Probabilities In Bingo

Author: Michael Shackleford

The following table shows the probability of forming a bingo, black out, or four corners within a specified number of calls. For example the probability of a single player forming a bingo within 25 calls is 0.06396106, or about 6.4%.

Probabilties in Bingo
Number of Calls	Bingo	Cover All	Four Corners	X
1	0.00000000	0.00000000	0.00000000	0.00000000
2	0.00000000	0.00000000	0.00000000	0.00000000
3	0.00000000	0.00000000	0.00000000	0.00000000
4	0.00000329	0.00000000	0.00000082	0.00000000
5	0.00001692	0.00000000	0.00000411	0.00000000
6	0.00005215	0.00000000	0.00001234	0.00000000
7	0.00012492	0.00000000	0.00002880	0.00000000
8	0.00025632	0.00000000	0.00005759	0.00000000
9	0.00047305	0.00000000	0.00010367	0.00000000
10	0.00080783	0.00000000	0.00017278	0.00000000
11	0.00129986	0.00000000	0.00027150	0.00000001
12	0.00199521	0.00000000	0.00040726	0.00000003
13	0.00294715	0.00000000	0.00058826	0.00000008
14	0.00421648	0.00000000	0.00082356	0.00000018
15	0.00587167	0.00000000	0.00112304	0.00000038
16	0.00798905	0.00000000	0.00149739	0.00000076
17	0.01065272	0.00000000	0.00195812	0.00000144
18	0.01395440	0.00000000	0.00251759	0.00000259
19	0.01799309	0.00000000	0.00318894	0.00000448
20	0.02287445	0.00000000	0.00398618	0.00000747
21	0.02871003	0.00000000	0.00492410	0.00001206
22	0.03561614	0.00000000	0.00601835	0.00001895
23	0.04371249	0.00000000	0.00728537	0.00002906
24	0.05312045	0.00000000	0.00874244	0.00004359
25	0.06396106	0.00000000	0.01040767	0.00006411
26	0.07635261	0.00000000	0.01229997	0.00009260
27	0.09040799	0.00000000	0.01443910	0.00013159
28	0.10623163	0.00000000	0.01684561	0.00018423
29	0.12391628	0.00000000	0.01954091	0.00025441
30	0.14353947	0.00000000	0.02254720	0.00034692
31	0.16515993	0.00000000	0.02588753	0.00046759
32	0.18881391	0.00000000	0.02958575	0.00062345
33	0.21451154	0.00000000	0.03366654	0.00082296
34	0.24223348	0.00000000	0.03815542	0.00107617
35	0.27192783	0.00000000	0.04307870	0.00139504
36	0.30350759	0.00000000	0.04846353	0.00179362
37	0.33684876	0.00000000	0.05433790	0.00228842
38	0.37178933	0.00000000	0.06073059	0.00289866
39	0.40812916	0.00000000	0.06767123	0.00364670
40	0.44563111	0.00000000	0.07519026	0.00455838
41	0.48402328	0.00000001	0.08331894	0.00566344
42	0.52300269	0.00000001	0.09208935	0.00699602
43	0.56224021	0.00000003	0.10153441	0.00859511
44	0.60138685	0.00000007	0.11168785	0.01050513
45	0.64008123	0.00000015	0.12258423	0.01277651
46	0.67795818	0.00000031	0.13425892	0.01546630
47	0.71465810	0.00000063	0.14674812	0.01863888
48	0.74983686	0.00000125	0.16008886	0.02236665
49	0.78317588	0.00000245	0.17431898	0.02673088
50	0.81439191	0.00000472	0.18947715	0.03182247
51	0.84324614	0.00000891	0.20560286	0.03774293
52	0.86955207	0.00001654	0.22273644	0.04460528
53	0.89318170	0.00003023	0.24091900	0.05253511
54	0.91406974	0.00005441	0.26019252	0.06167165
55	0.93221528	0.00009654	0.28059978	0.07216896
56	0.94768080	0.00016894	0.30218438	0.08419712
57	0.96058846	0.00029180	0.32499074	0.09794358
58	0.97111353	0.00049778	0.34906413	0.11361456
59	0.97947539	0.00083912	0.37445061	0.13143645
60	0.98592639	0.00139853	0.40119709	0.15165744
61	0.99073928	0.00230569	0.42935127	0.17454913
62	0.99419379	0.00376192	0.45896170	0.20040826
63	0.99656346	0.00607694	0.49007775	0.22955855
64	0.99810354	0.00972311	0.52274960	0.26235263
65	0.99904080	0.01541468	0.55702826	0.29917406
66	0.99956626	0.02422308	0.59296557	0.34043944
67	0.99983122	0.03774293	0.63061418	0.38660072
68	0.99994699	0.05832999	0.67002756	0.43814749
69	0.99998812	0.08943931	0.71126003	0.49560945
70	0.99999861	0.13610330	0.75436670	0.55955906
71	1.00000000	0.20560286	0.79940351	0.63061418
72	1.00000000	0.30840429	0.84642725	0.70944095
73	1.00000000	0.45945946	0.89549550	0.79675676
74	1.00000000	0.68000000	0.94666667	0.89333333
75	1.00000000	1.00000000	1.00000000	1.00000000

My method of analysis was entirely mathematical. The probability of x marks on the card given y calls is easily calculated as combin(24,x)*combin(51,y-x)/combin(75,y). The probability that x marks will form a bingo (five in a row) is more compicated and necessitated a computer program to run through all possible combinations and tabulate the results.

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