Last Updated: August 9, 2019

# Bingo Probabilities

## Introduction

This page is a follow-up to my main probabilities in bingo page.

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.

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

## Probability Density for One-Way Patterns

The next three tables show the probability of covering "one-way" patterns of 4 to 24 marks according to the exact number of calls. This table is only appropriate if there is only one way to make the pattern. For example the probability of covering the postage stamp pattern in exactly 50 calls is 1.52%, where the pattern is defined as covering the four numbers in the upper right corner of the card. This table is not appropriate, for example, if the player may cover the four numbers in any corner.

### Probability of Covering 4 to 10 Mark Patterns by Number of Calls ExactlyExpand

Calls 4 Marks 5 Marks 6 Marks 7 Marks 8 Marks 9 Marks 10 Marks
4 0.000000823 0 0 0 0 0 0
5 0.000003291 0.000000058 0 0 0 0 0
6 0.000008227 0.00000029 0.000000005 0 0 0 0
7 0.000016455 0.000000869 0.00000003 0.000000001 0 0 0
8 0.000028796 0.000002028 0.000000104 0.000000004 0 0 0
9 0.000046073 0.000004056 0.000000278 0.000000014 0 0 0
10 0.00006911 0.0000073 0.000000626 0.000000042 0.000000002 0 0
11 0.000098729 0.000012167 0.000001251 0.000000106 0.000000007 0 0
12 0.000135752 0.00001912 0.000002294 0.000000233 0.00000002 0.000000001 0
13 0.000181003 0.00002868 0.000003933 0.000000466 0.000000047 0.000000004 0
14 0.000235304 0.000041427 0.000006392 0.000000865 0.000000102 0.00000001 0.000000001
15 0.000299478 0.000057997 0.000009942 0.000001513 0.000000203 0.000000024 0.000000002
16 0.000374347 0.000079087 0.000014914 0.000002522 0.000000381 0.000000051 0.000000006
17 0.000460735 0.00010545 0.000021693 0.000004035 0.000000678 0.000000102 0.000000014
18 0.000559464 0.000137896 0.000030731 0.000006235 0.000001153 0.000000194 0.000000029
19 0.000671356 0.000177295 0.000042551 0.000009353 0.000001886 0.000000348 0.000000059
20 0.000797236 0.000224573 0.000057747 0.00001367 0.000002987 0.000000602 0.000000111
21 0.000937924 0.000280717 0.000076997 0.000019528 0.000004595 0.000001003 0.000000203
22 0.001094245 0.000346768 0.000101058 0.000027339 0.000006892 0.00000162 0.000000355
23 0.00126702 0.000423827 0.000130781 0.000037592 0.000010109 0.000002546 0.0000006
24 0.001457074 0.000513054 0.000167109 0.000050859 0.000014531 0.000003904 0.000000986
25 0.001665227 0.000615665 0.000211085 0.000067812 0.000020515 0.000005856 0.000001577
26 0.001892303 0.000732934 0.000263856 0.000089227 0.000028493 0.000008612 0.000002465
27 0.002139125 0.000866195 0.000326679 0.000115995 0.00003899 0.000012439 0.000003769
28 0.002406516 0.001016838 0.000400925 0.000149136 0.000052636 0.000017676 0.000005654
29 0.002695298 0.001186311 0.000488082 0.00018981 0.000070182 0.000024747 0.000008332
30 0.003006294 0.00137612 0.000589766 0.000239325 0.000092512 0.000034174 0.000012082
31 0.003340327 0.001587831 0.000707719 0.000299157 0.000120668 0.000046601 0.00001726
32 0.003698219 0.001823066 0.000843819 0.000370954 0.000155863 0.000062811 0.000024321
33 0.004080793 0.002083504 0.001000082 0.000456559 0.000199505 0.000083747 0.000033837
34 0.004488872 0.002370883 0.001178668 0.000558017 0.000253218 0.000110546 0.000046526
35 0.004923279 0.002687001 0.001381886 0.000677592 0.000318867 0.000144561 0.000063276
36 0.005384837 0.003033711 0.001612201 0.000817783 0.000398583 0.000187394 0.000085179
37 0.005874368 0.003412925 0.001872233 0.00098134 0.000494793 0.000240935 0.000113572
38 0.006392694 0.003826613 0.002164769 0.001171276 0.000610245 0.000307399 0.000150077
39 0.006940639 0.004276802 0.002492765 0.001390891 0.000748042 0.000389373 0.000196653
40 0.007519026 0.00476558 0.002859348 0.00164378 0.000911676 0.000489856 0.000255649
41 0.008128677 0.005295089 0.003267826 0.001933858 0.001105062 0.00061232 0.000329869
42 0.008770414 0.005867531 0.003721691 0.002265377 0.001332575 0.000760761 0.000422645
43 0.009445061 0.006485165 0.004224622 0.00264294 0.00159909 0.000939764 0.000537912
44 0.010153441 0.007150311 0.004780493 0.003071525 0.001910024 0.001154567 0.0006803
45 0.010896376 0.007865342 0.005393377 0.003556502 0.00227138 0.001411137 0.000855235
46 0.011674688 0.008632692 0.006067549 0.004103657 0.002689792 0.001716248 0.001069043
47 0.012489202 0.009454853 0.006807494 0.004719205 0.003172575 0.002077563 0.001329081
48 0.013340738 0.010334375 0.00761791 0.00540982 0.003727775 0.00250373 0.001643863
49 0.014230121 0.011273863 0.008503714 0.006182652 0.004364225 0.003004476 0.002023216
50 0.015158172 0.012275984 0.009470045 0.007045347 0.005091596 0.003590715 0.00247844
51 0.016125715 0.013343461 0.010522272 0.008006077 0.00592046 0.004274661 0.003022487
52 0.017133572 0.014479075 0.011665997 0.009073554 0.006862351 0.005069946 0.003670163
53 0.018182566 0.015685664 0.012907061 0.010257061 0.007929828 0.005991755 0.004438337
54 0.01927352 0.016966127 0.014251547 0.011566473 0.009136541 0.007056955 0.005346178
55 0.020407257 0.018323417 0.015705786 0.013012282 0.010497303 0.008284252 0.006415414
56 0.021584598 0.019760548 0.017276365 0.014605622 0.012028159 0.009694338 0.007670604
57 0.022806368 0.02128059 0.018970126 0.016358297 0.013746468 0.01131006 0.009139443
58 0.024073388 0.022886672 0.020794176 0.018282802 0.015670974 0.013156601 0.010853088
59 0.025386482 0.024581981 0.022755891 0.020392357 0.017821891 0.015261657 0.012846513
60 0.026746472 0.026369762 0.024862918 0.022700925 0.020220992 0.017655642 0.015158885
61 0.028154182 0.028253316 0.027123183 0.02522325 0.022891689 0.020371895 0.017833982
62 0.029610432 0.030236005 0.029544896 0.027974878 0.025859131 0.023446898 0.020920633
63 0.031116048 0.032321247 0.032136554 0.030972186 0.029150293 0.026920513 0.024473193
64 0.03267185 0.034512518 0.034906946 0.034232416 0.032794079 0.030836224 0.028552059
65 0.034278662 0.036813352 0.037865162 0.0377737 0.036821422 0.035241398 0.033224214
66 0.035937307 0.039227342 0.041020592 0.041615094 0.041265387 0.04018756 0.03856382
67 0.037648608 0.041758139 0.044382936 0.045776603 0.04616128 0.045730671 0.044652844
68 0.039413386 0.044409449 0.047962205 0.05027922 0.051546763 0.05193144 0.051581734
69 0.041232465 0.04718504 0.051768729 0.055144951 0.057461965 0.058855632 0.059450134
70 0.043106668 0.050088734 0.055813161 0.060396851 0.063949607 0.066574404 0.068367654
71 0.045036818 0.053124415 0.060106481 0.066059055 0.071055118 0.07516465 0.078454685
72 0.047023736 0.056296022 0.064660002 0.072156814 0.078826772 0.084709367 0.089843268
73 0.049068246 0.059607553 0.069485376 0.078716525 0.087315809 0.095298038 0.10267802
74 0.051171171 0.063063063 0.074594595 0.085765766 0.096576577 0.107027027 0.117117117
75 0.053333333 0.066666667 0.08 0.093333333 0.106666667 0.12 0.133333333
Total 1 1 1 1 1 1 1

### Probability of Covering 11 to 17 Mark Patterns by Number of Calls ExactlyExpand

Calls 11 Marks 12 Marks 13 Marks 14 Marks 15 Marks 16 Marks 17 Marks
11 0 0 0 0 0 0 0
12 0 0 0 0 0 0 0
13 0 0 0 0 0 0 0
14 0 0 0 0 0 0 0
15 0 0 0 0 0 0 0
16 0.000000001 0 0 0 0 0 0
17 0.000000002 0 0 0 0 0 0
18 0.000000004 0 0 0 0 0 0
19 0.000000009 0.000000001 0 0 0 0 0
20 0.000000019 0.000000003 0 0 0 0 0
21 0.000000038 0.000000006 0.000000001 0 0 0 0
22 0.000000072 0.000000014 0.000000002 0 0 0 0
23 0.000000132 0.000000027 0.000000005 0.000000001 0 0 0
24 0.000000234 0.000000052 0.000000011 0.000000002 0 0 0
25 0.0000004 0.000000096 0.000000021 0.000000004 0.000000001 0 0
26 0.000000667 0.000000171 0.000000041 0.000000009 0.000000002 0 0
27 0.000001084 0.000000296 0.000000076 0.000000019 0.000000004 0.000000001 0
28 0.000001722 0.000000499 0.000000137 0.000000036 0.000000009 0.000000002 0
29 0.000002679 0.000000822 0.00000024 0.000000067 0.000000018 0.000000004 0.000000001
30 0.000004089 0.000001324 0.00000041 0.000000121 0.000000034 0.000000009 0.000000002
31 0.000006134 0.000002091 0.000000683 0.000000214 0.000000064 0.000000018 0.000000005
32 0.000009055 0.000003241 0.000001115 0.000000368 0.000000116 0.000000035 0.00000001
33 0.000013171 0.000004939 0.000001784 0.00000062 0.000000207 0.000000066 0.00000002
34 0.000018897 0.000007408 0.000002803 0.000001022 0.000000359 0.000000121 0.000000039
35 0.000026771 0.000010952 0.000004331 0.000001655 0.000000611 0.000000217 0.000000074
36 0.000037479 0.000015971 0.000006591 0.000002633 0.000001018 0.00000038 0.000000137
37 0.000051894 0.000022998 0.000009887 0.000004122 0.000001665 0.000000651 0.000000246
38 0.000071113 0.000032728 0.000014633 0.000006354 0.000002679 0.000001095 0.000000434
39 0.000096511 0.000046062 0.000021386 0.000009658 0.000004241 0.000001809 0.000000749
40 0.000129791 0.000064158 0.000030891 0.000014487 0.000006616 0.00000294 0.000001271
41 0.000173054 0.000088494 0.00004413 0.000021463 0.000010178 0.000004705 0.000002118
42 0.000228879 0.000120941 0.00006239 0.000031427 0.000015456 0.000007419 0.000003474
43 0.000300403 0.000163856 0.000087347 0.000045516 0.000023184 0.000011541 0.000005611
44 0.000391434 0.000220182 0.000121158 0.000065239 0.000034377 0.000017723 0.000008937
45 0.000506562 0.000293576 0.000166593 0.000092597 0.000050419 0.00002689 0.000014043
46 0.000651294 0.000388556 0.000227172 0.000130215 0.000073189 0.000040335 0.000021791
47 0.000832209 0.000510674 0.00030735 0.000181512 0.000105209 0.000059852 0.000033413
48 0.00105713 0.000666713 0.000412727 0.000250913 0.000149843 0.000087908 0.000050659
49 0.001335323 0.000864925 0.000550303 0.00034411 0.000211543 0.000127866 0.000075988
50 0.001677713 0.001115298 0.000728779 0.000468372 0.00029616 0.000184277 0.000112831
51 0.002097141 0.001429869 0.00095892 0.000632935 0.000411334 0.000263253 0.000165928
52 0.002608639 0.001823083 0.001253972 0.000849465 0.000566973 0.000372942 0.00024178
53 0.003229744 0.002312203 0.001630164 0.00113262 0.000775858 0.000524135 0.000349238
54 0.003980847 0.00291778 0.002107285 0.001500722 0.001054371 0.000731031 0.00050026
55 0.004885585 0.003664188 0.002709367 0.00197656 0.001423401 0.001012196 0.000710896
56 0.00597127 0.004580236 0.003465469 0.002588353 0.001909441 0.00139177 0.001002546
57 0.007269372 0.005699849 0.004410597 0.003370878 0.002545921 0.001900954 0.001403565
58 0.008816047 0.007062856 0.005586756 0.00436682 0.003374825 0.002579867 0.001951297
59 0.010652724 0.008715865 0.007044171 0.005628345 0.004448633 0.00347982 0.002694649
60 0.012826749 0.01071325 0.008842683 0.007218964 0.005832653 0.004666122 0.003697309
61 0.015392099 0.013118266 0.011053354 0.009215699 0.007607808 0.006221496 0.005041785
62 0.018410157 0.016004284 0.013760297 0.011711618 0.009873964 0.008250245 0.006834419
63 0.021950572 0.019456189 0.017062769 0.014818782 0.01275387 0.010883302 0.009211608
64 0.026092189 0.023571921 0.021077538 0.018671665 0.016397832 0.014284334 0.012347475
65 0.030924076 0.028464206 0.025941585 0.023431109 0.020989225 0.018657089 0.0164633
66 0.036546635 0.034262471 0.031815151 0.029288886 0.026750973 0.024254216 0.021839072
67 0.04307282 0.041114965 0.038885185 0.036472953 0.033953159 0.031387809 0.028827574
68 0.050629456 0.049191119 0.047369225 0.045253478 0.042921917 0.040441984 0.037871519
69 0.059358672 0.058684142 0.057519774 0.055949755 0.054049822 0.051887829 0.049524294
70 0.069419464 0.069813893 0.0696292 0.068938091 0.067807958 0.066301115 0.064475025
71 0.080989374 0.082830042 0.084035241 0.084660814 0.084759948 0.084383237 0.083578736
72 0.094266321 0.09801555 0.101127155 0.103636513 0.105578181 0.10698589 0.10789255
73 0.109470566 0.115690485 0.121352585 0.126471677 0.131062569 0.135140072 0.138718993
74 0.126846847 0.136216216 0.145225225 0.153873874 0.162162162 0.17009009 0.177657658
75 0.146666667 0.16 0.173333333 0.186666667 0.2 0.213333333 0.226666667
Total 1 1 1 1 1 1 1

### Probability of Covering 18 to 24 Mark Patterns by Number of Calls ExactlyExpand

Calls 18 Marks 19 Marks 20 Marks 21 Marks 22 Marks 23 Marks 24 Marks
18 0 0 0 0 0 0 0
19 0 0 0 0 0 0 0
20 0 0 0 0 0 0 0
21 0 0 0 0 0 0 0
22 0 0 0 0 0 0 0
23 0 0 0 0 0 0 0
24 0 0 0 0 0 0 0
25 0 0 0 0 0 0 0
26 0 0 0 0 0 0 0
27 0 0 0 0 0 0 0
28 0 0 0 0 0 0 0
29 0 0 0 0 0 0 0
30 0.000000001 0 0 0 0 0 0
31 0.000000001 0 0 0 0 0 0
32 0.000000003 0.000000001 0 0 0 0 0
33 0.000000006 0.000000002 0 0 0 0 0
34 0.000000012 0.000000004 0.000000001 0 0 0 0
35 0.000000024 0.000000008 0.000000002 0.000000001 0 0 0
36 0.000000047 0.000000016 0.000000005 0.000000002 0 0 0
37 0.00000009 0.000000032 0.000000011 0.000000003 0.000000001 0 0
38 0.000000166 0.000000062 0.000000022 0.000000008 0.000000002 0.000000001 0
39 0.000000301 0.000000117 0.000000044 0.000000016 0.000000006 0.000000002 0.000000001
40 0.000000534 0.000000217 0.000000086 0.000000033 0.000000012 0.000000004 0.000000001
41 0.000000928 0.000000395 0.000000163 0.000000066 0.000000025 0.00000001 0.000000003
42 0.000001585 0.000000705 0.000000305 0.000000128 0.000000052 0.000000021 0.000000008
43 0.000002663 0.000001233 0.000000556 0.000000244 0.000000104 0.000000043 0.000000017
44 0.000004405 0.000002121 0.000000997 0.000000457 0.000000204 0.000000088 0.000000037
45 0.000007178 0.000003589 0.000001754 0.000000837 0.00000039 0.000000177 0.000000078
46 0.000011537 0.000005982 0.000003036 0.000001507 0.000000731 0.000000346 0.00000016
47 0.000018299 0.000009827 0.000005172 0.000002666 0.000001345 0.000000663 0.000000319
48 0.000028669 0.000015927 0.000008682 0.000004641 0.000002431 0.000001247 0.000000625
49 0.000044391 0.000025484 0.00001437 0.000007956 0.000004322 0.000002302 0.000001201
50 0.000067973 0.00004028 0.000023472 0.000013443 0.000007563 0.000004177 0.000002263
51 0.00010299 0.000062938 0.000037858 0.000022405 0.00001304 0.000007459 0.000004191
52 0.000154484 0.000097268 0.000060335 0.000036859 0.000022168 0.000013118 0.000007634
53 0.00022952 0.000148763 0.000095074 0.000059897 0.000037184 0.000022738 0.000013688
54 0.000337904 0.000225269 0.000148204 0.000096198 0.000061587 0.000038875 0.000024183
55 0.000493157 0.000337904 0.000228657 0.000152784 0.000100778 0.000065601 0.000042125
56 0.00071378 0.00050229 0.000349337 0.00024009 0.000163024 0.000109335 0.000072402
57 0.001024915 0.000740217 0.000528726 0.000373473 0.000260838 0.000180081 0.000122865
58 0.001460505 0.001081855 0.000793089 0.00057535 0.000412994 0.000293275 0.000205979
59 0.00206608 0.00156869 0.001179466 0.000878166 0.000647396 0.000472499 0.000341337
60 0.00290235 0.002257383 0.001739713 0.001328508 0.001005167 0.000753444 0.000559414
61 0.00404979 0.003224833 0.002545921 0.001992762 0.001546411 0.001189649 0.000907157
62 0.005614482 0.004574763 0.003697647 0.002964841 0.002358277 0.001860733 0.001456226
63 0.007735509 0.006446257 0.005331491 0.004376669 0.003566175 0.002884136 0.002315026
64 0.010594284 0.00902476 0.007633726 0.00641233 0.005349263 0.004431722 0.003646166
65 0.014426259 0.012556188 0.010856855 0.009327025 0.007961693 0.0067531 0.005691576
66 0.019535559 0.017364941 0.015341207 0.013472369 0.011761592 0.010208175 0.008808391
67 0.026313202 0.023876794 0.021542972 0.019329921 0.017250336 0.015312262 0.013519857
68 0.03525969 0.032647861 0.030070399 0.02755542 0.025125489 0.022798256 0.020587054
69 0.047012921 0.044401092 0.041730349 0.039036845 0.036351771 0.03370177 0.031109327
70 0.062382529 0.060072065 0.057587882 0.054970251 0.052255671 0.049477067 0.04666399
71 0.08239202 0.080866242 0.079042191 0.076958351 0.074650958 0.072154056 0.069499559
72 0.108330248 0.108330248 0.107922991 0.107138097 0.106004361 0.104549755 0.102801432
73 0.141814143 0.144440331 0.146612366 0.148345057 0.149653215 0.150551648 0.151055165
74 0.184864865 0.191711712 0.198198198 0.204324324 0.21009009 0.215495495 0.220540541
75 0.24 0.253333333 0.266666667 0.28 0.293333333 0.306666667 0.32
Total 1 1 1 1 1 1 1

## Mean Number of Calls to Cover Pattern

The next table shows the mean number of calls to cover a pattern of 1 to 24 marks. This table is only appropriate if there is only one way to cover the pattern.

Marks Expected
Calls
1 38
2 50.666667
3 57
4 60.8
5 63.333333
6 65.142857
7 66.5
8 67.555556
9 68.4
10 69.090909
11 69.666667
12 70.153846
13 70.571429
14 70.933333
15 71.25
16 71.529412
17 71.777778
18 72
19 72.2
20 72.380952
21 72.545455
22 72.695652
23 72.833333
24 72.96

## Multi-Player Bingo

The next three tables concern multi-player bingo. It is not accurate to say that if the probability of a single player achieving a bingo in x calls is p that the probability of at least one player out of n will do so is 1-(1-p)n. This is because the probability of winning between cards are correlated, because every hard must have five numbers in the range of 1 to 15, 16 to 30, 46 to 60, and 61 to 75, as well as four in the range from 31 to 45. Unlike the tables above, which were calculated using exact probabilities, the multi-player tables were determined by random simulation.

The next table shows the probability that a bingo will be called in exactly 4 to 31 calls by the number of cards in play. For example in a 200-card game the probability of the first bingo in exactly 15 calls is 11.77%. This table is based on a random simulation. Very low probabilities should be taken with a grain of salt, because they may be based on as little as one occurence in the sample.

### Probability of Bingo by Number of Calls Exactly

Calls 100 Cards 200 Cards 500 Cards 1000 Cards
4 0.000333167 0.000639132 0.001545351 0.002983166
5 0.001341463 0.002625 0.006396723 0.011673257
6 0.003404038 0.006691083 0.015624365 0.028503103
7 0.006963215 0.013373607 0.03056466 0.054277042
8 0.012340564 0.023433519 0.051918191 0.086568549
9 0.019871351 0.037010947 0.076838161 0.120499356
10 0.029717013 0.053648288 0.103894182 0.145935527
11 0.041651539 0.072544586 0.126298908 0.15385821
12 0.055417233 0.091149084 0.13820249 0.140720391
13 0.069777089 0.107159236 0.13611471 0.110260937
14 0.08362415 0.116721736 0.117300559 0.072856976
15 0.095122551 0.11774383 0.087937627 0.040533943
16 0.102117953 0.108574045 0.056048018 0.018943822
17 0.103352359 0.090687301 0.030212144 0.00801996
18 0.097540284 0.067779658 0.013738567 0.003046216
19 0.085478209 0.044590565 0.005132749 0.000995289
20 0.069016393 0.025538416 0.001649517 0.000279221
21 0.050929028 0.012566083 0.00046875 0.000039631
22 0.033866054 0.005165804 0.000095274 0.000004504
23 0.02017523 0.001741441 0.000016514 0.000000901
24 0.010526889 0.000481091 0.000002541 0
25 0.00477439 0.000112858 0 0
26 0.001839564 0.000019506 0 0
27 0.000604958 0.000002588 0 0
28 0.00017433 0.000000597 0 0
29 0.000033287 0 0 0
30 0.000007197 0 0 0
31 0.0000005 0 0 0
Total 1 1 1 1

The next table shows the probability that a bingo will be called in 4 to 31 calls or less by the number of cards in play. For example in a 200-card game the probability of the first bingo in 15 calls or less is 64.27%. Very low probabilities should be taken with a grain of salt, because they may be based on as little as one occurence in the sample.

### Probability of Bingo by Number of Calls or Less

Calls 100 Cards 200 Cards 500 Cards 1000 Cards
4 0.000333167 0.000639132 0.001545351 0.002983166
5 0.00167463 0.003264132 0.007942073 0.014656423
6 0.005078669 0.009955215 0.023566438 0.043159526
7 0.012041883 0.023328822 0.054131098 0.097436567
8 0.024382447 0.046762341 0.106049289 0.184005116
9 0.044253798 0.083773288 0.182887449 0.304504472
10 0.073970812 0.137421576 0.286781631 0.450439999
11 0.115622351 0.209966162 0.413080539 0.604298208
12 0.171039584 0.301115247 0.551283028 0.7450186
13 0.240816673 0.408274482 0.687397739 0.855279537
14 0.324440824 0.524996218 0.804698298 0.928136512
15 0.419563375 0.642740048 0.892635925 0.968670456
16 0.521681327 0.751314092 0.948683943 0.987614278
17 0.625033687 0.842001393 0.978896087 0.995634238
18 0.72257397 0.909781051 0.992634654 0.998680454
19 0.808052179 0.954371616 0.997767403 0.999675743
20 0.877068573 0.979910032 0.999416921 0.999954964
21 0.927997601 0.992476115 0.999885671 0.999994596
22 0.961863655 0.997641919 0.999980945 0.999999099
23 0.982038884 0.99938336 0.999997459 1
24 0.992565774 0.999864451 1 1
25 0.997340164 0.999977309 1 1
26 0.999179728 0.999996815 1 1
27 0.999784686 0.999999403 1 1
28 0.999959016 1 1 1
29 0.999992303 1 1 1
30 0.9999995 1 1 1
31 1 1 1 1

Ties are common in bingo. The more cards the greater the number of people will call bingo at the same time. The following table shows the expected number of winners according to the exact number of calls and cards. For example in a 200-card game if bingo is called on the 20th call then the expected number of players calling bingo will be 1.66. Very low probabilities should be taken with a grain of salt, because they may be based on as little as one occurence in the sample.

### Expected Number of Players to Call Bingo

Calls 100 Cards 200 Cards 500 Cards 1000 Cards
4 1.0090009 1.02335721 1.061652281 1.114432367
5 1.015275708 1.029496512 1.069307914 1.121296296
6 1.022258765 1.042122799 1.083987154 1.146942645
7 1.028581682 1.048192412 1.104964568 1.190889479
8 1.033890891 1.061522127 1.132701248 1.239306635
9 1.043170534 1.077518379 1.164762676 1.302551913
10 1.052359825 1.094201366 1.207151634 1.389465628
11 1.063636058 1.116077308 1.260499384 1.502997342
12 1.076579112 1.141551275 1.324602686 1.647857033
13 1.093521954 1.174362146 1.405741511 1.836531471
14 1.113105085 1.212457155 1.508972374 2.093635644
15 1.135955427 1.255469998 1.643348814 2.449646682
16 1.161564153 1.311716739 1.802746991 2.885650437
17 1.19272741 1.377605556 2.010154312 3.418463612
18 1.230036493 1.454971001 2.284419787 3.982554701
19 1.271820227 1.549211465 2.629625046 4.328506787
20 1.322227855 1.660278243 3.078167116 4.719354839
21 1.382000573 1.804489007 3.447154472 6.772727273
22 1.449972845 1.961545871 4.026666667 3.6
23 1.52832292 2.178420391 5.153846154 2
24 1.615738147 2.376086057 4.75 0
25 1.722860792 2.726631393 0 0
26 1.855784383 2.714285714 0 0
27 2.020819564 3.461538462 0 0
28 2.170298165 4.666666667 0 0
29 2.21021021 0 0 0
30 2.569444444 0 0 0
31 2.6 0 0 0
Overall 1.201004098 1.263574841 1.401860391 1.598345388

The 100-card bingo probabilites are based on a sample size of 10,004,000 games. For 200-cards the sample size was 5,024,000. For 500-cards the sample size was 5,574,400.For 1000-cards the sample size was 1,110,230.

## Multi-Player Coverall

The next three tables concern a coverall game (covering the entire card) with 100, 200, 500, and 1000 players.

The next table shows the probability that a coverall will be called in exactly 24 to 75 calls by the number of cards in play. For example in a 200-card game the probability of the first coverall in exactly 60 calls is 8.88%. Very low probabilities should be taken with a grain of salt, because they may be based on as little as one occurence in the sample.

### Probability of Coverall by Number of Calls Exactly

Calls 100 Cards 200 Cards 500 Cards 1000 Cards
24 0 0 0 0
25 0 0 0 0
26 0 0 0 0
27 0 0 0 0
28 0 0 0 0
29 0 0 0 0
30 0 0 0 0
31 0 0 0 0
32 0 0 0 0
33 0 0 0 0
34 0 0 0 0
35 0 0 0 0
36 0 0 0 0
37 0 0 0 0
38 0.000000081 0 0.000000556 0
39 0 0.000000451 0 0
40 0.000000244 0.000000451 0.000001668 0.00000335
41 0.000000812 0.000000677 0.000001112 0
42 0.000000812 0.000002481 0.000003336 0.000005584
43 0.0000013 0.00000406 0.000008341 0.000023453
44 0.000004387 0.000006316 0.000017794 0.000040205
45 0.000007392 0.000011954 0.000035587 0.000067009
46 0.000016653 0.000031127 0.0000873 0.000161939
47 0.000032331 0.000061126 0.000171819 0.000329462
48 0.000063444 0.000131273 0.000310832 0.000617601
49 0.000124939 0.000240217 0.000598866 0.001111235
50 0.000221852 0.000450885 0.001129893 0.002188966
51 0.000418197 0.000823052 0.002054604 0.004050704
52 0.000773924 0.001495433 0.003847309 0.007561983
53 0.001392283 0.002724033 0.00671597 0.013308019
54 0.002404224 0.004761024 0.011786588 0.02302323
55 0.004186596 0.008286004 0.020299155 0.038641948
56 0.00714078 0.014069246 0.033530916 0.062962922
57 0.011965475 0.023529942 0.054423376 0.096555729
58 0.019776442 0.037942709 0.083837856 0.136793612
59 0.031830382 0.059312281 0.120524911 0.17127094
60 0.04982039 0.08881606 0.157332629 0.180108331
61 0.075076767 0.124190143 0.177556161 0.147070583
62 0.106797563 0.156943949 0.161671486 0.082063882
63 0.140753859 0.172727416 0.107064613 0.027109672
64 0.164937206 0.152701928 0.045642794 0.004566674
65 0.163299594 0.099422578 0.01031528 0.000350681
66 0.126231113 0.04129559 0.000993661 0.000012285
67 0.067797238 0.009152588 0.000035587 0
68 0.021547035 0.000845833 0 0
69 0.003220227 0.000019172 0 0
70 0.000154427 0 0 0
71 0.000002031 0 0 0
72 0 0 0 0
73 0 0 0 0
74 0 0 0 0
75 0 0 0 0
Total 1 1 1 1

In a 100-player game the expected number of calls for a coverall is 63.43, in a 200-player game it is 62.00, in a 500-player game it is 60.18, and in a 1000-player game it is 58.85. Very low probabilities should be taken with a grain of salt, because they may be based on as little as one occurence in the sample.

The next table shows the probability that a coverall will be called in 24 to 75 calls or less by the number of cards in play. For example in a 200-card game the probability of the first coverall in 60 calls or less is 36.69%.

### Probability of Coverall by Number of Calls or Less

Calls 100 Cards 200 Cards 500 Cards 1000 Cards
24 0 0 0 0
25 0 0 0 0
26 0 0 0 0
27 0 0 0 0
28 0 0 0 0
29 0 0 0 0
30 0 0 0 0
31 0 0 0 0
32 0 0 0 0
33 0 0 0 0
34 0 0 0 0
35 0 0 0 0
36 0 0 0 0
37 0 0 0 0
38 0.000000081 0 0.000000556 0
39 0.000000081 0.000000451 0.000000556 0
40 0.000000325 0.000000902 0.000002224 0.00000335
41 0.000001137 0.000001579 0.000003336 0.00000335
42 0.00000195 0.00000406 0.000006673 0.000008935
43 0.000003249 0.00000812 0.000015013 0.000032388
44 0.000007636 0.000014436 0.000032807 0.000072593
45 0.000015028 0.00002639 0.000068394 0.000139602
46 0.000031682 0.000057517 0.000155694 0.000301541
47 0.000064013 0.000118642 0.000327513 0.000631003
48 0.000127457 0.000249915 0.000638345 0.001248604
49 0.000252396 0.000490132 0.001237211 0.002359839
50 0.000474249 0.000941017 0.002367104 0.004548805
51 0.000892445 0.001764069 0.004421708 0.008599509
52 0.001666369 0.003259502 0.008269017 0.016161492
53 0.003058652 0.005983534 0.014984987 0.029469511
54 0.005462876 0.010744558 0.026771575 0.052492741
55 0.009649472 0.019030563 0.04707073 0.091134688
56 0.016790252 0.033099808 0.080601646 0.15409761
57 0.028755727 0.056629751 0.135025022 0.250653339
58 0.048532169 0.09457246 0.218862878 0.387446951
59 0.080362551 0.153884741 0.339387789 0.558717891
60 0.130182941 0.242700801 0.496720418 0.738826223
61 0.205259708 0.366890944 0.674276579 0.885896806
62 0.312057271 0.523834893 0.835948065 0.967960688
63 0.452811129 0.69656231 0.943012678 0.99507036
64 0.617748335 0.849264238 0.988655472 0.999637034
65 0.781047929 0.948686816 0.998970752 0.999987715
66 0.907279041 0.989982407 0.999964413 1
67 0.975076279 0.999134995 1 1
68 0.996623314 0.999980828 1 1
69 0.999843542 1 1 1
70 0.999997969 1 1 1
71 1 1 1 1
72 1 1 1 1
73 1 1 1 1
74 1 1 1 1
75 1 1 1 1

## 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.

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.

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