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75 Number Bingo

Mean Number of Calls to Cover Pattern

The next table shows the mean number of calls to cover a pattern of 1 to 24 marks. This table is only appropriate if there is only one way to cover the pattern.

Expected Calls to Cover Pattern of x Marks
Marks	Expected Calls
1	38
2	50.666667
3	57
4	60.8
5	63.333333
6	65.142857
7	66.5
8	67.555556
9	68.4
10	69.090909
11	69.666667
12	70.153846
13	70.571429
14	70.933333
15	71.25
16	71.529412
17	71.777778
18	72
19	72.2
20	72.380952
21	72.545455
22	72.695652
23	72.833333
24	72.96

The next table shows the probability that a coverall will be called in 24 to 75 calls or less by the number of cards in play. For example in a 200-card game the probability of the first coverall in 60 calls or less is 36.69%.

Probability of Coverall by Number of Calls or Less
Calls	100 Cards	200 Cards	500 Cards	1000 Cards
24	0	0	0	0
25	0	0	0	0
26	0	0	0	0
27	0	0	0	0
28	0	0	0	0
29	0	0	0	0
30	0	0	0	0
31	0	0	0	0
32	0	0	0	0
33	0	0	0	0
34	0	0	0	0
35	0	0	0	0
36	0	0	0	0
37	0	0	0	0
38	0.000000081	0	0.000000556	0
39	0.000000081	0.000000451	0.000000556	0
40	0.000000325	0.000000902	0.000002224	0.00000335
41	0.000001137	0.000001579	0.000003336	0.00000335
42	0.00000195	0.00000406	0.000006673	0.000008935
43	0.000003249	0.00000812	0.000015013	0.000032388
44	0.000007636	0.000014436	0.000032807	0.000072593
45	0.000015028	0.00002639	0.000068394	0.000139602
46	0.000031682	0.000057517	0.000155694	0.000301541
47	0.000064013	0.000118642	0.000327513	0.000631003
48	0.000127457	0.000249915	0.000638345	0.001248604
49	0.000252396	0.000490132	0.001237211	0.002359839
50	0.000474249	0.000941017	0.002367104	0.004548805
51	0.000892445	0.001764069	0.004421708	0.008599509
52	0.001666369	0.003259502	0.008269017	0.016161492
53	0.003058652	0.005983534	0.014984987	0.029469511
54	0.005462876	0.010744558	0.026771575	0.052492741
55	0.009649472	0.019030563	0.04707073	0.091134688
56	0.016790252	0.033099808	0.080601646	0.15409761
57	0.028755727	0.056629751	0.135025022	0.250653339
58	0.048532169	0.09457246	0.218862878	0.387446951
59	0.080362551	0.153884741	0.339387789	0.558717891
60	0.130182941	0.242700801	0.496720418	0.738826223
61	0.205259708	0.366890944	0.674276579	0.885896806
62	0.312057271	0.523834893	0.835948065	0.967960688
63	0.452811129	0.69656231	0.943012678	0.99507036
64	0.617748335	0.849264238	0.988655472	0.999637034
65	0.781047929	0.948686816	0.998970752	0.999987715
66	0.907279041	0.989982407	0.999964413	1
67	0.975076279	0.999134995	1	1
68	0.996623314	0.999980828	1	1
69	0.999843542	1	1	1
70	0.999997969	1	1	1
71	1	1	1	1
72	1	1	1	1
73	1	1	1	1
74	1	1	1	1
75	1	1	1	1

Ties are common in all bingo games, including coveralls. The more cards the greater the number of people will call bingo at the same time. The following table shows the expected number of winners according to the exact number of calls and cards. If the cell is empty then the situation never occured in the simulation. For example in a 200-card game if coverall is called on the 65th call then the expected number of players calling bingo will be 1.67. Very low probabilities should be taken with a grain of salt, because they may be based on as little as one occurence in the sample.

Expected Number of Players to Call Bingo
Calls	100 Cards	200 Cards	500 Cards	1000 Cards
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38	1		1
39		1
40	1	1	1	1
41	1	1	1
42	1	1	1	1
43	1	1	1	1
44	1	1	1	1
45	1	1	1	1
46	1	1	1	1
47	1	1	1.003236246	1
48	1	1	1	1.001808318
49	1	1	1	1.001005025
50	1	1	1.000492126	1.00255102
51	1.00019425	1.000548095	1.00135318	1.003584229
52	1.000734754	1.000603318	1.001878884	1.005759858
53	1.000700158	1.001490436	1.003974168	1.007888553
54	1.001115016	1.00298465	1.007312356	1.014115935
55	1.002270214	1.004273737	1.011532351	1.024768786
56	1.003856524	1.008192253	1.020397333	1.04070809
57	1.006856988	1.013803681	1.033072797	1.069480429
58	1.010840097	1.02308895	1.05640893	1.114716088
59	1.018615121	1.037111967	1.093267882	1.189865411
60	1.03024186	1.060601982	1.153491643	1.314908631
61	1.048313397	1.097973654	1.252194528	1.522245932
62	1.078082998	1.158364843	1.413967326	1.865010887
63	1.124174112	1.255501856	1.678308967	2.411675043
64	1.198830069	1.413148224	2.113724965	3.240890193
65	1.318086894	1.674488986	2.844914021	4.541401274
66	1.510948224	2.105814817	3.968662563	5.454545455
67	1.831760494	2.804647839	6.109375
68	2.371556001	3.994133333
69	3.304129563	5.682352941
70	4.933193056
71	7.56
72
73
74
75
Overall	1.2751013	1.285061464	1.30028581	1.313246594

The 100-card coverall probabilites are based on a sample size of 12,310,000 games. For 200-cards the sample size was 4,435,500. For 500-cards the sample size was 1,798,400. For 1000-cards the sample size was 895,400.

Ties are common in bingo. The more cards the greater the number of people will call bingo at the same time. The following table shows the expected number of winners according to the exact number of calls and cards. For example in a 200-card game if bingo is called on the 20th call then the expected number of players calling bingo will be 1.66. Very low probabilities should be taken with a grain of salt, because they may be based on as little as one occurence in the sample.

Expected Number of Players to Call Bingo
Calls	100 Cards	200 Cards	500 Cards	1000 Cards
4	1.0090009	1.02335721	1.061652281	1.114432367
5	1.015275708	1.029496512	1.069307914	1.121296296
6	1.022258765	1.042122799	1.083987154	1.146942645
7	1.028581682	1.048192412	1.104964568	1.190889479
8	1.033890891	1.061522127	1.132701248	1.239306635
9	1.043170534	1.077518379	1.164762676	1.302551913
10	1.052359825	1.094201366	1.207151634	1.389465628
11	1.063636058	1.116077308	1.260499384	1.502997342
12	1.076579112	1.141551275	1.324602686	1.647857033
13	1.093521954	1.174362146	1.405741511	1.836531471
14	1.113105085	1.212457155	1.508972374	2.093635644
15	1.135955427	1.255469998	1.643348814	2.449646682
16	1.161564153	1.311716739	1.802746991	2.885650437
17	1.19272741	1.377605556	2.010154312	3.418463612
18	1.230036493	1.454971001	2.284419787	3.982554701
19	1.271820227	1.549211465	2.629625046	4.328506787
20	1.322227855	1.660278243	3.078167116	4.719354839
21	1.382000573	1.804489007	3.447154472	6.772727273
22	1.449972845	1.961545871	4.026666667	3.6
23	1.52832292	2.178420391	5.153846154	2
24	1.615738147	2.376086057	4.75	0
25	1.722860792	2.726631393	0	0
26	1.855784383	2.714285714	0	0
27	2.020819564	3.461538462	0	0
28	2.170298165	4.666666667	0	0
29	2.21021021	0	0	0
30	2.569444444	0	0	0
31	2.6	0	0	0
Overall	1.201004098	1.263574841	1.401860391	1.598345388

Source: The Wizard of Odds