90 Number Bingo - Analysis

Introduction

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.

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The following table shows the probability of covering 0 to 3 rows exactly by number of balls drawn.

Probabilities in 90-Number Bingo

Calls Zero Rows One Row Two Rows Three Rows
5 0.99999993 0.00000007 0.00000000 0.00000000
6 0.99999959 0.00000041 0.00000000 0.00000000
7 0.99999857 0.00000143 0.00000000 0.00000000
8 0.99999618 0.00000382 0.00000000 0.00000000
9 0.99999140 0.00000860 0.00000000 0.00000000
10 0.99998280 0.00001720 0.00000000 0.00000000
11 0.99996846 0.00003154 0.00000000 0.00000000
12 0.99994594 0.00005406 0.00000000 0.00000000
13 0.99991215 0.00008785 0.00000000 0.00000000
14 0.99986334 0.00013666 0.00000000 0.00000000
15 0.99979502 0.00020498 0.00000000 0.00000000
16 0.99970184 0.00029815 0.00000000 0.00000000
17 0.99957761 0.00042238 0.00000001 0.00000000
18 0.99941517 0.00058481 0.00000002 0.00000000
19 0.99920632 0.00079364 0.00000005 0.00000000
20 0.99894179 0.00105812 0.00000010 0.00000000
21 0.99861115 0.00138866 0.00000018 0.00000000
22 0.99820277 0.00179689 0.00000034 0.00000000
23 0.99770370 0.00229570 0.00000060 0.00000000
24 0.99709968 0.00289929 0.00000103 0.00000000
25 0.99637503 0.00362325 0.00000171 0.00000000
26 0.99551261 0.00448461 0.00000279 0.00000000
27 0.99449375 0.00550182 0.00000442 0.00000000
28 0.99329824 0.00669488 0.00000688 0.00000000
29 0.99190422 0.00808528 0.00001050 0.00000000
30 0.99028822 0.00969603 0.00001575 0.00000000
31 0.98842504 0.01155172 0.00002324 0.00000001
32 0.98628779 0.01367841 0.00003379 0.00000001
33 0.98384784 0.01610367 0.00004847 0.00000002
34 0.98107483 0.01885649 0.00006864 0.00000004
35 0.97793665 0.02196722 0.00009606 0.00000007
36 0.97439951 0.02546744 0.00013293 0.00000012
37 0.97042791 0.02938983 0.00018206 0.00000020
38 0.96598475 0.03376802 0.00024690 0.00000034
39 0.96103137 0.03863633 0.00033175 0.00000055
40 0.95552768 0.04402955 0.00044189 0.00000088
41 0.94943224 0.04998261 0.00058377 0.00000139
42 0.94270245 0.05653021 0.00076518 0.00000215
43 0.93529473 0.06370641 0.00099556 0.00000331
44 0.92716472 0.07154411 0.00128615 0.00000502
45 0.91826755 0.08007453 0.00165039 0.00000753
46 0.90855815 0.08932650 0.00210418 0.00001117
47 0.89799157 0.09932579 0.00266623 0.00001641
48 0.88652342 0.11009427 0.00335844 0.00002387
49 0.87411026 0.12164899 0.00420635 0.00003440
50 0.86071014 0.13400121 0.00523950 0.00004915
51 0.84628315 0.14715527 0.00649196 0.00006963
52 0.83079206 0.16110738 0.00800271 0.00009786
53 0.81420297 0.17584435 0.00981620 0.00013648
54 0.79648609 0.19134220 0.01198273 0.00018898
55 0.77761658 0.20756463 0.01455894 0.00025984
56 0.75757538 0.22446152 0.01760820 0.00035491
57 0.73635018 0.24196726 0.02120090 0.00048166
58 0.71393646 0.25999913 0.02541473 0.00064968
59 0.69033853 0.27845558 0.03033472 0.00087116
60 0.66557064 0.29721460 0.03605320 0.00116155
61 0.63965818 0.31613208 0.04266942 0.00154032
62 0.61263880 0.33504034 0.05028895 0.00203191
63 0.58456365 0.35374681 0.05902266 0.00266688
64 0.55549858 0.37203294 0.06898520 0.00348328
65 0.52552523 0.38965352 0.08029298 0.00452826
66 0.49474217 0.40633638 0.09306135 0.00586010
67 0.46326585 0.42178271 0.10740092 0.00755051
68 0.43123143 0.43566818 0.12341295 0.00968745
69 0.39879339 0.44764485 0.14118334 0.01237841
70 0.36612594 0.45734441 0.16077531 0.01575434
71 0.33342294 0.46438259 0.18222022 0.01997425
72 0.30089756 0.46836541 0.20550639 0.02523064
73 0.26878130 0.46889735 0.23056555 0.03175580
74 0.23732239 0.46559188 0.25725642 0.03982931
75 0.20678340 0.45808485 0.28534510 0.04978664
76 0.17743793 0.44605116 0.31448165 0.06202926
77 0.14956616 0.42922523 0.34417227 0.07703633
78 0.12344911 0.40742607 0.37374651 0.09537832
79 0.09936129 0.38058747 0.40231862 0.11773261
80 0.07756165 0.34879432 0.42874235 0.14490167
81 0.05828228 0.31232578 0.45155806 0.17783387
82 0.04171481 0.27170652 0.46893125 0.21764743
83 0.02799390 0.22776704 0.47858117 0.26565789
84 0.01717756 0.18171454 0.47769830 0.32340960
85 0.00922370 0.13521556 0.46284907 0.39271166
86 0.00396252 0.09049229 0.42986627 0.47567891
87 0.00106401 0.05043412 0.37372319 0.57477869
88 0.00000000 0.01872659 0.28838951 0.69288390
89 0.00000000 0.00000000 0.16666667 0.83333333
90 0.00000000 0.00000000 0.00000000 1.00000000

Methodology -- Part 1

Following is how I did the math for the table above. First, let me define some variables.

• n = number of balls drawn.
• a = probability all three rows covered.
• b = probability at least two specific rows covered.
• c = probability at least one specific row covered.

Here are formulas for a, b, and c:

• a = combin(a,15)/combin(90,15)
• b = combin(a,10)/combin(90,10)
• c = combin(a,5)/combin(90,5)

Here are the formulas for exactly zero to three rows covered. For one and two rows, they can be any one or two.

• Exactly three rows covered = a.
• Exactly two rows covered = 3×(b-a).
• Exactly one row covered = 3×(c-2b+a).
• Exactly zero rows covered = 1 - (3c-3b+a).

Methodology -- Part 2

This section shows another way to get the probabilities in the table above.

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

Urgent Games

Urgent Games is a provider of games for Internet casinos, which offer 90-nuumber bingo. The player may choose between a 45, 55, or 65 number draw. The following three tables examine each option. Please excuse the limit of 15 significant digits in Excel.

45 Numbers

The following table shows the game by Urgent Games software with a 45 ball draw. The lower right cell shows an expected return of 48.36%.

45 Numbers

Rows/th> Pays Combinations Probability Return
3 100 781,879,430,625,944,000,000 0.000008 0.000753
2 50 171,356,221,250,483,000,000,000 0.001650 0.082520
1 5 8,313,931,625,579,050,000,000,000 0.080075 0.400373
0 0 95,341,351,561,293,300,000,000,000 0.918268 0.000000
Total   103,827,421,287,553,000,000,000,000 1.000000 0.483645

55 Numbers

The following table shows the game by Urgent Games software with a 55 ball draw. The lower right cell shows an expected return of 57.37%.

65 Numbers

Rows/th> Pays Combinations Probability Return
3 50 2,942,618,815,403,660,000,000 0.000260 0.012992
2 10 164,874,003,096,150,000,000,000 0.014559 0.145589
1 2 2,350,584,069,921,270,000,000,000 0.207565 0.415129
0 0 8,806,187,820,277,450,000,000,000 0.777617 0.000000
Total 0 11,324,588,512,110,300,000,000,000 1.000000 0.573711

65 Numbers

The following table shows the game by Urgent Games software with a 65 ball draw. The lower right cell shows an expected return of 64.08%.

65 Numbers

Rows/th> Pays Combinations Probability Return
3 20 52,588,547,141,148,900,000 0.004528 0.090565
2 2 932,475,551,941,065,000,000 0.080293 0.160586
1 1 4,525,207,169,948,350,000,000 0.389654 0.389654
0 0 6,103,141,366,229,710,000,000 0.525525 0.000000
Total 0 11,613,412,635,260,300,000,000 1.000000 0.640805