li><li><a href="/tools/t-test-calculator.html"><span class="tool-icon"></span><span class="tool-label">t-Test Calculator</span></a></li><li><a href="/tools/chi-square-calculator.html"><span class="tool-icon"></span><span class="tool-label">Chi-Square Test</span></a></li><li><a href="/tools/confidence-interval-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M3.5 4v8M3.5 4H5M3.5 12H5"/><circle cx="8" cy="8" r="1.4" fill="currentColor" stroke="none"/><path d="M12.5 4v8M12.5 4H11M12.5 12H11"/></svg></span><span class="tool-label">Confidence Interval</span></a></li><li><a href="/tools/bootstrap-ci-calculator.html"><span class="tool-icon"></span><span class="tool-label">Bootstrap CI</span></a></li><li><a href="/tools/effect-size-converter.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2.5 5.5h11M11 3l2.5 2.5L11 8"/><path d="M13.5 10.5h-11M5 8l-2.5 2.5L5 13"/></svg></span><span class="tool-label">Effect Size Converter</span></a></li><li><a href="/tools/power-analysis.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M9 2L4 9h3.5L7 14l5-7H8.5z"/></svg></span><span class="tool-label">Power Analysis</span></a></li><li><a href="/tools/survival-power-calculator.html"><span class="tool-icon"></span><span class="tool-label">Survival Power</span></a></li><li><a href="/tools/type-i-ii-error-visualizer.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2 12c1.5 0 2-2 3.5-2S7 12 8 12s2-8 3.5-8S13 12 14.5 12"/><line x1="8" y1="2" x2="8" y2="14" stroke-dasharray="2 2"/></svg></span><span class="tool-label">Type I / II Error</span></a></li><li><a href="/tools/z-score-percentile.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2 13c2 0 3-1 4-3s1.5-6 2-6 .5 6 2 6 2.5 0 4 0"/><path d="M2 13.5h12"/></svg></span><span class="tool-label">Z-Score &amp; Percentile</span></a></li><li><a href="/tools/equivalence-noninferiority-calculator.html"><span class="tool-icon"></span><span class="tool-label">Equivalence / NI</span></a></li><li><a href="/tools/outlier-detection-calculator.html"><span class="tool-icon"></span><span class="tool-label">Outlier Detection</span></a></li><li><a href="/tools/roc-auc-calculator.html"><span class="tool-icon"></span><span class="tool-label">ROC / AUC</span></a></li><li class="sidebar-divider"><span class="subsec-chevron">&#9660;</span> Bayesian</li><li><a href="/tools/bayes-theorem-calculator.html"><span class="tool-icon"></span><span class="tool-label">Bayes Theorem</span></a></li><li><a href="/tools/bayes-factor-calculator.html"><span class="tool-icon"></span><span class="tool-label">Bayes Factor</span></a></li><li class="sidebar-divider"><span class="subsec-chevron">&#9660;</span> Interpreters</li><li><a href="/tools/lm-output-interpreter.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2 13.5L14 3"/><circle cx="4" cy="11.5" r="1" fill="currentColor" stroke="none"/><circle cx="7.5" cy="9" r="1" fill="currentColor" stroke="none"/><circle cx="11" cy="6.5" r="1" fill="currentColor" stroke="none"/></svg></span><span class="tool-label">lm() Output</span></a></li><li><a href="/tools/glm-output-interpreter.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2 13C5 13 5 3 8 3s3 10 6 10"/><path d="M2 13.5h12"/></svg></span><span class="tool-label">glm() Output</span></a></li><li><a href="/tools/anova-output-interpreter.html"><span class="tool-icon"></span><span class="tool-label">ANOVA Output</span></a></li><li><a href="/tools/vif-interpreter.html"><span class="tool-icon"></span><span class="tool-label">VIF / Multicollinearity</span></a></li><li><a href="/tools/confusion-matrix-interpreter.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="2.5" y="2.5" width="4.5" height="4.5"/><rect x="9" y="2.5" width="4.5" height="4.5"/><rect x="2.5" y="9" width="4.5" height="4.5"/><rect x="9" y="9" width="4.5" height="4.5"/></svg></span><span class="tool-label">Confusion Matrix</span></a></li><li><a href="/tools/diagnostic-plot-interpreter.html"><span class="tool-icon"></span><span class="tool-label">Diagnostic Plots</span></a></li><li class="sidebar-divider"><span class="subsec-chevron">&#9660;</span> Pickers</li><li><a href="/tools/normality-test-picker.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2 13c2 0 3-1 4-3s1.5-6 2-6 .5 6 2 6 2.5 0 4 0"/><circle cx="13" cy="4" r="1.6" fill="currentColor" stroke="none"/></svg></span><span class="tool-label">Normality Test</span></a></li><li><a href="/tools/nonparametric-test-picker.html"><span class="tool-icon"></span><span class="tool-label">Non-Parametric Test</span></a></li><li><a href="/tools/multiple-testing-correction.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2.5 4l2 2 3-3M2.5 8l2 2 3-3M2.5 12l2 2 3-3"/><path d="M10.5 4h3M10.5 8h3M10.5 12h3"/></svg></span><span class="tool-label">Multiple Testing</span></a></li><li class="sidebar-divider"><span class="subsec-chevron">&#9660;</span> Time series</li><li><a href="/tools/ts-stationarity-calculator.html"><span class="tool-icon"></span><span class="tool-label">TS Stationarity</span></a></li><li class="sidebar-divider"><span class="subsec-chevron">&#9660;</span> Utilities</li><li><a href="/tools/dag-confounder-picker.html"><span class="tool-icon"></span><span class="tool-label">DAG Confounder Picker</span></a></li><li><a href="/tools/reprex-builder.html"><span class="tool-icon"></span><span class="tool-label">Reprex Builder</span></a></li></ul></div></div>
        </div>

        <main id="content" class="col-xs-12 col-sm-7">


          <nav class="breadcrumb-nav" aria-label="Breadcrumb"><a href="/">Home</a> <span class="breadcrumb-sep">&rsaquo;</span> <span>Statistics</span> <span class="breadcrumb-sep">&rsaquo;</span> <span class="breadcrumb-current">Conditional Probability in R: P(A|B), Independence, and Bayes, With Real Examples</span></nav>
<!-- md2html:generated -->

<h1>Conditional Probability in R: P(A|B), Independence, and Bayes, With Real Examples</h1>

<p class="lead"><a class="auto-link" href="Bayes-Theorem-in-R.html" title="Bayes' Theorem in R: Why Medical Tests Mislead You, A Simulation That Shows Why">Conditional probability</a> measures how the chance of one event changes when you know another event already happened, written P(A|B) and read "the probability of A given B." In this tutorial you'll simulate conditional probabilities in R, prove when events are independent, and apply Bayes' theorem to a medical-testing scenario where the "obvious" answer is wrong.</p>
<div class="post-byline" style="color:#6b7280;font-size:14px;margin:2px 0 18px 0;line-height:1.5;">By <strong>Selva Prabhakaran</strong> &nbsp;&middot;&nbsp; Published May 10, 2026 &nbsp;&middot;&nbsp; Last updated May 10, 2026</div>

<div class="engagement-header" data-difficulty="Intermediate" data-time="40" data-exercises="9" data-xp="135"></div>

<h2>What is conditional probability and how do you calculate P(A|B)?</h2>

<p>Imagine you roll a fair die and it lands on an even number. What's the probability the roll is also greater than 4? Knowing the die is even shrinks the possible outcomes from six to three, {2, 4, 6}, and only one of those (6) is greater than 4. That gives us 1/3 instead of the unconditional 2/6.</p>

<p>Let's confirm this with a simulation of 100,000 rolls.</p>

<div class="webr-container" data-block-title="Simulate P greater than 4 given even">
  <div class="webr-code-block">
    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Simulate P greater than 4 given even</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># Simulate conditional probability: P(&gt;4 | even)</span></span>
<span class="cl"><span class="nf">set.seed</span>(<span class="m">42</span>)</span>
<span class="cl">rolls <span class="o">&lt;-</span> <span class="nf">sample</span>(<span class="m">1</span><span class="o">:</span><span class="m">6</span>, size <span class="o">=</span> <span class="m">100000</span>, replace <span class="o">=</span> <span class="kc">TRUE</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Filter to only even rolls (our &quot;given&quot; event)</span></span>
<span class="cl">even_rolls <span class="o">&lt;-</span> rolls[rolls <span class="o">%%</span> <span class="m">2</span> <span class="o">==</span> <span class="m">0</span>]</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Among even rolls, what fraction are &gt; 4?</span></span>
<span class="cl">p_gt4_given_even <span class="o">&lt;-</span> <span class="nf">mean</span>(even_rolls <span class="o">&gt;</span> <span class="m">4</span>)</span>
<span class="cl">p_gt4_given_even</span>
<span class="cl"><span class="c1">#&gt; [1] 0.3332</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Compare to unconditional P(&gt; 4)</span></span>
<span class="cl">p_gt4 <span class="o">&lt;-</span> <span class="nf">mean</span>(rolls <span class="o">&gt;</span> <span class="m">4</span>)</span>
<span class="cl">p_gt4</span>
<span class="cl"><span class="c1">#&gt; [1] 0.3336</span></span></div>
    <div class="webr-buttons">
      <button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run</button>
      <button class="btn btn-sm btn-default webr-reset-btn" onclick="resetWebR(this)">&#8634; Reset</button>
    </div>
    <pre class="webr-output"></pre>
  </div>
  <div class="webr-plot-output"></div>
</div>

<p>The simulation gives us roughly 0.333, exactly 1/3. Notice that P(>4) unconditionally is also about 1/3 in this case, but that's a coincidence of these particular events. The key idea is that we restricted our universe to even rolls before asking the question.</p>

<p>The formal formula captures this "restrict and re-normalise" logic:</p>

<p>$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$</p>

<p>Where:</p>

<ul>

<li>$P(A|B)$ = probability of A given B has occurred</li>

<li>$P(A \cap B)$ = probability both A and B happen (joint probability)</li>

<li>$P(B)$ = probability of the conditioning event</li>

</ul>

<p>Let's verify the formula matches our simulation.</p>

<div class="webr-container" data-block-title="Verify with joint formula">
  <div class="webr-code-block">
    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Verify with joint formula</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># Verify with the formula: P(A ∩ B) / P(B)</span></span>
<span class="cl"><span class="c1"># A = roll &gt; 4, B = roll is even</span></span>
<span class="cl"><span class="c1"># A ∩ B = roll is 6 (only number that&#39;s both &gt; 4 and even)</span></span>
<span class="cl">p_a_and_b <span class="o">&lt;-</span> <span class="nf">mean</span>(rolls <span class="o">==</span> <span class="m">6</span>)</span>
<span class="cl">p_b <span class="o">&lt;-</span> <span class="nf">mean</span>(rolls <span class="o">%%</span> <span class="m">2</span> <span class="o">==</span> <span class="m">0</span>)</span>
<span class="cl"></span>
<span class="cl">p_formula <span class="o">&lt;-</span> p_a_and_b <span class="o">/</span> p_b</span>
<span class="cl">p_formula</span>
<span class="cl"><span class="c1">#&gt; [1] 0.3332</span></span></div>
    <div class="webr-buttons">
      <button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run</button>
      <button class="btn btn-sm btn-default webr-reset-btn" onclick="resetWebR(this)">&#8634; Reset</button>
    </div>
    <pre class="webr-output"></pre>
  </div>
  <div class="webr-plot-output"></div>
</div>

<p>Both approaches, simulation and formula, give the same answer. That's the power of Monte Carlo: you can always check a formula by simulating the experiment thousands of times.</p>

<div class="callout callout-insight"><div class="callout-label">Key Insight</div><div class="callout-body"><strong>Conditioning shrinks the sample space.</strong> P(A|B) doesn't change event A, it changes the universe of outcomes you're considering. You throw away everything where B didn't happen, then ask "what fraction of what's left is A?"</div></div>

<section class="tryit-block">
<p><strong>Try it:</strong> Given a die roll is odd, what's the probability it's less than or equal to 3? Simulate with 100,000 rolls and verify the answer is 2/3.</p>
<div class="webr-container" data-block-title="Exercise: P at most 3 given odd">
  <div class="webr-code-block">
    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Exercise: P at most 3 given odd</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># Try it: P(≤ 3 | odd)</span></span>
<span class="cl"><span class="nf">set.seed</span>(<span class="m">99</span>)</span>
<span class="cl">ex_rolls <span class="o">&lt;-</span> <span class="nf">sample</span>(<span class="m">1</span><span class="o">:</span><span class="m">6</span>, size <span class="o">=</span> <span class="m">100000</span>, replace <span class="o">=</span> <span class="kc">TRUE</span>)</span>
<span class="cl">ex_odd <span class="o">&lt;-</span> ex_rolls[ex_rolls <span class="o">%%</span> <span class="m">2</span> <span class="o">==</span> <span class="m">1</span>]</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># your code here</span></span>
<span class="cl"><span class="nf">mean</span>(ex_odd <span class="o">&lt;=</span> <span class="m">3</span>)</span>
<span class="cl"><span class="c1">#&gt; Expected: ~0.667</span></span></div>
    <div class="webr-buttons">
      <button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run</button>
      <button class="btn btn-sm btn-default webr-reset-btn" onclick="resetWebR(this)">&#8634; Reset</button>
    </div>
    <pre class="webr-output"></pre>
  </div>
  <div class="webr-plot-output"></div>
</div>
<details>
<summary>Click to reveal solution</summary>

<div class="webr-container" data-block-title="Odd roll solution">
  <div class="webr-code-block">
    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Odd roll solution</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl"><span class="nf">set.seed</span>(<span class="m">99</span>)</span>
<span class="cl">ex_rolls <span class="o">&lt;-</span> <span class="nf">sample</span>(<span class="m">1</span><span class="o">:</span><span class="m">6</span>, size <span class="o">=</span> <span class="m">100000</span>, replace <span class="o">=</span> <span class="kc">TRUE</span>)</span>
<span class="cl">ex_odd <span class="o">&lt;-</span> ex_rolls[ex_rolls <span class="o">%%</span> <span class="m">2</span> <span class="o">==</span> <span class="m">1</span>]</span>
<span class="cl"><span class="nf">mean</span>(ex_odd <span class="o">&lt;=</span> <span class="m">3</span>)</span>
<span class="cl"><span class="c1">#&gt; [1] 0.6676</span></span></div>
    <div class="webr-buttons">
      <button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run</button>
      <button class="btn btn-sm btn-default webr-reset-btn" onclick="resetWebR(this)">&#8634; Reset</button>
    </div>
    <pre class="webr-output"></pre>
  </div>
  <div class="webr-plot-output"></div>
</div>

<p><strong>Explanation:</strong> The odd outcomes are {1, 3, 5}. Two of those three (1 and 3) are ≤ 3, so the answer is 2/3 ≈ 0.667.</p>

</details>
</section>

<h2>How does joint probability connect to conditional probability?</h2>

<p>The conditional probability formula can be rearranged into the <strong><a class="auto-link" href="Counting-Principles-in-R.html" title="Counting Principles in R: Permutations, Combinations & Birthday Problem">multiplication rule</a></strong>, a way to compute the probability that two events both happen:</p>

<p>$$P(A \cap B) = P(A|B) \times P(B)$$</p>

<p>This says: the chance of A <em>and</em> B equals the chance of B times the chance of A <em>given</em> B. Think of it as a two-step process, first B happens, then A happens in that restricted world.</p>

<p>Let's test this with playing cards. What's the probability of drawing a card that is both red and a face card?</p>

<div class="webr-container" data-block-title="Joint card probability simulation">
  <div class="webr-code-block">
    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Joint card probability simulation</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># Joint probability: P(Red ∩ Face) via simulation</span></span>
<span class="cl"><span class="nf">set.seed</span>(<span class="m">123</span>)</span>
<span class="cl">n_sims <span class="o">&lt;-</span> <span class="m">50000</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Simulate card draws (1-52)</span></span>
<span class="cl">draws <span class="o">&lt;-</span> <span class="nf">sample</span>(<span class="m">1</span><span class="o">:</span><span class="m">52</span>, size <span class="o">=</span> n_sims, replace <span class="o">=</span> <span class="kc">TRUE</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># In a standard deck: cards 1-26 are red, 27-52 are black</span></span>
<span class="cl"><span class="c1"># Face cards (J, Q, K) are cards 11-13, 24-26, 37-39, 50-52</span></span>
<span class="cl">face_positions <span class="o">&lt;-</span> <span class="nf">c</span>(<span class="m">11</span><span class="o">:</span><span class="m">13</span>, <span class="m">24</span><span class="o">:</span><span class="m">26</span>, <span class="m">37</span><span class="o">:</span><span class="m">39</span>, <span class="m">50</span><span class="o">:</span><span class="m">52</span>)</span>
<span class="cl"></span>
<span class="cl">is_red <span class="o">&lt;-</span> draws <span class="o">&lt;=</span> <span class="m">26</span></span>
<span class="cl">is_face <span class="o">&lt;-</span> draws <span class="o">%in%</span> face_positions</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Empirical joint probability</span></span>
<span class="cl">p_red_and_face <span class="o">&lt;-</span> <span class="nf">mean</span>(is_red <span class="o">&amp;</span> is_face)</span>
<span class="cl">p_red_and_face</span>
<span class="cl"><span class="c1">#&gt; [1] 0.1155</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Verify via multiplication rule: P(Face|Red) × P(Red)</span></span>
<span class="cl">p_red <span class="o">&lt;-</span> <span class="nf">mean</span>(is_red)</span>
<span class="cl">p_face_given_red <span class="o">&lt;-</span> <span class="nf">mean</span>(is_face[is_red])</span>
<span class="cl">p_multiplication <span class="o">&lt;-</span> p_face_given_red <span class="o">*</span> p_red</span>
<span class="cl">p_multiplication</span>
<span class="cl"><span class="c1">#&gt; [1] 0.1155</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Theoretical: 6 red face cards / 52 = 0.1154</span></span>
<span class="cl"><span class="m">6</span> <span class="o">/</span> <span class="m">52</span></span>
<span class="cl"><span class="c1">#&gt; [1] 0.1154</span></span></div>
    <div class="webr-buttons">
      <button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run</button>
      <button class="btn btn-sm btn-default webr-reset-btn" onclick="resetWebR(this)">&#8634; Reset</button>
    </div>
    <pre class="webr-output"></pre>
  </div>
  <div class="webr-plot-output"></div>
</div>

<p>The simulation, the multiplication rule, and the theoretical calculation all converge on about 0.115. There are 6 red face cards (Jack, Queen, King of Hearts and Diamonds) out of 52 total cards.</p>

<div class="callout callout-tip"><div class="callout-label">Tip</div><div class="callout-body"><strong>The multiplication rule is just the conditional probability formula rearranged.</strong> If you know P(A|B) and P(B), multiply them to get P(A ∩ B). If you know P(A ∩ B) and P(B), divide to get P(A|B). One formula, two directions.</div></div>

<section class="tryit-block">
<p><strong>Try it:</strong> In a standard 52-card deck, what's P(Heart ∩ King)? There's exactly 1 King of Hearts. Compute it using the multiplication rule, P(King|Heart) × P(Heart), and verify with simulation.</p>
<div class="webr-container" data-block-title="Exercise: Heart and King joint">
  <div class="webr-code-block">
    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Exercise: Heart and King joint</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># Try it: P(Heart ∩ King)</span></span>
<span class="cl"><span class="nf">set.seed</span>(<span class="m">200</span>)</span>
<span class="cl">ex_draws <span class="o">&lt;-</span> <span class="nf">sample</span>(<span class="m">1</span><span class="o">:</span><span class="m">52</span>, size <span class="o">=</span> <span class="m">50000</span>, replace <span class="o">=</span> <span class="kc">TRUE</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Hearts are cards 1-13, Kings are cards 13, 26, 39, 52</span></span>
<span class="cl">ex_is_heart <span class="o">&lt;-</span> ex_draws <span class="o">&lt;=</span> <span class="m">13</span></span>
<span class="cl">ex_is_king <span class="o">&lt;-</span> ex_draws <span class="o">%in%</span> <span class="nf">c</span>(<span class="m">13</span>, <span class="m">26</span>, <span class="m">39</span>, <span class="m">52</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># your code here: compute empirical P(Heart ∩ King)</span></span>
<span class="cl"><span class="c1"># and verify via P(King|Heart) × P(Heart)</span></span>
<span class="cl"><span class="c1">#&gt; Expected: ~0.0192 (1/52)</span></span></div>
    <div class="webr-buttons">
      <button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run</button>
      <button class="btn btn-sm btn-default webr-reset-btn" onclick="resetWebR(this)">&#8634; Reset</button>
    </div>
    <pre class="webr-output"></pre>
  </div>
  <div class="webr-plot-output"></div>
</div>
<details>
<summary>Click to reveal solution</summary>

<div class="webr-container" data-block-title="Heart King solution">
  <div class="webr-code-block">
    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Heart King solution</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl"><span class="nf">set.seed</span>(<span class="m">200</span>)</span>
<span class="cl">ex_draws <span class="o">&lt;-</span> <span class="nf">sample</span>(<span class="m">1</span><span class="o">:</span><span class="m">52</span>, size <span class="o">=</span> <span class="m">50000</span>, replace <span class="o">=</span> <span class="kc">TRUE</span>)</span>
<span class="cl">ex_is_heart <span class="o">&lt;-</span> ex_draws <span class="o">&lt;=</span> <span class="m">13</span></span>
<span class="cl">ex_is_king <span class="o">&lt;-</span> ex_draws <span class="o">%in%</span> <span class="nf">c</span>(<span class="m">13</span>, <span class="m">26</span>, <span class="m">39</span>, <span class="m">52</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Empirical joint</span></span>
<span class="cl"><span class="nf">mean</span>(ex_is_heart <span class="o">&amp;</span> ex_is_king)</span>
<span class="cl"><span class="c1">#&gt; [1] 0.0194</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Multiplication rule</span></span>
<span class="cl"><span class="nf">mean</span>(ex_is_king[ex_is_heart]) <span class="o">*</span> <span class="nf">mean</span>(ex_is_heart)</span>
<span class="cl"><span class="c1">#&gt; [1] 0.0194</span></span></div>
    <div class="webr-buttons">
      <button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run</button>
      <button class="btn btn-sm btn-default webr-reset-btn" onclick="resetWebR(this)">&#8634; Reset</button>
    </div>
    <pre class="webr-output"></pre>
  </div>
  <div class="webr-plot-output"></div>
</div>

<p><strong>Explanation:</strong> P(Heart) = 13/52 = 1/4, P(King|Heart) = 1/13 (one King among 13 Hearts), so P(Heart ∩ King) = (1/13) × (1/4) = 1/52 ≈ 0.0192.</p>

</details>
</section>

<h2>When are two events independent, and how can you prove it in R?</h2>

<p>Two events are <strong>independent</strong> when knowing one tells you nothing about the other. Formally:</p>

<p>$$P(A|B) = P(A)$$</p>

<p>This is equivalent to saying $P(A \cap B) = P(A) \times P(B)$, the joint probability equals the product of the individual probabilities.</p>

<p>The classic example: rolling two separate dice. The outcome of die 1 can't possibly influence die 2.</p>

<div class="webr-container" data-block-title="Two-dice independence check">
  <div class="webr-code-block">
    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Two-dice independence check</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># Independence proof: two dice</span></span>
<span class="cl"><span class="nf">set.seed</span>(<span class="m">77</span>)</span>
<span class="cl">n <span class="o">&lt;-</span> <span class="m">100000</span></span>
<span class="cl">die1 <span class="o">&lt;-</span> <span class="nf">sample</span>(<span class="m">1</span><span class="o">:</span><span class="m">6</span>, n, replace <span class="o">=</span> <span class="kc">TRUE</span>)</span>
<span class="cl">die2 <span class="o">&lt;-</span> <span class="nf">sample</span>(<span class="m">1</span><span class="o">:</span><span class="m">6</span>, n, replace <span class="o">=</span> <span class="kc">TRUE</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># P(die1 = 6)</span></span>
<span class="cl">p_die1_six <span class="o">&lt;-</span> <span class="nf">mean</span>(die1 <span class="o">==</span> <span class="m">6</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># P(die1 = 6 | die2 = 6)</span></span>
<span class="cl">p_die1_six_given_die2_six <span class="o">&lt;-</span> <span class="nf">mean</span>(die1[die2 <span class="o">==</span> <span class="m">6</span>] <span class="o">==</span> <span class="m">6</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="nf">cat</span>(<span class="s">&quot;P(die1 = 6)          =&quot;</span>, <span class="nf">round</span>(p_die1_six, <span class="m">4</span>), <span class="s">&quot;\n&quot;</span>)</span>
<span class="cl"><span class="c1">#&gt; P(die1 = 6)          = 0.1667</span></span>
<span class="cl"><span class="nf">cat</span>(<span class="s">&quot;P(die1 = 6 | die2=6) =&quot;</span>, <span class="nf">round</span>(p_die1_six_given_die2_six, <span class="m">4</span>), <span class="s">&quot;\n&quot;</span>)</span>
<span class="cl"><span class="c1">#&gt; P(die1 = 6 | die2=6) = 0.1644</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Nearly identical, knowing die2 = 6 doesn&#39;t change die1&#39;s probability</span></span></div>
    <div class="webr-buttons">
      <button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run</button>
      <button class="btn btn-sm btn-default webr-reset-btn" onclick="resetWebR(this)">&#8634; Reset</button>
    </div>
    <pre class="webr-output"></pre>
  </div>
  <div class="webr-plot-output"></div>
</div>

<p>The two probabilities are nearly identical (both around 0.167 = 1/6). Knowing die 2 landed on 6 tells you nothing about die 1. That's independence.</p>

<p>Now here's the contrast, <strong>dependent events</strong>. When you draw cards <em>without</em> replacement, the first draw changes the deck for the second draw.</p>

<div class="webr-container" data-block-title="Dependent card draws">
  <div class="webr-code-block">
    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Dependent card draws</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># Dependent events: cards without replacement</span></span>
<span class="cl"><span class="nf">set.seed</span>(<span class="m">55</span>)</span>
<span class="cl">n_sims <span class="o">&lt;-</span> <span class="m">100000</span></span>
<span class="cl"></span>
<span class="cl">first_ace <span class="o">&lt;-</span> <span class="m">0</span></span>
<span class="cl">second_ace_given_first <span class="o">&lt;-</span> <span class="m">0</span></span>
<span class="cl">second_ace_overall <span class="o">&lt;-</span> <span class="m">0</span></span>
<span class="cl"></span>
<span class="cl"><span class="kr">for</span> (i <span class="kr">in</span> <span class="m">1</span><span class="o">:</span>n_sims) {</span>
<span class="cl">  hand <span class="o">&lt;-</span> <span class="nf">sample</span>(<span class="m">1</span><span class="o">:</span><span class="m">52</span>, size <span class="o">=</span> <span class="m">2</span>, replace <span class="o">=</span> <span class="kc">FALSE</span>)</span>
<span class="cl">  <span class="c1"># Aces are cards 1, 14, 27, 40</span></span>
<span class="cl">  aces <span class="o">&lt;-</span> <span class="nf">c</span>(<span class="m">1</span>, <span class="m">14</span>, <span class="m">27</span>, <span class="m">40</span>)</span>
<span class="cl">  card1_ace <span class="o">&lt;-</span> hand[<span class="m">1</span>] <span class="o">%in%</span> aces</span>
<span class="cl">  card2_ace <span class="o">&lt;-</span> hand[<span class="m">2</span>] <span class="o">%in%</span> aces</span>
<span class="cl"></span>
<span class="cl">  <span class="kr">if</span> (card1_ace) {</span>
<span class="cl">    first_ace <span class="o">&lt;-</span> first_ace <span class="o">+</span> <span class="m">1</span></span>
<span class="cl">    <span class="kr">if</span> (card2_ace) second_ace_given_first <span class="o">&lt;-</span> second_ace_given_first <span class="o">+</span> <span class="m">1</span></span>
<span class="cl">  }</span>
<span class="cl">  <span class="kr">if</span> (card2_ace) second_ace_overall <span class="o">&lt;-</span> second_ace_overall <span class="o">+</span> <span class="m">1</span></span>
<span class="cl">}</span>
<span class="cl"></span>
<span class="cl"><span class="nf">cat</span>(<span class="s">&quot;P(2nd Ace)           =&quot;</span>, <span class="nf">round</span>(second_ace_overall <span class="o">/</span> n_sims, <span class="m">4</span>), <span class="s">&quot;\n&quot;</span>)</span>
<span class="cl"><span class="c1">#&gt; P(2nd Ace)           = 0.0770</span></span>
<span class="cl"><span class="nf">cat</span>(<span class="s">&quot;P(2nd Ace | 1st Ace) =&quot;</span>, <span class="nf">round</span>(second_ace_given_first <span class="o">/</span> first_ace, <span class="m">4</span>), <span class="s">&quot;\n&quot;</span>)</span>
<span class="cl"><span class="c1">#&gt; P(2nd Ace | 1st Ace) = 0.0588</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Different! The first draw changes the second draw&#39;s odds</span></span>
<span class="cl"><span class="c1"># Theoretical: 3/51 ≈ 0.0588 vs 4/52 ≈ 0.0769</span></span></div>
    <div class="webr-buttons">
      <button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run</button>