Bingo Probabilities I

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.

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.

The Wizard's information about bingo:

Bingo Probabilities

Coast Casinos bingo

Analysis of the Jumbo Progressive at Station Casinos

Analysis of 90-Number Bingo

See important note about Bodog payouts & deposits.
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