| .g
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 |
|
