Methodology: The probability of covering m marks in c calls is combin(15,m)*combin(75,c-m)/combin(90,m). Using that you can find the probability of covering a card as combin(75,90-m)/combin(90,m). To get the probability of covering 1 or 2 rows I determined the probability that m marks would cover 1 or 2 rows. The chart below shows those probabilities, which is based on basic probability.

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90 Number Bingo -- Analysis

Last Update: Feb 23, 2005

Unlike American bingo with a 5 by 5 card with numbers from 1 to 75, in Europe and South America bingo is often played with a 3 by 9 card with numbers from 1 to 90. Below is an example.

As the example shows the card contains 3 rows and 9 columns. On each row are exactly 5 numbers. The other four cells in each row are blank, or free squares. From other examples I have seen the first row contains the numbers 1 to 10, the second 11 to 20, and so on, but mathematically this doesn't matter. Winning events I have heard of all are based on covering rows only, so mathematically speaking the game could played on a 3 by 5 card with all numbers covered, the odds would be the same.

The purpose of this appendix is to show the probability of covering (1) at least one row, (2) at least 2 rows, and (3) all three rows, in any given number of calls from 5 to 90. For example, the table shows the probability of covering at least one row in 50 calls on any one card is 0.139289864, or 13.93%.

Calls		5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90

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90 Number Bingo Probabilities

Two Rows
Probability

Three Rows
Probability

One Row Inverse
Probability

Two Rows Inverse
Probability

Three Rows Inverse
Probability

0.0000000683

1 in 14649756

0.0000004096

1 in 2441626

0.0000014335

1 in 697607

0.0000038226

1 in 261603

0.0000086008

1 in 116268

0.0000172017

1 in 58134

1 in 1906881827301

0.0000315364

1 in 31709

1 in 173352893391

0.0000540623

1 in 18497

1 in 28892148899

0.0000878511

0.0000000001

1 in 11383

1 in 6667418977

0.000136657

0.0000000005

1 in 7318

1 in 1904976850

0.0002049848

0.0000000016

1 in 4878

1 in 634992301

1 in 45795673964460800

0.0002981578

0.0000000042

1 in 3354

1 in 238122146

1 in 2862229622778800

0.0004223859

0.0000000102

1 in 2368

1 in 98050336

1 in 336732896797506

0.0005848332

0.0000000229

1 in 1710

1 in 43577969

1 in 56122149466251

0.0007936849

0.0000000484

1 in 1260

1 in 20642236

1 in 11815189361316

0.0010582143

0.0000000969

1 in 945

1 in 10321154

1 in 2953797340329

0.0013888484

0.000000185

1 in 720

1 in 5406350

1 in 843942097237

0.0017972335

0.0000003391

1 in 556

1 in 2948945

1 in 268527030939

0.0022962984

0.0000005999

1 in 435

1 in 1666818

1 in 93400706414

0.0029003168

0.0000010285

1 in 345

1 in 972330

1 in 35025264905

0.0036249674

0.000001714

0.0000000001

1 in 276

1 in 583414

1 in 14010105962

0.0044873918

0.0000027852

0.0000000002

1 in 223

1 in 359038

1 in 5927352522

0.0055062482

0.0000044234

0.0000000004

1 in 182

1 in 226072

1 in 2634378899

0.006701763

0.0000068803

0.0000000008

1 in 149

1 in 145342

1 in 1223104489

0.008095776

0.0000105007

0.0000000017

1 in 124

1 in 95232

1 in 590464236

0.0097117813

0.0000157493

0.0000000034

1 in 103

1 in 63495

1 in 295232118

0.0115749612

0.0000232459

0.0000000066

1 in 86

1 in 43018

1 in 152377867

0.0137122121

0.0000338066

0.0000000124

1 in 73

1 in 29580

1 in 80950742

0.0161521615

0.0000484952

0.0000000226

1 in 62

1 in 20621

1 in 44154950

0.0189251748

0.0000686847

0.0000000405

1 in 53

1 in 14559

1 in 24674825

0.0220633488

0.0000961302

0.0000000709

1 in 45

1 in 10403

1 in 14099900

0.0256004928

0.0001330566

0.0000001216

1 in 39

1 in 7516

1 in 8224942

0.0295720915

0.0001822611

0.0000002045

1 in 34

1 in 5487

1 in 4890506

0.0340152517

0.0002472336

0.0000003378

1 in 29

1 in 4045

1 in 2960043

0.0389686274

0.0003322973

0.000000549

1 in 26

1 in 3009

1 in 1821565

0.0444723213

0.0004427703

0.0000008784

1 in 22

1 in 2259

1 in 1138478

0.0505677613

0.0005851526

0.0000013851

1 in 20

1 in 1709

1 in 721962

0.0572975481

0.0007673395

0.0000021546

1 in 17

1 in 1303

1 in 464118

0.0647052697

0.0009988639

0.0000033089

1 in 15

1 in 1001

1 in 302217

0.0728352824

0.0012911709

0.0000050204

1 in 14

1 in 774

1 in 199188

0.081732452

0.0016579252

0.0000075306

1 in 12

1 in 603

1 in 132792

0.0914418539

0.002115356

0.0000111744

1 in 11

1 in 473

1 in 89490

0.1020084273

0.0026826381

0.0000164124

1 in 10

1 in 373

1 in 60930

0.1134765801

0.0033823132

0.0000238726

1 in 8.8

1 in 296

1 in 41889

0.1258897424

0.0042407513

0.0000344046

1 in 7.9

1 in 236

1 in 29066

0.1392898636

0.0052886518

0.0000491494

1 in 7.2

1 in 189

1 in 20346

0.1537168505

0.0065615844

0.0000696283

1 in 6.5

1 in 152

1 in 14362

0.1692079444

0.0081005673

0.000097856

1 in 5.9

1 in 123

1 in 10219

0.1857970345

0.0099526798

0.0001364834

1 in 5.4

1 in 100

1 in 7327

0.203513905

0.0121717032

0.000188977

1 in 4.9

1 in 82

1 in 5292

0.2223834172

0.0148187832

0.0002598433

1 in 4.5

1 in 67

1 in 3848

0.2424246244

0.0179631039

0.000354908

1 in 4.1

1 in 56

1 in 2818

0.2636498229

0.0216825599

0.0004816608

1 in 3.8

1 in 46

1 in 2076

0.2860635412

0.0260644094

0.000649682

1 in 3.5

1 in 38

1 in 1539

0.309661472

0.031205888

0.0008711645

1 in 3.2

1 in 32

1 in 1148

0.3344293554

0.037214755

0.0011615527

1 in 3

1 in 27

1 in 861

0.3603418208

0.0442097423

0.0015403199

1 in 2.8

1 in 23

1 in 649

0.387361203

0.0523208636

0.0020319114

1 in 2.6

1 in 19

1 in 492

0.4154363465

0.0616895391

0.0026668837

1 in 2.4

1 in 16

1 in 375

0.4445014202

0.0724684766

0.0034832766

1 in 2.2

1 in 14

1 in 287

0.4744747683

0.0848212433

0.0045282596

1 in 2.1

1 in 12

1 in 221

0.5052578274

0.0989214474

0.0058601006

1 in 2

1 in 10

1 in 171

0.5367341493

0.1149514356

0.0075505143

1 in 1.86

1 in 8.7

1 in 132

0.568768574

0.1331003983

0.0096874523

1 in 1.76

1 in 7.5

1 in 103

0.6012066069

0.153561752

0.0123784113

1 in 1.66

1 in 6.5

1 in 81

0.6338740649

0.1765296538

0.0157543416

1 in 1.58

1 in 5.7

1 in 63

0.6665770642

0.2021944733

0.0199742546

1 in 1.5

1 in 4.9

1 in 50

0.6991024401

0.2307370275

0.0252306373

1 in 1.43

1 in 4.3

1 in 40

0.7312186968

0.262321349

0.0317558022

1 in 1.37

1 in 3.8

1 in 31

0.7626776074

0.2970857299

0.0398293112

1 in 1.31

1 in 3.4

1 in 25

0.7932165977

0.3351317439

0.049786639

1 in 1.26

1 in 3

1 in 20

0.8225620687

0.3765109088

0.0620292551

1 in 1.22

1 in 2.7

1 in 16

0.8504338369

0.4212086067

0.077036333

1 in 1.18

1 in 2.4

1 in 13

0.8765508925

0.4691248258

0.095378317

1 in 1.14

1 in 2.1

1 in 10

0.9006387073

0.5200512338

0.1177326101

1 in 1.11

1 in 1.92

1 in 8.5

0.9224383526

0.5736440281

0.1449016739

1 in 1.08

1 in 1.74

1 in 6.9

0.941717722

0.6293919373

0.1778338726

1 in 1.06

1 in 1.59

1 in 5.6

0.9582851926

0.686578675

0.2176474261

1 in 1.04

1 in 1.46

1 in 4.6

0.9720060987

0.7442390568

0.2656578878

1 in 1.03

1 in 1.34

1 in 3.8

0.9828224403

0.801107902

0.3234096025

1 in 1.02

1 in 1.25

1 in 3.1

0.9907762969

0.8555607343

0.3927116602

1 in 1.01

1 in 1.17

1 in 2.5

0.9960374767

0.9055451845

0.4756789123

1 in 1.004

1 in 1.1

1 in 2.1

0.9989359891

0.9485018727

0.5747786857

1 in 1.001

1 in 1.05

1 in 1.74

0.9812734082

0.6928838951

1 in 1

1 in 1.02

1 in 1.44

0.8333333333

1 in 1

1 in 1.2

1 in 1

Marks	0 Rows	1 Row	2 Rows	3 Rows	Total
Rows Covered by Number of Marks
5	0.999001	0.000999	0	0	1
6	0.994006	0.005994	0	0	1
7	0.979021	0.020979	0	0	1
8	0.944056	0.055944	0	0	1
9	0.874126	0.125874	0	0	1
10	0.749251	0.24975	0.000999	0	1
11	0.549451	0.43956	0.010989	0	1
12	0.274725	0.659341	0.065934	0	1
13	0	0.714286	0.285714	0	1
14	0	0	1	0	1
15	0	0	0	1	1