Bingo Probabilities

Last Update: Sep 29, 2007

Every table in this document is based on American bingo, which is based on a 24-number card (plus a free square) and 75 balls.

Average Balls Drawn

The following table shows the average number of balls drawn by game type and number of cards:

Average Number of Balls Drawn
Game	Cards
	2000	4000	6000	8000	10000
Single Bingo	8.62	8.05	7.82	7.71	7.56
Double Bingo	19.32	18.04	17.22	16.79	16.53
Triple Bingo	27.13	25.77	25.03	24.49	24.08
Single Hardway	11.41	10.33	9.79	9.49	9.36
Double Hardway	24.56	23.07	22.25	21.76	21.28
Triple Hardway	33.44	31.95	31.09	30.64	30.02
Six Pack	9.51	8.9	8.55	8.37	8.26
Nine Pack	21.79	20.27	19.6	18.95	18.65
Coverall	57.57	56.38	55.56	55.08	54.79

Jackpot Sharing

Ties are common in all bingo games, including coveralls. The greater the number of cards, and the easier the pattern is to cover, the more ties you will see. The following table shows the averge number of people that will call bingo accoring to the pattern and number of cards. HW stands for Hard Way, meaning the player can not make use of the free square.

Expected Number of Players to Call Bingo
Game	Cards
	2000	4000	6000	8000	10000
Single Bingo	2.62	4.11	5.72	7.11	8.2
Double Bingo	1.3	1.34	1.37	1.39	1.42
Triple Bingo	1.27	1.31	1.33	1.34	1.33
Single HW Bingo	1.49	1.78	2.01	2.32	2.6
Double HW Bingo	1.27	1.3	1.33	1.35	1.4
Triple HW Bingo	1.26	1.27	1.29	1.31	1.31
Six Pack	1.96	2.54	3.08	3.68	4.21
Nine Pack	1.35	1.43	1.47	1.53	1.55
Coverall	1.32	1.34	1.34	1.35	1.38

A major frustration in bingo is having to share a jackpot. In my opinion, many players would pay a premium to receive a jackpot in full, regardless of the number of other players that bingo at the same time. The table above could be used to base a fair premium for such jackpot-sharing insurance. For example, in a coverall game with 10,000 cards, the expected number of winners is 1.38. A fair premium for jackpot sharing insurance would be 38% of the price per card.

I have a patent pending on this concept of jackpot sharing insurance. I welcome any bingo parlor to try out this concept. Please contact me with expressions of interest.

Coverall Probabilities for Single Card

The next table shows the coverall probabilities for a single card. The Density column is the probability of achieving a coverall in exactly the given many balls drawn. The Distribution column is the probability of achieving a coverall in the given number of balls drawn drawn or less. For example, the probbility of getting a coverall in exactly 60 balls is 0.000559. The probability of getting a coverall in 60 balls or less is 0.001399. By the way, the expected number of calls to get a coverall on one card is 72.96.

Coverall Probabilities for Single Card
Balls	Density	Distribution
24	0.00000000000000000004	0.00000000000000000004
25	0.00000000000000000093	0.00000000000000000097
26	0.00000000000000001164	0.00000000000000001261
27	0.00000000000000010086	0.00000000000000011347
28	0.00000000000000068079	0.00000000000000079426
29	0.00000000000000381245	0.00000000000000460671
30	0.00000000000001842684	0.00000000000002303355
31	0.00000000000007897218	0.00000000000010200573
32	0.00000000000030601718	0.00000000000040802291
33	0.00000000000108806109	0.00000000000149608400
34	0.00000000000359060160	0.00000000000508668560
35	0.00000000001109822313	0.00000000001618490874
36	0.00000000003236981747	0.00000000004855472621
37	0.00000000008963949453	0.00000000013819422074
38	0.00000000023690437841	0.00000000037509859915
39	0.00000000060015775864	0.00000000097525635779
40	0.00000000146288453669	0.00000000243814089448
41	0.00000000344208126279	0.00000000588022215727
42	0.00000000784029620969	0.00000001372051836696
43	0.00000001733118109511	0.00000003105169946207
44	0.00000003726203935449	0.00000006831373881656
45	0.00000007807284436178	0.00000014638658317834
46	0.00000015969445437637	0.00000030608103755472
47	0.00000031938890875275	0.00000062546994630747
48	0.00000062546994630747	0.00000125093989261493
49	0.00000120090229691033	0.00000245184218952526
50	0.00000226323894417717	0.00000471508113370243
51	0.00000419118322995771	0.00000890626436366014
52	0.00000763394088313726	0.00001654020524679740
53	0.00001368844572148750	0.00003022865096828490
54	0.00002418292077462790	0.00005441157174291290
55	0.00004212508780096470	0.00009653665954387760
56	0.00007240249465790830	0.00016893915420178600
57	0.00012286483941948100	0.00029180399362126600
58	0.00020597928961501200	0.00049778328323627800
59	0.00034133710850487700	0.00083912039174115500
60	0.00055941359449410300	0.00139853398623526000
61	0.00090715718026070800	0.00230569116649597000
62	0.00145622599989219000	0.00376191716638816000
63	0.00231502594854655000	0.00607694311493471000
64	0.00364616586896083000	0.00972310898389554000
65	0.00569157599057300000	0.01541468497446850000
66	0.00880839141398202000	0.02422307638845060000
67	0.01351985658890260000	0.03774293297735320000
68	0.02058705435128360000	0.05832998732863680000
69	0.03110932657527300000	0.08943931390390970000
70	0.04666398986290940000	0.13610330376681900000
71	0.06949955937029060000	0.20560286313711000000
72	0.10280143156855500000	0.30840429470566500000
73	0.15105516475379500000	0.45945945945945900000
74	0.22054054054054100000	0.68000000000000000000
75	0.32000000000000000000	1.00000000000000000000

The next table shows the probability that a coverall will be hit in exactly the given number of balls and number of cards in play. For example, the probability that with 6000 cards a coverall will be hit in exactly 50 balls is 0.012944. The last row shows the number of sessions in the sample size.

Average Number of Balls Drawn for Coverall
Game	Cards
	2000	4000	6000	8000	10000
40 or Less	0	0	0	0	0
41	0	0.00004	0	0.00009	0
42	0.00004	0.00004	0.000063	0	0.000112
43	0	0.00004	0	0.00018	0.000112
44	0.00004	0.00028	0.000127	0.00027	0.000448
45	0.00012	0.00048	0.000508	0.00054	0.00056
46	0.000241	0.00048	0.000952	0.000989	0.001121
47	0.000482	0.001039	0.002284	0.003238	0.002914
48	0.001084	0.002118	0.003617	0.004047	0.005155
49	0.002571	0.004077	0.006409	0.010073	0.012104
50	0.004338	0.008593	0.012944	0.017178	0.020733
51	0.008274	0.015508	0.022525	0.0286	0.035974
52	0.014018	0.028338	0.043464	0.053422	0.065785
53	0.026148	0.049043	0.071447	0.087418	0.101984
54	0.042355	0.081418	0.113135	0.135264	0.151294
55	0.073263	0.124625	0.153934	0.179243	0.19489
56	0.10865	0.167073	0.187056	0.194622	0.194329
57	0.152692	0.190495	0.186485	0.161435	0.132691
58	0.180025	0.168832	0.124492	0.089756	0.06119
59	0.179945	0.108318	0.056091	0.02797	0.016026
60	0.128853	0.03969	0.012437	0.005216	0.002466
61	0.059245	0.008194	0.002094	0.00054	0.000336
62	0.015344	0.001319	0	0.00009	0
63	0.002229	0.00008	0	0.00009	0
64	0.00008	0	0	0	0
65 or More	0	0	0	0	0
Total	1	1	1	1	1
Average	57.57741	56.316	55.594289	55.12672	54.768912
Sample Size	49793	25019	15760	11119	8923

For more bingo probability tables, please see my Probabilities in Bingo II page.

Another good source on bingo probabilities is Durango Bill's Bingo Probabilities . He has the same probabilities I do but goes into more depth on how they were calculated.

More Bingo Information:

• Bingo Probabilities
• Coast Casinos Bingo
• Analysis of the Jumbo Progressive at Station Casinos
• Analysis of 90-Number Bingo