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Bingo Probabilities - Part One

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Introduction

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.

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Single Bingo

Single Bingo

Balls Bingo on
Last Ball		 Bingo on or Before
Last Ball
4 0.0000032910 0.0000032910
5 0.0000136274 0.0000169183
6 0.0000352272 0.0000521455
7 0.0000727720 0.0001249175
8 0.0001314050 0.0002563225
9 0.0002167256 0.0004730480
10 0.0003347781 0.0008078261
11 0.0004920317 0.0012998578
12 0.0006953500 0.0019952077
13 0.0009519466 0.0029471543
14 0.0012693244 0.0042164788
15 0.0016551953 0.0058716740
16 0.0021173771 0.0079890511
17 0.0026636664 0.0106527175
18 0.0033016835 0.0139544010
19 0.0040386878 0.0179930888
20 0.0048813646 0.0228744533
21 0.0058355802 0.0287100336
22 0.0069061107 0.0356161443
23 0.0080963437 0.0437124880
24 0.0094079610 0.0531204490
25 0.0108406072 0.0639610563
26 0.0123915550 0.0763526113
27 0.0140553776 0.0904079889
28 0.0158236430 0.1062316319
29 0.0176846454 0.1239162772
30 0.0196231923 0.1435394695
31 0.0216204653 0.1651599348
32 0.0236539757 0.1888139106
33 0.0256976321 0.2145115427
34 0.0277219404 0.2422334831
35 0.0296943496 0.2719278328
36 0.0315797565 0.3035075893
37 0.0333411743 0.3368487636
38 0.0349405659 0.3717893296
39 0.0363398328 0.4081291624
40 0.0375019431 0.4456311055
41 0.0383921711 0.4840232766
42 0.0389794115 0.5230026881
43 0.0392375219 0.5622402100
44 0.0391466357 0.6013868457
45 0.0386943841 0.6400812298
46 0.0378769549 0.6779581847
47 0.0366999203 0.7146581050
48 0.0351787594 0.7498368644
49 0.0333390155 0.7831758799
50 0.0312160322 0.8143919121
51 0.0288542319 0.8432461440
52 0.0263059212 0.8695520652
53 0.0236296327 0.8931816978
54 0.0208880426 0.9140697405
55 0.0181455366 0.9322152770
56 0.0154655273 0.9476808043
57 0.0129076599 0.9605884642
58 0.0105250674 0.9711135316
59 0.0083618555 0.9794753870
60 0.0064510069 0.9859263939
61 0.0048128866 0.9907392806
62 0.0034545076 0.9941937882
63 0.0023696766 0.9965634648
64 0.0015400756 0.9981035404
65 0.0009372604 0.9990408007
66 0.0005254596 0.9995662603
67 0.0002649610 0.9998312213
68 0.0001157686 0.9999469899
69 0.0000411308 0.9999881208
70 0.0000104887 0.9999986095
71 0.0000013905 1.0000000000

 

On average it will take 41.36857386 balls to get a single bingo per card.

Double Bingo

 

Double Bingo

Balls Bingo on
Last Ball		 Bingo on or Before
Last Ball
8 0.0000000018 0.0000000018
9 0.0000000144 0.0000000162
10 0.0000000657 0.0000000819
11 0.0000002221 0.0000003041
12 0.0000006189 0.0000009230
13 0.0000015047 0.0000024277
14 0.0000033014 0.0000057291
15 0.0000066830 0.0000124121
16 0.0000126749 0.0000250870
17 0.0000227751 0.0000478622
18 0.0000390983 0.0000869605
19 0.0000645402 0.0001515006
20 0.0001029613 0.0002544619
21 0.0001593834 0.0004138453
22 0.0002401923 0.0006540376
23 0.0003533348 0.0010073725
24 0.0005084980 0.0015158704
25 0.0007172522 0.0022331226
26 0.0009931393 0.0032262619
27 0.0013516829 0.0045779448
28 0.0018102975 0.0063882423
29 0.0023880712 0.0087763135
30 0.0031053998 0.0118817133
31 0.0039834503 0.0158651635
32 0.0050434409 0.0209086044
33 0.0063057275 0.0272143319
34 0.0077887008 0.0350030328
35 0.0095075097 0.0445105425
36 0.0114726396 0.0559831821
37 0.0136883951 0.0696715772
38 0.0161513504 0.0858229276
39 0.0188488500 0.1046717776
40 0.0217576605 0.1264294381
41 0.0248428835 0.1512723216
42 0.0280572541 0.1793295757
43 0.0313409465 0.2106705222
44 0.0346220068 0.2452925289
45 0.0378175163 0.2831100452
46 0.0408355604 0.3239456056
47 0.0435780438 0.3675236494
48 0.0459443381 0.4134679875
49 0.0478356920 0.4613036795
50 0.0491602623 0.5104639418
51 0.0498385487 0.5603024906
52 0.0498089462 0.6101114368
53 0.0490330530 0.6591444898
54 0.0475003236 0.7066448133
55 0.0452316183 0.7518764316
56 0.0422811998 0.7941576315
57 0.0387367615 0.8328943930
58 0.0347171547 0.8676115476
59 0.0303676144 0.8979791620
60 0.0258524685 0.9238316305
61 0.0213455485 0.9451771790
62 0.0170187944 0.9621959734
63 0.0130298378 0.9752258112
64 0.0095096326 0.9847354438
65 0.0065514457 0.9912868895
66 0.0042026630 0.9954895525
67 0.0024608692 0.9979504217
68 0.0012754404 0.9992258621
69 0.0005554048 0.9997812669
70 0.0001835059 0.9999647728
71 0.0000352272 1.0000000000

 

On average it will take 49.77068987 balls to get a double bingo per card.

Triple Bingo

 

Triple Bingo

Balls Bingo on
Last Ball		 Bingo on or Before
Last Ball
12 0.0000000001 0.0000000001
13 0.0000000007 0.0000000008
14 0.0000000031 0.0000000039
15 0.0000000113 0.0000000152
16 0.0000000349 0.0000000501
17 0.0000000958 0.0000001458
18 0.0000002391 0.0000003849
19 0.0000005519 0.0000009368
20 0.0000011929 0.0000021297
21 0.0000024372 0.0000045669
22 0.0000047417 0.0000093086
23 0.0000088368 0.0000181454
24 0.0000158509 0.0000339962
25 0.0000274733 0.0000614696
26 0.0000461616 0.0001076311
27 0.0000753959 0.0001830270
28 0.0001199818 0.0003030088
29 0.0001863972 0.0004894060
30 0.0002831733 0.0007725792
31 0.0004212947 0.0011938739
32 0.0006145917 0.0018084656
33 0.0008800920 0.0026885576
34 0.0012382870 0.0039268446
35 0.0017132610 0.0056401056
36 0.0023326212 0.0079727267
37 0.0031271645 0.0110998912
38 0.0041302140 0.0152301051
39 0.0053765636 0.0206066687
40 0.0069009811 0.0275076498
41 0.0087362399 0.0362438897
42 0.0109106775 0.0471545673
43 0.0134453197 0.0605998870
44 0.0163506541 0.0769505411
45 0.0196231930 0.0965737341
46 0.0232420258 0.1198157599
47 0.0271656188 0.1469813787
48 0.0313291799 0.1783105586
49 0.0356429468 0.2139535054
50 0.0399917842 0.2539452896
51 0.0442364714 0.2981817609
52 0.0482170241 0.3463987851
53 0.0517583069 0.3981570920
54 0.0546780588 0.4528351508
55 0.0567972642 0.5096324150
56 0.0579525572 0.5675849722
57 0.0580100610 0.6255950332
58 0.0568797442 0.6824747774
59 0.0545290603 0.7370038377
60 0.0509943448 0.7879981825
61 0.0463882422 0.8343864247
62 0.0409013669 0.8752877916
63 0.0347965403 0.9100843320
64 0.0283943555 0.9384786874
65 0.0220495722 0.9605282596
66 0.0161189715 0.9766472311
67 0.0109228231 0.9875700543
68 0.0067039829 0.9942740371
69 0.0035907078 0.9978647449
70 0.0015712722 0.9994360171
71 0.0004899362 0.9999259534
72 0.0000740466 1.0000000000
73 0.0000000000 1.0000000000
74 0.0000000000 1.0000000000
75 0.0000000000 1.0000000000

 

On average it will take 54.78946365 balls to get a triple bingo per card.

Single Bingo with No Free Square

 

Single Bingo with No Free Square

Balls Bingo on
Last Ball		 Bingo on or Before
Last Ball
5 0.0000006953 0.0000006953
6 0.0000034764 0.0000041716
7 0.0000104291 0.0000146007
8 0.0000243346 0.0000389353
9 0.0000486688 0.0000876041
10 0.0000876012 0.0001752053
11 0.0001459908 0.0003211960
12 0.0002293786 0.0005505746
13 0.0003439739 0.0008945485
14 0.0004966322 0.0013911807
15 0.0006948216 0.0020860023
16 0.0009465740 0.0030325764
17 0.0012604186 0.0042929950
18 0.0016452913 0.0059382863
19 0.0021104183 0.0080487046
20 0.0026651680 0.0107138726
21 0.0033188678 0.0140327404
22 0.0040805825 0.0181133230
23 0.0049588523 0.0230721752
24 0.0059613873 0.0290335625
25 0.0070947220 0.0361282845
26 0.0083638302 0.0444921147
27 0.0097717060 0.0542638207
28 0.0113189209 0.0655827415
29 0.0130031666 0.0785859081
30 0.0148188010 0.0934047091
31 0.0167564152 0.1101611242
32 0.0188024453 0.1289635695
33 0.0209388544 0.1499024239
34 0.0231429138 0.1730453377
35 0.0253871137 0.1984324514
36 0.0276392322 0.2260716836
37 0.0298625942 0.2559342778
38 0.0320165421 0.2879508199
39 0.0340571399 0.3220079598
40 0.0359381210 0.3579460808
41 0.0376120797 0.3955581605
42 0.0390318944 0.4345900549
43 0.0401523535 0.4747424084
44 0.0409319406 0.5156743491
45 0.0413347161 0.5570090652
46 0.0413322169 0.5983412821
47 0.0409052786 0.6392465606
48 0.0400456748 0.6792922355
49 0.0387574563 0.7180496918
50 0.0370578736 0.7551075654
51 0.0349777718 0.7900853372
52 0.0325613591 0.8226466963
53 0.0298652766 0.8525119729
54 0.0269569311 0.8794689040
55 0.0239120954 0.9033809993
56 0.0208118339 0.9241928332
57 0.0177388714 0.9419317046
58 0.0147735798 0.9567052845
59 0.0119898201 0.9686951045
60 0.0094509236 0.9781460281
61 0.0072061336 0.9853521617
62 0.0052878380 0.9906399996
63 0.0037099080 0.9943499076
64 0.0024674013 0.9968173089
65 0.0015377958 0.9983551047
66 0.0008837793 0.9992388840
67 0.0004574502 0.9996963342
68 0.0002055778 0.9999019120
69 0.0000753625 0.9999772745
70 0.0000199444 0.9999972189
71 0.0000027811 1.0000000000
72 0.0000000000 1.0000000000
73 0.0000000000 1.0000000000
74 0.0000000000 1.0000000000
75 0.0000000000 1.0000000000

On average it will take 43.545581 balls to get a single bingo on a card with no free square.

Coverall

Coverall

Balls Bingo on
Last Ball		 Bingo on or Before
Last Ball
24 0.00000000000000000004 0.00000000000000000004
25 0.00000000000000000093 0.00000000000000000097
26 0.00000000000000001164 0.00000000000000001261
27 0.00000000000000010086 0.00000000000000011347
28 0.00000000000000068079 0.00000000000000079426
29 0.00000000000000381245 0.00000000000000460671
30 0.00000000000001842684 0.00000000000002303355
31 0.00000000000007897218 0.00000000000010200573
32 0.00000000000030601718 0.00000000000040802291
33 0.00000000000108806109 0.00000000000149608400
34 0.00000000000359060160 0.00000000000508668560
35 0.00000000001109822313 0.00000000001618490874
36 0.00000000003236981747 0.00000000004855472621
37 0.00000000008963949453 0.00000000013819422074
38 0.00000000023690437841 0.00000000037509859915
39 0.00000000060015775864 0.00000000097525635779
40 0.00000000146288453669 0.00000000243814089448
41 0.00000000344208126279 0.00000000588022215727
42 0.00000000784029620969 0.00000001372051836696
43 0.00000001733118109511 0.00000003105169946207
44 0.00000003726203935449 0.00000006831373881656
45 0.00000007807284436178 0.00000014638658317834
46 0.00000015969445437637 0.00000030608103755472
47 0.00000031938890875275 0.00000062546994630747
48 0.00000062546994630747 0.00000125093989261493
49 0.00000120090229691033 0.00000245184218952526
50 0.00000226323894417717 0.00000471508113370243
51 0.00000419118322995771 0.00000890626436366014
52 0.00000763394088313726 0.00001654020524679740
53 0.00001368844572148750 0.00003022865096828490
54 0.00002418292077462790 0.00005441157174291290
55 0.00004212508780096470 0.00009653665954387760
56 0.00007240249465790830 0.00016893915420178600
57 0.00012286483941948100 0.00029180399362126600
58 0.00020597928961501200 0.00049778328323627800
59 0.00034133710850487700 0.00083912039174115500
60 0.00055941359449410300 0.00139853398623526000
61 0.00090715718026070800 0.00230569116649597000
62 0.00145622599989219000 0.00376191716638816000
63 0.00231502594854655000 0.00607694311493471000
64 0.00364616586896083000 0.00972310898389554000
65 0.00569157599057300000 0.01541468497446850000
66 0.00880839141398202000 0.02422307638845060000
67 0.01351985658890260000 0.03774293297735320000
68 0.02058705435128360000 0.05832998732863680000
69 0.03110932657527300000 0.08943931390390970000
70 0.04666398986290940000 0.13610330376681900000
71 0.06949955937029060000 0.20560286313711000000
72 0.10280143156855500000 0.30840429470566500000
73 0.15105516475379500000 0.45945945945945900000
74 0.22054054054054100000 0.68000000000000000000
75 0.32000000000000000000 1.00000000000000000000

On average it will take 72.96 balls to get a coverall per card.

Single Hard Way Bingo

Single Hard Way

Balls Bingo on
Last Ball		 Bingo on or Before
Last Ball
5 0.0000004635 0.0000004635
6 0.0000023176 0.0000027811
7 0.0000069527 0.0000097338
8 0.0000162231 0.0000259569
9 0.0000324460 0.0000584029
10 0.0000584018 0.0001168047
11 0.0000973324 0.0002141371
12 0.0001529384 0.0003670755
13 0.0002293741 0.0005964496
14 0.0003312395 0.0009276891
15 0.0004635678 0.0013912569
16 0.0006318070 0.0020230639
17 0.0008417943 0.0028648582
18 0.0010997211 0.0039645793
19 0.0014120871 0.0053766664
20 0.0017856405 0.0071623069
21 0.0022273027 0.0093896096
22 0.0027440754 0.0121336850
23 0.0033429263 0.0154766113
24 0.0040306530 0.0195072643
25 0.0048137229 0.0243209872
26 0.0056980862 0.0300190734
27 0.0066889635 0.0367080368
28 0.0077906066 0.0444986434
29 0.0090060345 0.0535046779
30 0.0103367469 0.0638414249
31 0.0117824200 0.0756238448
32 0.0133405897 0.0889644345
33 0.0150063323 0.1039707668
34 0.0167719509 0.1207427177
35 0.0186266827 0.1393694004
36 0.0205564399 0.1599258403
37 0.0225436038 0.1824694442
38 0.0245668899 0.2070363340
39 0.0266013047 0.2336376387
40 0.0286182169 0.2622558556
41 0.0305855616 0.2928414172
42 0.0324682002 0.3253096174
43 0.0342284508 0.3595380682
44 0.0358268037 0.3953648719
45 0.0372228271 0.4325876991
46 0.0383762640 0.4709639630
47 0.0392483071 0.5102122701
48 0.0398030322 0.5500153023
49 0.0400089519 0.5900242542
50 0.0398406418 0.6298648960
51 0.0392803740 0.6691452700
52 0.0383196783 0.7074649483
53 0.0369607397 0.7444256880
54 0.0352175273 0.7796432154
55 0.0331165455 0.8127597609
56 0.0306970947 0.8434568556
57 0.0280109358 0.8714677914
58 0.0251212678 0.8965890592
59 0.0221009525 0.9186900117
60 0.0190299571 0.9377199688
61 0.0159920384 0.9537120072
62 0.0130707515 0.9667827587
63 0.0103449428 0.9771277015
64 0.0078839692 0.9850116707
65 0.0057429755 0.9907546462
66 0.0039586442 0.9947132904
67 0.0025459092 0.9972591996
68 0.0014961718 0.9987553714
69 0.0007775636 0.9995329350
70 0.0003377441 0.9998706791
71 0.0001095751 0.9999802542
72 0.0000197458 1.0000000000

On average it will take 46.4899160402 balls to get a single hard way bingo per card.

Double Hard Way Bingo

Double Hard Way

Balls Bingo on
Last Ball		 Bingo on or Before
Last Ball
9 0.0000000001 0.0000000001
10 0.0000000012 0.0000000013
11 0.0000000059 0.0000000072
12 0.0000000218 0.0000000290
13 0.0000000662 0.0000000952
14 0.0000001743 0.0000002695
15 0.0000004115 0.0000006810
16 0.0000008919 0.0000015729
17 0.0000018038 0.0000033767
18 0.0000034442 0.0000068208
19 0.0000062642 0.0000130850
20 0.0000109277 0.0000240127
21 0.0000183836 0.0000423963
22 0.0000299553 0.0000723516
23 0.0000474471 0.0001197987
24 0.0000732685 0.0001930673
25 0.0001105764 0.0003036437
26 0.0001634331 0.0004670768
27 0.0002369779 0.0007040547
28 0.0003376065 0.0010416613
29 0.0004731509 0.0015148122
30 0.0006530511 0.0021678633
31 0.0008885042 0.0030563675
32 0.0011925775 0.0042489450
33 0.0015802642 0.0058292091
34 0.0020684627 0.0078976718
35 0.0026758547 0.0105735265
36 0.0034226567 0.0139961832
37 0.0043302203 0.0183264035
38 0.0054204565 0.0237468600
39 0.0067150636 0.0304619236
40 0.0082345436 0.0386964672
41 0.0099970019 0.0486934692
42 0.0120167345 0.0607102036
43 0.0143026248 0.0750128284
44 0.0168563903 0.0918692187
45 0.0196707382 0.1115399569
46 0.0227275180 0.1342674749
47 0.0259959788 0.1602634537
48 0.0294312706 0.1896947243
49 0.0329733459 0.2226680702
50 0.0365464411 0.2592145113
51 0.0400593272 0.2992738385
52 0.0434065180 0.3426803565
53 0.0464706113 0.3891509678
54 0.0491259048 0.4382768726
55 0.0512433712 0.4895202438
56 0.0526969993 0.5422172431
57 0.0533713964 0.5955886396
58 0.0531704122 0.6487590518
59 0.0520263822 0.7007854340
60 0.0499094077 0.7506948417
61 0.0468358962 0.7975307378
62 0.0428754009 0.8404061387
63 0.0381546396 0.8785607783
64 0.0328574729 0.9114182512
65 0.0272196204 0.9386378716
66 0.0215170339 0.9601549055
67 0.0160471970 0.9762021025
68 0.0111032396 0.9873053421
69 0.0069417249 0.9942470670
70 0.0037463731 0.9979934401
71 0.0015918987 0.9995853388
72 0.0004146612 1.0000000000

On average it will take 55.0035663910 balls to get a double hard way bingo per card.

Triple Hard Way Bingo

Triple Hard Way

Balls Bingo on
Last Ball		 Bingo on or Before
Last Ball
16 0.0000000002 0.0000000002
17 0.0000000007 0.0000000009
18 0.0000000023 0.0000000032
19 0.0000000070 0.0000000103
20 0.0000000191 0.0000000294
21 0.0000000477 0.0000000771
22 0.0000001111 0.0000001882
23 0.0000002439 0.0000004321
24 0.0000005085 0.0000009406
25 0.0000010134 0.0000019540
26 0.0000019399 0.0000038939
27 0.0000035823 0.0000074762
28 0.0000064043 0.0000138805
29 0.0000111166 0.0000249971
30 0.0000187830 0.0000437801
31 0.0000309585 0.0000747386
32 0.0000498678 0.0001246064
33 0.0000786284 0.0002032348
34 0.0001215231 0.0003247579
35 0.0001843249 0.0005090828
36 0.0002746708 0.0007837536
37 0.0004024805 0.0011862341
38 0.0005804049 0.0017666390
39 0.0008242836 0.0025909227
40 0.0011535803 0.0037445030
41 0.0015917504 0.0053362533
42 0.0021664871 0.0075027404
43 0.0029097745 0.0104125149
44 0.0038576673 0.0142701822
45 0.0050497038 0.0193198860
46 0.0065278566 0.0258477426
47 0.0083349214 0.0341826640
48 0.0105122593 0.0446949233
49 0.0130968243 0.0577917477
50 0.0161174496 0.0739091973
51 0.0195904165 0.0934996138
52 0.0235144079 0.1170140217
53 0.0278650399 0.1448790616
54 0.0325892844 0.1774683460
55 0.0376002256 0.2150685716
56 0.0427727408 0.2578413124
57 0.0479408366 0.3057821490
58 0.0528975070 0.3586796560
59 0.0573980671 0.4160777231
60 0.0611679493 0.4772456724
61 0.0639158709 0.5411615433
62 0.0653530654 0.6065146087
63 0.0652188485 0.6717334572
64 0.0633121256 0.7350455828
65 0.0595274582 0.7945730410
66 0.0538929599 0.8484660009
67 0.0466055215 0.8950715224
68 0.0380566599 0.9331281822
69 0.0288396509 0.9619678332
70 0.0197256102 0.9816934434
71 0.0115929937 0.9932864371
72 0.0052918672 0.9985783043
73 0.0014216957 1.0000000000

On average it will take 60.1705499275 balls to get a triple hard way bingo per card.

Six Pack

Six Pack

Balls Bingo on
Last Ball		 Bingo on or Before
Last Ball
5 0.0000006953 0.0000006953
6 0.0000035360 0.0000042312
7 0.0000107665 0.0000149978
8 0.0000254448 0.0000404426
9 0.0000514410 0.0000918835
10 0.0000934159 0.0001852994
11 0.0001567803 0.0003420797
12 0.0002476339 0.0005897136
13 0.0003726862 0.0009623997
14 0.0005391587 0.0015015585
15 0.0007546703 0.0022562287
16 0.0010271047 0.0032833335
17 0.0013644645 0.0046477979
18 0.0017747082 0.0064225062
19 0.0022655766 0.0086880828
20 0.0028444054 0.0115324882
21 0.0035179293 0.0150504175
22 0.0042920780 0.0193424954
23 0.0051717657 0.0245142611
24 0.0061606794 0.0306749406
25 0.0072610663 0.0379360069
26 0.0084735255 0.0464095324
27 0.0097968078 0.0562063402
28 0.0112276284 0.0674339685
29 0.0127604972 0.0801944657
30 0.0143875725 0.0945820382
31 0.0160985450 0.1106805832
32 0.0178805576 0.1285611408
33 0.0197181689 0.1482793097
34 0.0215933674 0.1698726771
35 0.0234856425 0.1933583196
36 0.0253721218 0.2187304415
37 0.0272277775 0.2459582189
38 0.0290257109 0.2749839298
39 0.0307375174 0.3057214472
40 0.0323337352 0.3380551824
41 0.0337843784 0.3718395608
42 0.0350595517 0.4068991126
43 0.0361301432 0.4430292558
44 0.0369685838 0.4799978396
45 0.0375496635 0.5175475031
46 0.0378513839 0.5553988870
47 0.0378558282 0.5932547151
48 0.0375500188 0.6308047340
49 0.0369267336 0.6677314675
50 0.0359852422 0.7037167097
51 0.0347319252 0.7384486349
52 0.0331807316 0.7716293664
53 0.0313534309 0.8029827973
54 0.0292796174 0.8322624147
55 0.0269964237 0.8592588384
56 0.0245479097 0.8838067481
57 0.0219840999 0.9057908480
58 0.0193596534 0.9251505014
59 0.0167321721 0.9418826735
60 0.0141601669 0.9560428404
61 0.0117007332 0.9677435736
62 0.0094070129 0.9771505865
63 0.0073255539 0.9844761404
64 0.0054937147 0.9899698550
65 0.0039372960 0.9939071511
66 0.0026686169 0.9965757679
67 0.0016852805 0.9982610485
68 0.0009698942 0.9992309426
69 0.0004910075 0.9997219501
70 0.0002055097 0.9999274598
71 0.0000626673 0.9999901271
72 0.0000098729 1.0000000000

On average it will take 44.4584604938 balls to get a six pack per card.

Nine Pack

Nine Pack

Balls Bingo on
Last Ball		 Bingo on or Before
Last Ball
8 0.0000000005 0.0000000005
9 0.0000000043 0.0000000048
10 0.0000000192 0.0000000240
11 0.0000000640 0.0000000880
12 0.0000001760 0.0000002640
13 0.0000004223 0.0000006864
14 0.0000009147 0.0000016011
15 0.0000018284 0.0000034295
16 0.0000034254 0.0000068549
17 0.0000060830 0.0000129379
18 0.0000103267 0.0000232646
19 0.0000168686 0.0000401333
20 0.0000266511 0.0000667844
21 0.0000408951 0.0001076795
22 0.0000611531 0.0001688326
23 0.0000893658 0.0002581984
24 0.0001279218 0.0003861202
25 0.0001797192 0.0005658394
26 0.0002482265 0.0008140659
27 0.0003375429 0.0011516088
28 0.0004524530 0.0016040618
29 0.0005984757 0.0022025375
30 0.0007819018 0.0029844393
31 0.0010098184 0.0039942577
32 0.0012901149 0.0052843726
33 0.0016314663 0.0069158389
34 0.0020432891 0.0089591280
35 0.0025356648 0.0114947928
36 0.0031192249 0.0146140178
37 0.0038049929 0.0184190107
38 0.0046041776 0.0230231883
39 0.0055279132 0.0285511015
40 0.0065869424 0.0351380439
41 0.0077912383 0.0429292822
42 0.0091495650 0.0520788473
43 0.0106689768 0.0627478241
44 0.0123542576 0.0751020816
45 0.0142073078 0.0893093894
46 0.0162264874 0.1055358768
47 0.0184059274 0.1239418043
48 0.0207348299 0.1446766342
49 0.0231967785 0.1678734126
50 0.0257690891 0.1936425017
51 0.0284222367 0.2220647384
52 0.0311194001 0.2531841385
53 0.0338161723 0.2870003108
54 0.0364604925 0.3234608033
55 0.0389928587 0.3624536620
56 0.0413468848 0.4038005467
57 0.0434502700 0.4472508167
58 0.0452262455 0.4924770622
59 0.0465955616 0.5390726238
60 0.0474790709 0.5865516947
61 0.0478009470 0.6343526417
62 0.0474925628 0.6818452045
63 0.0464970194 0.7283422240
64 0.0447742836 0.7731165076
65 0.0423068418 0.8154233494
66 0.0391057196 0.8545290691
67 0.0352166431 0.8897457122
68 0.0307260310 0.9204717432
69 0.0257664064 0.9462381496
70 0.0205206988 0.9667588484
71 0.0152247791 0.9819836275
72 0.0101674277 0.9921510552
73 0.0056867827 0.9978378378
74 0.0021621622 1.0000000000

On average it will take 57.6752567694 balls to get a nine-pack bingo per card.

Dual Daub

The next table is for a cover-all on a dual-daub card. A dual-daub card is one with two numbers on each square. If at least one of the two numbers is called then that square on the card is considered marked.

Coverall on Dual-Duab Card

Balls Bingo on
Last Ball		 Bingo on or Before
Last Ball
24 0.000000000001 0.000000000001
25 0.000000000012 0.000000000012
26 0.000000000111 0.000000000123
27 0.000000000722 0.000000000846
28 0.000000003653 0.000000004499
29 0.000000015282 0.000000019781
30 0.000000055038 0.000000074819
31 0.000000175275 0.000000250093
32 0.000000503302 0.000000753396
33 0.000001322379 0.000002075775
34 0.000003215539 0.000005291314
35 0.000007302455 0.000012593769
36 0.000015603066 0.000028196835
37 0.000031559360 0.000059756195
38 0.000060735751 0.000120491947
39 0.000111696679 0.000232188626
40 0.000197027019 0.000429215645
41 0.000334418464 0.000763634108
42 0.000547697933 0.001311332042
43 0.000867629575 0.002178961616
44 0.001332288975 0.003511250591
45 0.001986796244 0.005498046835
46 0.002882211528 0.008380258363
47 0.004073447014 0.012453705377
48 0.005616133603 0.018069838980
49 0.007562492655 0.025632331635
50 0.009956392660 0.035588724295
51 0.012827902127 0.048416626422
52 0.016187765772 0.064604392195
53 0.020022313729 0.084626705924
54 0.024289348154 0.108916054078
55 0.028915528796 0.137831582874
56 0.033795695989 0.171627278863
57 0.038794431200 0.210421710063
58 0.043749973813 0.254171683876
59 0.048480406433 0.302652090309
60 0.052791811272 0.355443901581
61 0.056487909803 0.411931811385
62 0.059380547344 0.471312358729
63 0.061300289862 0.532612648591
64 0.062106371734 0.594719020325
65 0.061695273082 0.656414293407
66 0.060007308706 0.716421602113
67 0.057030766168 0.773452368281
68 0.052803322100 0.826255690381
69 0.047410674090 0.873666364471
70 0.040982531000 0.914648895471
71 0.033686289029 0.948335184500
72 0.025718869554 0.974054054054
73 0.017297297297 0.991351351351
74 0.008648648649 1.000000000000

On average it will take 62.46183333 balls to get a cover-all on a dual-daub card.

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Champagne Glass

The next table is for any 12-mark pattern, not counting the free square, like the champagne glass. The player must cover the pattern in a specific way (no orientations) to win.

Champagne Glass Example

B I N G O
O O O O O
  O O O  
    O    
    O    
  O O O  

Champagne Glass — 12-Mark Pattern

Balls Bingo on
Last Ball		 Bingo on or Before
Last Ball
12 0.00000000000004 0.00000000000004
13 0.00000000000046 0.00000000000050
14 0.00000000000299 0.00000000000348
15 0.00000000001393 0.00000000001742
16 0.00000000005225 0.00000000006967
17 0.00000000016720 0.00000000023687
18 0.00000000047374 0.00000000071061
19 0.00000000121820 0.00000000192881
20 0.00000000289321 0.00000000482202
21 0.00000000642936 0.00000001125139
22 0.00000001350166 0.00000002475305
23 0.00000002700333 0.00000005175638
24 0.00000005175638 0.00000010351276
25 0.00000009555024 0.00000019906301
26 0.00000017062544 0.00000036968844
27 0.00000029575075 0.00000066543920
28 0.00000049907940 0.00000116451860
29 0.00000082201313 0.00000198653172
30 0.00000132435448 0.00000331088620
31 0.00000209108602 0.00000540197222
32 0.00000324118333 0.00000864315556
33 0.00000493894603 0.00001358210159
34 0.00000740841905 0.00002099052064
35 0.00001095157599 0.00003194209663
36 0.00001597104832 0.00004791314495
37 0.00002299830957 0.00007091145452
38 0.00003272836363 0.00010363981815
39 0.00004606214140 0.00014970195955
40 0.00006415798266 0.00021385994221
41 0.00008849376919 0.00030235371140
42 0.00012094148456 0.00042329519596
43 0.00016385620489 0.00058715140084
44 0.00022018177532 0.00080733317616
45 0.00029357570042 0.00110090887658
46 0.00038855607409 0.00148946495067
47 0.00051067369737 0.00200013864805
48 0.00066671288268 0.00266685153073
49 0.00086492482078 0.00353177635150
50 0.00111529779521 0.00464707414671
51 0.00142986896822 0.00607694311493
52 0.00182308293448 0.00790002604942
53 0.00231220274617 0.01021222879559
54 0.00291777965588 0.01313000845147
55 0.00366418840506 0.01679419685653
56 0.00458023550633 0.02137443236285
57 0.00569984863009 0.02707428099295
58 0.00706285591120 0.03413713690415
59 0.00871586474149 0.04285300164564
60 0.01071325041141 0.05356625205705
61 0.01311826580989 0.06668451786693
62 0.01600428428806 0.08268880215500
63 0.01945618874235 0.10214499089735
64 0.02357192097631 0.12571691187366
65 0.02846420646196 0.15418111833562
66 0.03426247074125 0.18844358907687
67 0.04111496488950 0.22955855396637
68 0.04919111870708 0.27874967267345
69 0.05868414161546 0.33743381428892
70 0.06981389261150 0.40724770690042
71 0.08283004208144 0.49007774898186
72 0.09801554979637 0.58809329877823
73 0.11569048500555 0.70378378378378
74 0.13621621621622 0.84000000000000
75 0.16000000000000 1.00000000000000

On average it will take 70.15384615 balls to get any 12-mark pattern (not counting the free square), like the champagne glass, on a single card.

Please note that you can find the probabilities for covering any pattern by just counting the number of marks required and using the tables in my More Bingo Probabilities page.

For more bingo probability tables, please see my Probabilities in Bingo II page.

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.

The Wizard's Information About Bingo

Written by: Michael Shackleford