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Last Updated: February 23, 2005

90 Number Bingo - Analysis


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 ProbabilitiesExpand

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

The Wizard's Information About Bingo