90-Number Bingo

Last update: Feburary 23, 2005

Unlike American bingo with a 5 by 5 card with numbers from 1 to 75, in Europe and South America bingo is often played with a 3 by 9 card with numbers from 1 to 90. Below is an example.

As the example shows the card contains 3 rows and 9 columns. On each row are exactly 5 numbers. The other four cells in each row are blank, or free squares. From other examples I have seen the first row contains the numbers 1 to 10, the second 11 to 20, and so on, but mathematically this doesn't matter. Winning events I have heard of all are based on covering rows only, so mathematically speaking the game could played on a 3 by 5 card with all numbers covered, the odds would be the same.

The purpose of this appendix is to show the probability of covering (1) at least one row, (2) at least 2 rows, and (3) all three rows, in any given number of calls from 5 to 90. For example, the table shows the probability of covering at least one row in 50 calls on any one card is 0.139289864, or 13.93%.

90 Number Bingo Probabilities

Calls One Row
Probability Two Rows
Probability Three Rows
Probability One Row Inverse
Probability Two Rows Inverse
Probability Three Rows Inverse
Probability

5 0.0000000683 0 0 1 in 14649756

6 0.0000004096 0 0 1 in 2441626

7 0.0000014335 0 0 1 in 697607

8 0.0000038226 0 0 1 in 261603

9 0.0000086008 0 0 1 in 116268

10 0.0000172017 0 0 1 in 58134 1 in 1906881827301

11 0.0000315364 0 0 1 in 31709 1 in 173352893391

12 0.0000540623 0 0 1 in 18497 1 in 28892148899

13 0.0000878511 0.0000000001 0 1 in 11383 1 in 6667418977

14 0.000136657 0.0000000005 0 1 in 7318 1 in 1904976850

15 0.0002049848 0.0000000016 0 1 in 4878 1 in 634992301 1 in 45795673964460800

16 0.0002981578 0.0000000042 0 1 in 3354 1 in 238122146 1 in 2862229622778800

17 0.0004223859 0.0000000102 0 1 in 2368 1 in 98050336 1 in 336732896797506

18 0.0005848332 0.0000000229 0 1 in 1710 1 in 43577969 1 in 56122149466251

19 0.0007936849 0.0000000484 0 1 in 1260 1 in 20642236 1 in 11815189361316

20 0.0010582143 0.0000000969 0 1 in 945 1 in 10321154 1 in 2953797340329

21 0.0013888484 0.000000185 0 1 in 720 1 in 5406350 1 in 843942097237

22 0.0017972335 0.0000003391 0 1 in 556 1 in 2948945 1 in 268527030939

23 0.0022962984 0.0000005999 0 1 in 435 1 in 1666818 1 in 93400706414

24 0.0029003168 0.0000010285 0 1 in 345 1 in 972330 1 in 35025264905

25 0.0036249674 0.000001714 0.0000000001 1 in 276 1 in 583414 1 in 14010105962

26 0.0044873918 0.0000027852 0.0000000002 1 in 223 1 in 359038 1 in 5927352522

27 0.0055062482 0.0000044234 0.0000000004 1 in 182 1 in 226072 1 in 2634378899

28 0.006701763 0.0000068803 0.0000000008 1 in 149 1 in 145342 1 in 1223104489

29 0.008095776 0.0000105007 0.0000000017 1 in 124 1 in 95232 1 in 590464236

30 0.0097117813 0.0000157493 0.0000000034 1 in 103 1 in 63495 1 in 295232118

31 0.0115749612 0.0000232459 0.0000000066 1 in 86 1 in 43018 1 in 152377867

32 0.0137122121 0.0000338066 0.0000000124 1 in 73 1 in 29580 1 in 80950742

33 0.0161521615 0.0000484952 0.0000000226 1 in 62 1 in 20621 1 in 44154950

34 0.0189251748 0.0000686847 0.0000000405 1 in 53 1 in 14559 1 in 24674825

35 0.0220633488 0.0000961302 0.0000000709 1 in 45 1 in 10403 1 in 14099900

36 0.0256004928 0.0001330566 0.0000001216 1 in 39 1 in 7516 1 in 8224942

37 0.0295720915 0.0001822611 0.0000002045 1 in 34 1 in 5487 1 in 4890506

38 0.0340152517 0.0002472336 0.0000003378 1 in 29 1 in 4045 1 in 2960043

39 0.0389686274 0.0003322973 0.000000549 1 in 26 1 in 3009 1 in 1821565

40 0.0444723213 0.0004427703 0.0000008784 1 in 22 1 in 2259 1 in 1138478

41 0.0505677613 0.0005851526 0.0000013851 1 in 20 1 in 1709 1 in 721962

42 0.0572975481 0.0007673395 0.0000021546 1 in 17 1 in 1303 1 in 464118

43 0.0647052697 0.0009988639 0.0000033089 1 in 15 1 in 1001 1 in 302217

44 0.0728352824 0.0012911709 0.0000050204 1 in 14 1 in 774 1 in 199188

45 0.081732452 0.0016579252 0.0000075306 1 in 12 1 in 603 1 in 132792

46 0.0914418539 0.002115356 0.0000111744 1 in 11 1 in 473 1 in 89490

47 0.1020084273 0.0026826381 0.0000164124 1 in 10 1 in 373 1 in 60930

48 0.1134765801 0.0033823132 0.0000238726 1 in 8.8 1 in 296 1 in 41889

49 0.1258897424 0.0042407513 0.0000344046 1 in 7.9 1 in 236 1 in 29066

50 0.1392898636 0.0052886518 0.0000491494 1 in 7.2 1 in 189 1 in 20346

51 0.1537168505 0.0065615844 0.0000696283 1 in 6.5 1 in 152 1 in 14362

52 0.1692079444 0.0081005673 0.000097856 1 in 5.9 1 in 123 1 in 10219

53 0.1857970345 0.0099526798 0.0001364834 1 in 5.4 1 in 100 1 in 7327

54 0.203513905 0.0121717032 0.000188977 1 in 4.9 1 in 82 1 in 5292

55 0.2223834172 0.0148187832 0.0002598433 1 in 4.5 1 in 67 1 in 3848

56 0.2424246244 0.0179631039 0.000354908 1 in 4.1 1 in 56 1 in 2818

57 0.2636498229 0.0216825599 0.0004816608 1 in 3.8 1 in 46 1 in 2076

58 0.2860635412 0.0260644094 0.000649682 1 in 3.5 1 in 38 1 in 1539

59 0.309661472 0.031205888 0.0008711645 1 in 3.2 1 in 32 1 in 1148

60 0.3344293554 0.037214755 0.0011615527 1 in 3 1 in 27 1 in 861

61 0.3603418208 0.0442097423 0.0015403199 1 in 2.8 1 in 23 1 in 649

62 0.387361203 0.0523208636 0.0020319114 1 in 2.6 1 in 19 1 in 492

63 0.4154363465 0.0616895391 0.0026668837 1 in 2.4 1 in 16 1 in 375

64 0.4445014202 0.0724684766 0.0034832766 1 in 2.2 1 in 14 1 in 287

65 0.4744747683 0.0848212433 0.0045282596 1 in 2.1 1 in 12 1 in 221

66 0.5052578274 0.0989214474 0.0058601006 1 in 2 1 in 10 1 in 171

67 0.5367341493 0.1149514356 0.0075505143 1 in 1.86 1 in 8.7 1 in 132

68 0.568768574 0.1331003983 0.0096874523 1 in 1.76 1 in 7.5 1 in 103

69 0.6012066069 0.153561752 0.0123784113 1 in 1.66 1 in 6.5 1 in 81

70 0.6338740649 0.1765296538 0.0157543416 1 in 1.58 1 in 5.7 1 in 63

71 0.6665770642 0.2021944733 0.0199742546 1 in 1.5 1 in 4.9 1 in 50

72 0.6991024401 0.2307370275 0.0252306373 1 in 1.43 1 in 4.3 1 in 40

73 0.7312186968 0.262321349 0.0317558022 1 in 1.37 1 in 3.8 1 in 31

74 0.7626776074 0.2970857299 0.0398293112 1 in 1.31 1 in 3.4 1 in 25

75 0.7932165977 0.3351317439 0.049786639 1 in 1.26 1 in 3 1 in 20

76 0.8225620687 0.3765109088 0.0620292551 1 in 1.22 1 in 2.7 1 in 16

77 0.8504338369 0.4212086067 0.077036333 1 in 1.18 1 in 2.4 1 in 13

78 0.8765508925 0.4691248258 0.095378317 1 in 1.14 1 in 2.1 1 in 10

79 0.9006387073 0.5200512338 0.1177326101 1 in 1.11 1 in 1.92 1 in 8.5

80 0.9224383526 0.5736440281 0.1449016739 1 in 1.08 1 in 1.74 1 in 6.9

81 0.941717722 0.6293919373 0.1778338726 1 in 1.06 1 in 1.59 1 in 5.6

82 0.9582851926 0.686578675 0.2176474261 1 in 1.04 1 in 1.46 1 in 4.6

83 0.9720060987 0.7442390568 0.2656578878 1 in 1.03 1 in 1.34 1 in 3.8

84 0.9828224403 0.801107902 0.3234096025 1 in 1.02 1 in 1.25 1 in 3.1

85 0.9907762969 0.8555607343 0.3927116602 1 in 1.01 1 in 1.17 1 in 2.5

86 0.9960374767 0.9055451845 0.4756789123 1 in 1.004 1 in 1.1 1 in 2.1

87 0.9989359891 0.9485018727 0.5747786857 1 in 1.001 1 in 1.05 1 in 1.74

88 1 0.9812734082 0.6928838951 1 in 1 1 in 1.02 1 in 1.44

89 1 1 0.8333333333 1 in 1 1 in 1 1 in 1.2

90 1 1 1 1 in 1 1 in 1 1 in 1

Methodology: The probability of covering m marks in c calls is combin(15,m)*combin(75,c-m)/combin(90,m). Using that you can find the probability of covering a card as combin(75,90-m)/combin(90,m). To get the probability of covering 1 or 2 rows I determined the probability that m marks would cover 1 or 2 rows. The chart below shows those probabilities, which is based on basic probability.

Rows Covered by Number of Marks

Marks 0 Rows 1 Row 2 Rows 3 Rows Total

5 0.999001 0.000999 0 0 1

6 0.994006 0.005994 0 0 1

7 0.979021 0.020979 0 0 1

8 0.944056 0.055944 0 0 1

9 0.874126 0.125874 0 0 1

10 0.749251 0.24975 0.000999 0 1

11 0.549451 0.43956 0.010989 0 1

12 0.274725 0.659341 0.065934 0 1

13 0 0.714286 0.285714 0 1

14 0 0 1 0 1

15 0 0 0 1 1

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