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Probability in Bingo II

Last Update: Apr 09, 2010

This page is an addendum to my Bingo Probabilities I page. This tables in this page are more essoteric, and probably of less interest.

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.

Probability Density for Common Multiple-Way Patterns

The following table shows the probability of achieving various winning goals in exactly the given number of calls. "HW" stands for Hardway, meaning the player may not use the free square. A "Six Pack" is a 3 by 2 block of marks anywhere on the card, which may include the free square. A "Nine Pack" is a 3 by 3 block of marks anywhere on the card, including the free square.

Probability Density for Common Multiple-Way Patterns
CallsSingleDoubleTripleSingle HWDouble HWTriple HWSix PackNine Pack
4 0.00000329 0 0 0 0 0 0 0
5 0.00001363 0 0 0.00000046 0 0 0.0000007 0
6 0.00003523 0 0 0.00000232 0 0 0.00000354 0
7 0.00007277 0 0 0.00000695 0 0 0.00001077 0
8 0.0001314 0 0 0.00001622 0 0 0.00002544 0
9 0.00021673 0.00000001 0 0.00003245 0 0 0.00005144 0
10 0.00033478 0.00000007 0 0.0000584 0 0 0.00009342 0.00000002
11 0.00049203 0.00000022 0 0.00009733 0.00000001 0 0.00015678 0.00000006
12 0.00069535 0.00000062 0 0.00015294 0.00000002 0 0.00024763 0.00000018
13 0.00095195 0.0000015 0 0.00022937 0.00000007 0 0.00037269 0.00000042
14 0.00126932 0.0000033 0 0.00033124 0.00000017 0 0.00053916 0.00000091
15 0.0016552 0.00000668 0.00000001 0.00046357 0.00000041 0 0.00075467 0.00000183
16 0.00211738 0.00001267 0.00000003 0.00063181 0.00000089 0 0.0010271 0.00000343
17 0.00266367 0.00002278 0.0000001 0.00084179 0.0000018 0 0.00136446 0.00000608
18 0.00330168 0.0000391 0.00000024 0.00109972 0.00000344 0 0.00177471 0.00001033
19 0.00403869 0.00006454 0.00000055 0.00141209 0.00000626 0.00000001 0.00226558 0.00001687
20 0.00488136 0.00010296 0.00000119 0.00178564 0.00001093 0.00000002 0.00284441 0.00002665
21 0.00583558 0.00015938 0.00000244 0.0022273 0.00001838 0.00000005 0.00351793 0.0000409
22 0.00690611 0.00024019 0.00000474 0.00274408 0.00002996 0.00000011 0.00429208 0.00006115
23 0.00809634 0.00035333 0.00000884 0.00334293 0.00004745 0.00000024 0.00517177 0.00008937
24 0.00940796 0.0005085 0.00001585 0.00403065 0.00007327 0.00000051 0.00616068 0.00012792
25 0.01084061 0.00071725 0.00002747 0.00481372 0.00011058 0.00000101 0.00726107 0.00017972
26 0.01239156 0.00099314 0.00004616 0.00569809 0.00016343 0.00000194 0.00847353 0.00024823
27 0.01405538 0.00135168 0.0000754 0.00668896 0.00023698 0.00000358 0.00979681 0.00033754
28 0.01582364 0.0018103 0.00011998 0.00779061 0.00033761 0.0000064 0.01122763 0.00045245
29 0.01768465 0.00238807 0.0001864 0.00900603 0.00047315 0.00001112 0.0127605 0.00059848
30 0.01962319 0.0031054 0.00028317 0.01033675 0.00065305 0.00001878 0.01438757 0.0007819
31 0.02162047 0.00398345 0.00042129 0.01178242 0.0008885 0.00003096 0.01609855 0.00100982
32 0.02365398 0.00504344 0.00061459 0.01334059 0.00119258 0.00004987 0.01788056 0.00129011
33 0.02569763 0.00630573 0.00088009 0.01500633 0.00158026 0.00007863 0.01971817 0.00163147
34 0.02772194 0.0077887 0.00123829 0.01677195 0.00206846 0.00012152 0.02159337 0.00204329
35 0.02969435 0.00950751 0.00171326 0.01862668 0.00267585 0.00018432 0.02348564 0.00253566
36 0.03157976 0.01147264 0.00233262 0.02055644 0.00342266 0.00027467 0.02537212 0.00311922
37 0.03334117 0.0136884 0.00312716 0.0225436 0.00433022 0.00040248 0.02722778 0.00380499
38 0.03494057 0.01615135 0.00413021 0.02456689 0.00542046 0.0005804 0.02902571 0.00460418
39 0.03633983 0.01884885 0.00537656 0.0266013 0.00671506 0.00082428 0.03073752 0.00552791
40 0.03750194 0.02175766 0.00690098 0.02861822 0.00823454 0.00115358 0.03233374 0.00658694
41 0.03839217 0.02484288 0.00873624 0.03058556 0.009997 0.00159175 0.03378438 0.00779124
42 0.03897941 0.02805725 0.01091068 0.0324682 0.01201673 0.00216649 0.03505955 0.00914957
43 0.03923752 0.03134095 0.01344532 0.03422845 0.01430262 0.00290977 0.03613014 0.01066898
44 0.03914664 0.03462201 0.01635065 0.0358268 0.01685639 0.00385767 0.03696858 0.01235426
45 0.03869438 0.03781752 0.01962319 0.03722283 0.01967074 0.0050497 0.03754966 0.01420731
46 0.03787695 0.04083556 0.02324203 0.03837626 0.02272752 0.00652786 0.03785138 0.01622649
47 0.03669992 0.04357804 0.02716562 0.03924831 0.02599598 0.00833492 0.03785583 0.01840593
48 0.03517876 0.04594434 0.03132918 0.03980303 0.02943127 0.01051226 0.03755002 0.02073483
49 0.03333902 0.04783569 0.03564295 0.04000895 0.03297335 0.01309682 0.03692673 0.02319678
50 0.03121603 0.04916026 0.03999178 0.03984064 0.03654644 0.01611745 0.03598524 0.02576909
51 0.02885423 0.04983855 0.04423647 0.03928037 0.04005933 0.01959042 0.03473193 0.02842224
52 0.02630592 0.04980895 0.04821702 0.03831968 0.04340652 0.02351441 0.03318073 0.0311194
53 0.02362963 0.04903305 0.05175831 0.03696074 0.04647061 0.02786504 0.03135343 0.03381617
54 0.02088804 0.04750032 0.05467806 0.03521753 0.0491259 0.03258928 0.02927962 0.03646049
55 0.01814554 0.04523162 0.05679726 0.03311655 0.05124337 0.03760023 0.02699642 0.03899286
56 0.01546553 0.0422812 0.05795256 0.03069709 0.052697 0.04277274 0.02454791 0.04134688
57 0.01290766 0.03873676 0.05801006 0.02801094 0.0533714 0.04794084 0.0219841 0.04345027
58 0.01052507 0.03471715 0.05687974 0.02512127 0.05317041 0.05289751 0.01935965 0.04522625
59 0.00836186 0.03036761 0.05452906 0.02210095 0.05202638 0.05739807 0.01673217 0.04659556
60 0.00645101 0.02585247 0.05099434 0.01902996 0.04990941 0.06116795 0.01416017 0.04747907
61 0.00481289 0.02134555 0.04638824 0.01599204 0.0468359 0.06391587 0.01170073 0.04780095
62 0.00345451 0.01701879 0.04090137 0.01307075 0.0428754 0.06535307 0.00940701 0.04749256
63 0.00236968 0.01302984 0.03479654 0.01034494 0.03815464 0.06521885 0.00732555 0.04649702
64 0.00154008 0.00950963 0.02839436 0.00788397 0.03285747 0.06331213 0.00549371 0.04477428
65 0.00093726 0.00655145 0.02204957 0.00574298 0.02721962 0.05952746 0.0039373 0.04230684
66 0.00052546 0.00420266 0.01611897 0.00395864 0.02151703 0.05389296 0.00266862 0.03910572
67 0.00026496 0.00246087 0.01092282 0.00254591 0.0160472 0.04660552 0.00168528 0.03521664
68 0.00011577 0.00127544 0.00670398 0.00149617 0.01110324 0.03805666 0.00096989 0.03072603
69 0.00004113 0.0005554 0.00359071 0.00077756 0.00694172 0.02883965 0.00049101 0.02576641
70 0.00001049 0.00018351 0.00157127 0.00033774 0.00374637 0.01972561 0.00020551 0.0205207
71 0.00000139 0.00003523 0.00048994 0.00010958 0.0015919 0.01159299 0.00006267 0.01522478
72 0 0 0.00007405 0.00001975 0.00041466 0.00529187 0.00000987 0.01016743
73 0 0 0 0 0 0.0014217 0 0.00568678
74 0 0 0 0 0 0 0 0.00216216
75 0 0 0 0 0 0 0 0
Total 1 1 1 1 1 1 1 1
Mean 41.36857386 49.77068987 54.78946365 46.48991604 55.00356639 60.17054993 44.45846049 57.67525677
Median 42 50 55 47 56 61 45 59
Mode 43 51 57 49 57 62 47 61

Probability Distribution for Common Multiple-Way Patterns

The following table shows the probability of achieving various winning goals in the given number of calls or less. "HW" stands for Hardway, meaning the player may not use the free square. A "Six Pack" is a 3 by 2 block of marks anywhere on the card, which may include the free square. A "Nine Pack" is a 3 by 3 block of marks anywhere on the card, including the free square.

Probability Distribution for Common Multiple-Way Patterns
CallsSingleDoubleTripleSingle HWDouble HWTriple HWSix PackNine Pack
4 0.00000329 0 0 0 0 0 0 0
5 0.00001692 0 0 0.00000046 0 0 0.0000007 0
6 0.00005215 0 0 0.00000278 0 0 0.00000423 0
7 0.00012492 0 0 0.00000973 0 0 0.000015 0
8 0.00025632 0 0 0.00002596 0 0 0.00004044 0
9 0.00047305 0.00000002 0 0.0000584 0 0 0.00009188 0
10 0.00080783 0.00000008 0 0.0001168 0 0 0.0001853 0.00000002
11 0.00129986 0.0000003 0 0.00021414 0.00000001 0 0.00034208 0.00000009
12 0.00199521 0.00000092 0 0.00036708 0.00000003 0 0.00058971 0.00000026
13 0.00294715 0.00000243 0 0.00059645 0.0000001 0 0.0009624 0.00000069
14 0.00421648 0.00000573 0 0.00092769 0.00000027 0 0.00150156 0.0000016
15 0.00587167 0.00001241 0.00000002 0.00139126 0.00000068 0 0.00225623 0.00000343
16 0.00798905 0.00002509 0.00000005 0.00202306 0.00000157 0 0.00328333 0.00000685
17 0.01065272 0.00004786 0.00000015 0.00286486 0.00000338 0 0.0046478 0.00001294
18 0.0139544 0.00008696 0.00000038 0.00396458 0.00000682 0 0.00642251 0.00002326
19 0.01799309 0.0001515 0.00000094 0.00537667 0.00001309 0.00000001 0.00868808 0.00004013
20 0.02287445 0.00025446 0.00000213 0.00716231 0.00002401 0.00000003 0.01153249 0.00006678
21 0.02871003 0.00041385 0.00000457 0.00938961 0.0000424 0.00000008 0.01505042 0.00010768
22 0.03561614 0.00065404 0.00000931 0.01213369 0.00007235 0.00000019 0.0193425 0.00016883
23 0.04371249 0.00100737 0.00001815 0.01547661 0.0001198 0.00000043 0.02451426 0.0002582
24 0.05312045 0.00151587 0.000034 0.01950726 0.00019307 0.00000094 0.03067494 0.00038612
25 0.06396106 0.00223312 0.00006147 0.02432099 0.00030364 0.00000195 0.03793601 0.00056584
26 0.07635261 0.00322626 0.00010763 0.03001907 0.00046708 0.00000389 0.04640953 0.00081407
27 0.09040799 0.00457794 0.00018303 0.03670804 0.00070405 0.00000748 0.05620634 0.00115161
28 0.10623163 0.00638824 0.00030301 0.04449864 0.00104166 0.00001388 0.06743397 0.00160406
29 0.12391628 0.00877631 0.00048941 0.05350468 0.00151481 0.000025 0.08019447 0.00220254
30 0.14353947 0.01188171 0.00077258 0.06384142 0.00216786 0.00004378 0.09458204 0.00298444
31 0.16515993 0.01586516 0.00119387 0.07562384 0.00305637 0.00007474 0.11068058 0.00399426
32 0.18881391 0.0209086 0.00180847 0.08896443 0.00424894 0.00012461 0.12856114 0.00528437
33 0.21451154 0.02721433 0.00268856 0.10397077 0.00582921 0.00020323 0.14827931 0.00691584
34 0.24223348 0.03500303 0.00392684 0.12074272 0.00789767 0.00032476 0.16987268 0.00895913
35 0.27192783 0.04451054 0.00564011 0.1393694 0.01057353 0.00050908 0.19335832 0.01149479
36 0.30350759 0.05598318 0.00797273 0.15992584 0.01399618 0.00078375 0.21873044 0.01461402
37 0.33684876 0.06967158 0.01109989 0.18246944 0.0183264 0.00118623 0.24595822 0.01841901
38 0.37178933 0.08582293 0.01523011 0.20703633 0.02374686 0.00176664 0.27498393 0.02302319
39 0.40812916 0.10467178 0.02060667 0.23363764 0.03046192 0.00259092 0.30572145 0.0285511
40 0.44563111 0.12642944 0.02750765 0.26225586 0.03869647 0.0037445 0.33805518 0.03513804
41 0.48402328 0.15127232 0.03624389 0.29284142 0.04869347 0.00533625 0.37183956 0.04292928
42 0.52300269 0.17932958 0.04715457 0.32530962 0.0607102 0.00750274 0.40689911 0.05207885
43 0.56224021 0.21067052 0.06059989 0.35953807 0.07501283 0.01041251 0.44302926 0.06274782
44 0.60138685 0.24529253 0.07695054 0.39536487 0.09186922 0.01427018 0.47999784 0.07510208
45 0.64008123 0.28311005 0.09657373 0.4325877 0.11153996 0.01931989 0.5175475 0.08930939
46 0.67795818 0.32394561 0.11981576 0.47096396 0.13426747 0.02584774 0.55539889 0.10553588
47 0.7146581 0.36752365 0.14698138 0.51021227 0.16026345 0.03418266 0.59325472 0.1239418
48 0.74983686 0.41346799 0.17831056 0.5500153 0.18969472 0.04469492 0.63080473 0.14467663
49 0.78317588 0.46130368 0.21395351 0.59002425 0.22266807 0.05779175 0.66773147 0.16787341
50 0.81439191 0.51046394 0.25394529 0.6298649 0.25921451 0.0739092 0.70371671 0.1936425
51 0.84324614 0.56030249 0.29818176 0.66914527 0.29927384 0.09349961 0.73844863 0.22206474
52 0.86955207 0.61011144 0.34639879 0.70746495 0.34268036 0.11701402 0.77162937 0.25318414
53 0.8931817 0.65914449 0.39815709 0.74442569 0.38915097 0.14487906 0.8029828 0.28700031
54 0.91406974 0.70664481 0.45283515 0.77964322 0.43827687 0.17746835 0.83226241 0.3234608
55 0.93221528 0.75187643 0.50963241 0.81275976 0.48952024 0.21506857 0.85925884 0.36245366
56 0.9476808 0.79415763 0.56758497 0.84345686 0.54221724 0.25784131 0.88380675 0.40380055
57 0.96058846 0.83289439 0.62559503 0.87146779 0.59558864 0.30578215 0.90579085 0.44725082
58 0.97111353 0.86761155 0.68247478 0.89658906 0.64875905 0.35867966 0.9251505 0.49247706
59 0.97947539 0.89797916 0.73700384 0.91869001 0.70078543 0.41607772 0.94188267 0.53907262
60 0.98592639 0.92383163 0.78799818 0.93771997 0.75069484 0.47724567 0.95604284 0.58655169
61 0.99073928 0.94517718 0.83438642 0.95371201 0.79753074 0.54116154 0.96774357 0.63435264
62 0.99419379 0.96219597 0.87528779 0.96678276 0.84040614 0.60651461 0.97715059 0.6818452
63 0.99656346 0.97522581 0.91008433 0.9771277 0.87856078 0.67173346 0.98447614 0.72834222
64 0.99810354 0.98473544 0.93847869 0.98501167 0.91141825 0.73504558 0.98996986 0.77311651
65 0.9990408 0.99128689 0.96052826 0.99075465 0.93863787 0.79457304 0.99390715 0.81542335
66 0.99956626 0.99548955 0.97664723 0.99471329 0.96015491 0.848466 0.99657577 0.85452907
67 0.99983122 0.99795042 0.98757005 0.9972592 0.9762021 0.89507152 0.99826105 0.88974571
68 0.99994699 0.99922586 0.99427404 0.99875537 0.98730534 0.93312818 0.99923094 0.92047174
69 0.99998812 0.99978127 0.99786474 0.99953293 0.99424707 0.96196783 0.99972195 0.94623815
70 0.99999861 0.99996477 0.99943602 0.99987068 0.99799344 0.98169344 0.99992746 0.96675885
71 1 1 0.99992595 0.99998025 0.99958534 0.99328644 0.99999013 0.98198363
72 1 1 1 1 1 0.9985783 1 0.99215106
73 1 1 1 1 1 1 1 0.99783784
74 1 1 1 1 1 1 1 1
75 1 1 1 1 1 1 1 1

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 Exactly
Calls4 Marks5 Marks6 Marks7 Marks8 Marks9 Marks10 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 Exactly
Calls11 Marks12 Marks13 Marks14 Marks15 Marks16 Marks17 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 Exactly
Calls18 Marks19 Marks20 Marks21 Marks22 Marks23 Marks24 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

Probability Distribution for Patterns

The next three tables show the probability of covering a pattern of 4 to 24 marks according to the given number of calls or less. 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 or less is 18.95%, 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 or Less
Calls4 Marks5 Marks6 Marks7 Marks8 Marks9 Marks10 Marks
4 0.000000823 0 0 0 0 0 0
5 0.000004114 0.000000058 0 0 0 0 0
6 0.000012341 0.000000348 0.000000005 0 0 0 0
7 0.000028796 0.000001217 0.000000035 0.000000001 0 0 0
8 0.000057592 0.000003245 0.000000139 0.000000004 0 0 0
9 0.000103665 0.0000073 0.000000417 0.000000018 0.000000001 0 0
10 0.000172776 0.000014601 0.000001043 0.00000006 0.000000003 0 0
11 0.000271504 0.000026768 0.000002294 0.000000166 0.00000001 0 0
12 0.000407257 0.000045888 0.000004589 0.000000399 0.000000029 0.000000002 0
13 0.000588259 0.000074568 0.000008522 0.000000865 0.000000076 0.000000006 0
14 0.000823563 0.000115995 0.000014914 0.000001729 0.000000178 0.000000016 0.000000001
15 0.001123041 0.000173992 0.000024856 0.000003242 0.000000381 0.00000004 0.000000004
16 0.001497388 0.00025308 0.00003977 0.000005764 0.000000763 0.000000091 0.00000001
17 0.001958123 0.000358529 0.000061462 0.000009798 0.000001441 0.000000194 0.000000023
18 0.002517586 0.000496425 0.000092193 0.000016034 0.000002594 0.000000387 0.000000053
19 0.003188942 0.00067372 0.000134744 0.000025387 0.00000448 0.000000736 0.000000111
20 0.003986178 0.000898294 0.000192491 0.000039056 0.000007467 0.000001337 0.000000223
21 0.004924102 0.00117901 0.000269488 0.000058584 0.000012061 0.00000234 0.000000426
22 0.006018347 0.001525778 0.000370546 0.000085924 0.000018954 0.00000396 0.00000078
23 0.007285368 0.001949605 0.000501327 0.000123515 0.000029062 0.000006507 0.00000138
24 0.008742441 0.002462659 0.000668436 0.000174375 0.000043594 0.00001041 0.000002366
25 0.010407668 0.003078324 0.000879521 0.000242187 0.000064108 0.000016266 0.000003943
26 0.012299971 0.003811259 0.001143378 0.000331414 0.000092601 0.000024878 0.000006408
27 0.014439097 0.004677454 0.001470057 0.000447409 0.000131591 0.000037317 0.000010177
28 0.016845613 0.005694292 0.001870982 0.000596545 0.000184227 0.000054993 0.000015831
29 0.019540911 0.006880602 0.002359064 0.000786355 0.000254409 0.00007974 0.000024164
30 0.022547205 0.008256723 0.00294883 0.00102568 0.000346921 0.000113914 0.000036245
31 0.025887531 0.009844554 0.003656549 0.001324836 0.000467589 0.000160516 0.000053505
32 0.02958575 0.01166762 0.004500368 0.001695791 0.000623452 0.000223326 0.000077826
33 0.033666543 0.013751123 0.005500449 0.00215235 0.000822957 0.000307074 0.000111663
34 0.038155416 0.016122007 0.006679117 0.002710366 0.001076175 0.00041762 0.000158189
35 0.043078695 0.018809008 0.008061003 0.003387958 0.001395041 0.000562181 0.000221465
36 0.048463532 0.021842719 0.009673204 0.004205741 0.001793625 0.000749575 0.000306644
37 0.0543379 0.025255643 0.011545437 0.00518708 0.002288418 0.000990509 0.000420216
38 0.060730594 0.029082256 0.013710206 0.006358357 0.002898663 0.001297909 0.000570293
39 0.067671233 0.033359058 0.016202971 0.007749247 0.003646705 0.001687281 0.000766946
40 0.075190259 0.038124638 0.019062319 0.009393027 0.004558381 0.002177137 0.001022595
41 0.083318935 0.043419727 0.022330145 0.011326885 0.005663443 0.002789457 0.001352464
42 0.09208935 0.049287258 0.026051836 0.013592262 0.006996017 0.003550218 0.001775109
43 0.101534411 0.055772423 0.030276458 0.016235202 0.008595107 0.004489981 0.002313021
44 0.111687852 0.062922734 0.035056952 0.019306727 0.010505131 0.005644548 0.002993321
45 0.122584228 0.070788075 0.040450329 0.022863229 0.01277651 0.007055685 0.003848555
46 0.134258916 0.079420767 0.046517878 0.026966886 0.015466302 0.008771933 0.004917599
47 0.146748118 0.088875621 0.053325372 0.031686091 0.018638877 0.010849496 0.006246679
48 0.160088856 0.099209995 0.060943283 0.037095911 0.022366652 0.013353225 0.007890542
49 0.174318977 0.110483858 0.069446997 0.043278563 0.026730877 0.016357701 0.009913758
50 0.189477148 0.122759843 0.078917042 0.050323911 0.031822473 0.019948416 0.012392198
51 0.205602863 0.136103304 0.089439314 0.058329987 0.037742933 0.024223076 0.015414685
52 0.222736435 0.150582379 0.101105311 0.067403541 0.044605284 0.029293023 0.019084848
53 0.240919001 0.166268043 0.114012372 0.077660601 0.052535113 0.035284777 0.023523185
54 0.260192521 0.18323417 0.128263919 0.089227074 0.061671654 0.042341733 0.028869363
55 0.280599778 0.201557587 0.143969705 0.102239356 0.072168957 0.050625985 0.035284777
56 0.302184376 0.221318135 0.16124607 0.116844978 0.084197116 0.060320322 0.042955381
57 0.324990744 0.242598725 0.180216195 0.133203275 0.097943584 0.071630383 0.052094824
58 0.349064133 0.265485397 0.201010372 0.151486077 0.113614558 0.084786984 0.062947912
59 0.374450615 0.290067378 0.223766263 0.171878434 0.131436449 0.100048641 0.075794425
60 0.401197087 0.316437139 0.248629181 0.194579359 0.151657442 0.117704283 0.09095331
61 0.429351269 0.344690455 0.275752364 0.219802609 0.174549131 0.138076178 0.108787292
62 0.458961701 0.37492646 0.305297261 0.247777487 0.200408261 0.161523076 0.129707925
63 0.490077749 0.407247707 0.337433814 0.278749673 0.229558554 0.188443589 0.154181118
64 0.522749599 0.441760224 0.372340761 0.312982089 0.262352633 0.219279813 0.182733177
65 0.557028261 0.478573576 0.410205923 0.350755789 0.299174055 0.254521211 0.215957391
66 0.592965568 0.517800919 0.451226515 0.392370883 0.340439442 0.294708771 0.254521211
67 0.630614176 0.559559057 0.495609451 0.438147486 0.386600723 0.340439442 0.299174055
68 0.670027562 0.603968506 0.543571656 0.488426705 0.438147486 0.392370883 0.350755789
69 0.711260027 0.651153546 0.595340385 0.543571656 0.495609451 0.451226515 0.410205923
70 0.754366695 0.70124228 0.651153546 0.603968506 0.559559057 0.517800919 0.478573576
71 0.799403513 0.754366695 0.711260027 0.670027562 0.630614176 0.592965568 0.557028261
72 0.846427249 0.810662718 0.77592003 0.742184376 0.709440948 0.677674935 0.646871529
73 0.895495495 0.87027027 0.845405405 0.820900901 0.796756757 0.772972973 0.74954955
74 0.946666667 0.933333333 0.92 0.906666667 0.893333333 0.88 0.866666667
75 1 1 1 1 1 1 1

Probability of Covering 11 to 17 Mark Patterns by Number of Calls or Less
Calls11 Marks12 Marks13 Marks14 Marks15 Marks16 Marks17 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 0.000000003 0 0 0 0 0 0
18 0.000000006 0.000000001 0 0 0 0 0
19 0.000000015 0.000000002 0 0 0 0 0
20 0.000000034 0.000000005 0.000000001 0 0 0 0
21 0.000000072 0.000000011 0.000000002 0 0 0 0
22 0.000000144 0.000000025 0.000000004 0.000000001 0 0 0
23 0.000000276 0.000000052 0.000000009 0.000000001 0 0 0
24 0.00000051 0.000000104 0.00000002 0.000000003 0.000000001 0 0
25 0.00000091 0.000000199 0.000000041 0.000000008 0.000000001 0 0
26 0.000001577 0.00000037 0.000000082 0.000000017 0.000000003 0.000000001 0
27 0.000002662 0.000000665 0.000000158 0.000000036 0.000000008 0.000000002 0
28 0.000004384 0.000001165 0.000000296 0.000000072 0.000000016 0.000000004 0.000000001
29 0.000007063 0.000001987 0.000000536 0.000000138 0.000000034 0.000000008 0.000000002
30 0.000011152 0.000003311 0.000000946 0.000000259 0.000000068 0.000000017 0.000000004
31 0.000017286 0.000005402 0.000001629 0.000000473 0.000000132 0.000000035 0.000000009
32 0.000026341 0.000008643 0.000002744 0.000000841 0.000000248 0.00000007 0.000000019
33 0.000039512 0.000013582 0.000004527 0.00000146 0.000000455 0.000000136 0.000000039
34 0.000058408 0.000020991 0.00000733 0.000002483 0.000000814 0.000000258 0.000000079
35 0.000085179 0.000031942 0.000011661 0.000004138 0.000001425 0.000000475 0.000000153
36 0.000122658 0.000047913 0.000018253 0.000006771 0.000002442 0.000000855 0.00000029
37 0.000174551 0.000070911 0.000028139 0.000010893 0.000004107 0.000001506 0.000000536
38 0.000245665 0.00010364 0.000042772 0.000017247 0.000006786 0.000002601 0.00000097
39 0.000342176 0.000149702 0.000064158 0.000026905 0.000011027 0.000004411 0.000001719
40 0.000471967 0.00021386 0.000095049 0.000041392 0.000017643 0.000007351 0.00000299
41 0.000645021 0.000302354 0.000139179 0.000062855 0.000027821 0.000012056 0.000005108
42 0.0008739 0.000423295 0.000201569 0.000094282 0.000043277 0.000019475 0.000008582
43 0.001174303 0.000587151 0.000288916 0.000139798 0.000066461 0.000031015 0.000014193
44 0.001565737 0.000807333 0.000410074 0.000205037 0.000100838 0.000048738 0.00002313
45 0.002072299 0.001100909 0.000576667 0.000297634 0.000151257 0.000075628 0.000037173
46 0.002723593 0.001489465 0.000803838 0.000427849 0.000224446 0.000115964 0.000058965
47 0.003555802 0.002000139 0.001111188 0.000609361 0.000329654 0.000175816 0.000092378
48 0.004612932 0.002666852 0.001523915 0.000860275 0.000479497 0.000263724 0.000143037
49 0.005948255 0.003531776 0.002074218 0.001204385 0.00069104 0.00039159 0.000219025
50 0.007625968 0.004647074 0.002802997 0.001672756 0.0009872 0.000575867 0.000331856
51 0.009723109 0.006076943 0.003761917 0.002305691 0.001398534 0.00083912 0.000497783
52 0.012331748 0.007900026 0.00501589 0.003155156 0.001965507 0.001212063 0.000739564
53 0.015561491 0.010212229 0.006646054 0.004287777 0.002741365 0.001736198 0.001088802
54 0.019542338 0.013130008 0.008753339 0.005788498 0.003795737 0.002467229 0.001589063
55 0.024427923 0.016794197 0.011462706 0.007765059 0.005219138 0.003479425 0.002299959
56 0.030399193 0.021374432 0.014928175 0.010353412 0.007128579 0.004871195 0.003302505
57 0.037668565 0.027074281 0.019338772 0.01372429 0.009674499 0.00677215 0.00470607
58 0.046484612 0.034137137 0.024925529 0.018091109 0.013049325 0.009352016 0.006657367
59 0.057137336 0.042853002 0.0319697 0.023719455 0.017497958 0.012831836 0.009352016
60 0.069964084 0.053566252 0.040812383 0.030938419 0.023330611 0.017497958 0.013049325
61 0.085356183 0.066684518 0.051865736 0.040154118 0.030938419 0.023719455 0.018091109
62 0.10376634 0.082688802 0.065626033 0.051865736 0.040812383 0.0319697 0.024925529
63 0.125716912 0.102144991 0.082688802 0.066684518 0.053566252 0.042853002 0.034137137
64 0.151809101 0.125716912 0.10376634 0.085356183 0.069964084 0.057137336 0.046484612
65 0.182733177 0.154181118 0.129707925 0.108787292 0.09095331 0.075794425 0.062947912
66 0.219279813 0.188443589 0.161523076 0.138076178 0.117704283 0.100048641 0.084786984
67 0.262352633 0.229558554 0.200408261 0.174549131 0.151657442 0.131436449 0.113614558
68 0.312982089 0.278749673 0.247777487 0.219802609 0.194579359 0.171878434 0.151486077
69 0.372340761 0.337433814 0.305297261 0.275752364 0.248629181 0.223766263 0.201010372
70 0.441760224 0.407247707 0.37492646 0.344690455 0.316437139 0.290067378 0.265485397
71 0.522749599 0.490077749 0.458961701 0.429351269 0.401197087 0.374450615 0.349064133
72 0.61701592 0.588093299 0.560088856 0.532987782 0.506775268 0.481436505 0.456956683
73 0.726486486 0.703783784 0.681441441 0.659459459 0.637837838 0.616576577 0.595675676
74 0.853333333 0.84 0.826666667 0.813333333 0.8 0.786666667 0.773333333
75 1 1 1 1 1 1 1

Probability of Covering 18 to 24 Mark Patterns by Number of Calls or Less
Calls18 Marks19 Marks20 Marks21 Marks22 Marks23 Marks24 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 0.000000002 0 0 0 0 0 0
32 0.000000005 0.000000001 0 0 0 0 0
33 0.000000011 0.000000003 0.000000001 0 0 0 0
34 0.000000023 0.000000006 0.000000002 0 0 0 0
35 0.000000047 0.000000014 0.000000004 0.000000001 0 0 0
36 0.000000095 0.00000003 0.000000009 0.000000003 0.000000001 0 0
37 0.000000185 0.000000062 0.00000002 0.000000006 0.000000002 0.000000001 0
38 0.000000351 0.000000123 0.000000042 0.000000014 0.000000004 0.000000001 0
39 0.000000652 0.00000024 0.000000086 0.00000003 0.00000001 0.000000003 0.000000001
40 0.000001186 0.000000458 0.000000172 0.000000062 0.000000022 0.000000007 0.000000002
41 0.000002114 0.000000853 0.000000335 0.000000128 0.000000047 0.000000017 0.000000006
42 0.000003699 0.000001558 0.00000064 0.000000256 0.0000001 0.000000038 0.000000014
43 0.000006363 0.000002791 0.000001196 0.0000005 0.000000204 0.000000081 0.000000031
44 0.000010767 0.000004911 0.000002193 0.000000957 0.000000408 0.000000169 0.000000068
45 0.000017946 0.000008501 0.000003947 0.000001794 0.000000797 0.000000346 0.000000146
46 0.000029482 0.000014483 0.000006983 0.000003301 0.000001528 0.000000692 0.000000306
47 0.000047782 0.00002431 0.000012155 0.000005967 0.000002873 0.000001355 0.000000625
48 0.000076451 0.000040237 0.000020837 0.000010608 0.000005304 0.000002602 0.000001251
49 0.000120841 0.000065721 0.000035207 0.000018564 0.000009626 0.000004904 0.000002452
50 0.000188814 0.000106001 0.000058679 0.000032007 0.000017189 0.000009081 0.000004715
51 0.000291804 0.000168939 0.000096537 0.000054412 0.000030229 0.00001654 0.000008906
52 0.000446288 0.000266207 0.000156872 0.000091271 0.000052396 0.000029658 0.00001654
53 0.000675808 0.00041497 0.000251946 0.000151168 0.000089581 0.000052396 0.000030229
54 0.001013712 0.000640239 0.00040015 0.000247365 0.000151168 0.000091271 0.000054412
55 0.00150687 0.000978144 0.000628807 0.00040015 0.000251946 0.000156872 0.000096537
56 0.00222065 0.001480433 0.000978144 0.000640239 0.00041497 0.000266207 0.000168939
57 0.003245566 0.00222065 0.00150687 0.001013712 0.000675808 0.000446288 0.000291804
58 0.00470607 0.003302505 0.002299959 0.001589063 0.001088802 0.000739564 0.000497783
59 0.00677215 0.004871195 0.003479425 0.002467229 0.001736198 0.001212063 0.00083912
60 0.009674499 0.007128579 0.005219138 0.003795737 0.002741365 0.001965507 0.001398534
61 0.01372429 0.010353412 0.007765059 0.005788498 0.004287777 0.003155156 0.002305691
62 0.019338772 0.014928175 0.011462706 0.008753339 0.006646054 0.00501589 0.003761917
63 0.027074281 0.021374432 0.016794197 0.013130008 0.010212229 0.007900026 0.006076943
64 0.037668565 0.030399193 0.024427923 0.019542338 0.015561491 0.012331748 0.009723109
65 0.052094824 0.042955381 0.035284777 0.028869363 0.023523185 0.019084848 0.015414685
66 0.071630383 0.060320322 0.050625985 0.042341733 0.035284777 0.029293023 0.024223076
67 0.097943584 0.084197116 0.072168957 0.061671654 0.052535113 0.044605284 0.037742933
68 0.133203275 0.116844978 0.102239356 0.089227074 0.077660601 0.067403541 0.058329987
69 0.180216195 0.16124607 0.143969705 0.128263919 0.114012372 0.101105311 0.089439314
70 0.242598725 0.221318135 0.201557587 0.18323417 0.166268043 0.150582379 0.136103304
71 0.324990744 0.302184376 0.280599778 0.260192521 0.240919001 0.222736435 0.205602863
72 0.433320992 0.410514624 0.388522769 0.367330618 0.346923362 0.32728619 0.308404295
73 0.575135135 0.554954955 0.535135135 0.515675676 0.496576577 0.477837838 0.459459459
74 0.76 0.746666667 0.733333333 0.72 0.706666667 0.693333333 0.68
75 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.

Expected Calls to Cover Pattern of x Marks
MarksExpected
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
Calls100 Cards200 Cards500 Cards1000 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
Calls100 Cards200 Cards500 Cards1000 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
Calls100 Cards200 Cards500 Cards1000 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
Calls100 Cards200 Cards500 Cards1000 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
Calls100 Cards200 Cards500 Cards1000 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
GameCards
200040006000800010000
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.

Double Action Coverall

Some bingo cards have two numbers per square. The player only needs to catch either number to cover that square. The same number will never appear twice on the same card. The following table shows the probability of a coverall in any given number of calls. The "density" column shows the probability of getting a coverall in exactly the given number of calls. The "distribution" column shows the proability of getting a coverall in the given number of calls or less. For example, the probability of getting a coverall in exactly 60 calls is 5.28%, and the probability of getting a coverall is 60 calls or less is 35.44%. The expected number of calls is 62.461833.

Double Action Coverall
CallsDensityDistribution
24 0.000000000000651 0.000000000000651
25 0.000000000011791 0.000000000012442
26 0.000000000110983 0.000000000123426
27 0.000000000722403 0.000000000845828
28 0.00000000365274 0.000000004498569
29 0.000000015282438 0.000000019781007
30 0.000000055037575 0.000000074818582
31 0.000000175274908 0.00000025009349
32 0.00000050330219 0.00000075339568
33 0.000001322379291 0.000002075774971
34 0.000003215539021 0.000005291313992
35 0.000007302455101 0.000012593769092
36 0.000015603065865 0.000028196834957
37 0.000031559360264 0.000059756195221
38 0.000060735751459 0.00012049194668
39 0.000111696678956 0.000232188625636
40 0.000197027019034 0.00042921564467
41 0.000334418463722 0.000763634108392
42 0.000547697933306 0.001311332041698
43 0.000867629574561 0.002178961616259
44 0.001332288974613 0.003511250590872
45 0.001986796243843 0.005498046834716
46 0.002882211528208 0.008380258362924
47 0.004073447014396 0.012453705377319
48 0.005616133603155 0.018069838980474
49 0.00756249265487 0.025632331635344
50 0.009956392660117 0.035588724295462
51 0.012827902126855 0.048416626422317
52 0.016187765772388 0.064604392194704
53 0.020022313728826 0.08462670592353
54 0.024289348154052 0.108916054077582
55 0.028915528796213 0.137831582873794
56 0.033795695989459 0.171627278863254
57 0.038794431200106 0.21042171006336
58 0.043749973813065 0.254171683876425
59 0.048480406432651 0.302652090309076
60 0.052791811272383 0.355443901581459
61 0.05648790980336 0.411931811384819
62 0.059380547344258 0.471312358729077
63 0.0613002898622 0.532612648591277
64 0.062106371733525 0.594719020324802
65 0.061695273082077 0.656414293406879
66 0.06000730870618 0.716421602113059
67 0.057030766168104 0.773452368281164
68 0.052803322099551 0.826255690380714
69 0.047410674090203 0.873666364470918
70 0.040982530999896 0.914648895470813
71 0.033686289028755 0.948335184499568
72 0.025718869554486 0.974054054054054
73 0.017297297297297 0.991351351351351
74 0.008648648648649 1
75 0 1

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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