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Durango Bill's

Bingo Probabilities



Bingo Probability Statistics Analysis
Data for a Complete Board Cover


   The 2nd column in the table below shows the single-board probability that all 24 cells (plus the center cell) will be filled when the "Qth" number is called. The 3rd column shows the single-board cumulative probability that all 24 cells (plus the center cell) will be filled on or before the "Qth" number is called.

   If there are 1,000 boards in play, the 4th column in the table below shows the probability that all 24 cells (plus the center cell) will be covered by at least one board when the "Qth" number is called. If there are 1,000 boards in play, the 5th column shows the cumulative probability that at least one board will have a complete cover on or before the time the "Qth" number is called.

For example, when the 64th number is called in a Bingo game, we get the following probabilities:

1) A single board has a 0.00364617 probability of completely covering the board when the 64th number is called.
2) A single board has a 0.00972311 probability of completely covering the board on or before the 64th number.
3) If 1,000 boards are in play, there is a 0.0021959 probability that at least one board will get a complete cover when the 64th number is called.
4) If 1,000 boards are in play, there is a 0.99994 probability that that at least one board will have a complete cover on or before the time the 64th number is called.

(Note, if 1,000 boards are in play, the game will probably end before the 64th number is called.)

To calculate the probability that at least 1 board will be completely covered on or before the "Qth" number is called when "N" boards are in play, go down to the "Qth" row and insert the value for "Cumulative Prob. Complete Cover This Turn 1 Board in the equation:

Prob. of a complete cover with N boards in play = 1.0 - (1.0 - Cum. Prob. 1 board)N

For example, the probability that at least 1 complete cover will occur on at least 1 board on or before the 60th call when 1000 boards are in play
= 1.0 - (1.0 - 0.00139853)1000
= 0.75328



Computer program and data by Bill Butler

Quantity    Prob. of a    Cumulative Prob.    Prob. of a    Cumulative Prob.
of Bingo  Complete Cover   Complete Cover   Complete Cover   Complete Cover
Numbers     This  Turn       This  Turn       This  Turn       This  Turn
Picked       1  Board         1  Board       1000  Boards      1000 Boards
---------------------------------------------------------------------------
    1                 0                0                0                0
    2                 0                0                0                0
    3                 0                0                0                0
    4                 0                0                0                0
    5                 0                0                0                0
    6                 0                0                0                0
    7                 0                0                0                0
    8                 0                0                0                0
    9                 0                0                0                0
   10                 0                0                0                0
   11                 0                0                0                0
   12                 0                0                0                0
   13                 0                0                0                0
   14                 0                0                0                0
   15                 0                0                0                0
   16                 0                0                0                0
   17                 0                0                0                0
   18                 0                0                0                0
   19                 0                0                0                0
   20                 0                0                0                0
   21                 0                0                0                0
   22                 0                0                0                0
   23                 0                0                0                0
   24      3.87917e-020     3.87917e-020      3.8792e-017      3.8792e-017
   25      9.31001e-019     9.69793e-019        9.31e-016      9.6979e-016
   26      1.16375e-017     1.26073e-017      1.1638e-014      1.2607e-014
   27      1.00858e-016     1.13466e-016      1.0086e-013      1.1347e-013
   28      6.80795e-016      7.9426e-016      6.8079e-013      7.9426e-013
   29      3.81245e-015     4.60671e-015      3.8124e-012      4.6067e-012
   30      1.84268e-014     2.30336e-014      1.8427e-011      2.3034e-011
   31      7.89722e-014     1.02006e-013      7.8972e-011      1.0201e-010
   32      3.06017e-013     4.08023e-013      3.0602e-010      4.0802e-010
   33      1.08806e-012     1.49608e-012      1.0881e-009      1.4961e-009
   34       3.5906e-012     5.08669e-012      3.5906e-009      5.0867e-009
   35      1.10982e-011     1.61849e-011      1.1098e-008      1.6185e-008
   36      3.23698e-011     4.85547e-011       3.237e-008      4.8555e-008
   37      8.96395e-011     1.38194e-010      8.9639e-008      1.3819e-007
   38      2.36904e-010     3.75099e-010       2.369e-007       3.751e-007
   39      6.00158e-010     9.75256e-010      6.0016e-007      9.7526e-007
   40      1.46288e-009     2.43814e-009      1.4629e-006      2.4381e-006
   41      3.44208e-009     5.88022e-009      3.4421e-006      5.8802e-006
   42       7.8403e-009     1.37205e-008      7.8402e-006       1.372e-005
   43      1.73312e-008     3.10517e-008      1.7331e-005      3.1051e-005
   44       3.7262e-008     6.83137e-008       3.726e-005      6.8311e-005
   45      7.80728e-008     1.46387e-007      7.8064e-005       0.00014638
   46      1.59694e-007     3.06081e-007       0.00015966       0.00030603
   47      3.19389e-007      6.2547e-007       0.00031924       0.00062527
   48       6.2547e-007     1.25094e-006       0.00062488        0.0012502
   49       1.2009e-006     2.45184e-006        0.0011987        0.0024488
   50      2.26324e-006     4.71508e-006        0.0022552         0.004704
   51      4.19118e-006     8.90626e-006        0.0041628        0.0088668
   52      7.63394e-006     1.65402e-005        0.0075375         0.016404
   53      1.36884e-005     3.02287e-005         0.013372         0.029777
   54      2.41829e-005     5.44116e-005         0.023182         0.052959
   55      4.21251e-005     9.65367e-005         0.039068         0.092028
   56      7.24025e-005      0.000168939         0.063424          0.15545
   57       0.000122865      0.000291804         0.097665          0.25312
   58       0.000205979      0.000497783           0.1391          0.39220
   59       0.000341337       0.00083912           0.1759          0.56806
   60       0.000559414       0.00139853           0.1852          0.75328
   61       0.000907157       0.00230569           0.1473          0.90057
   62        0.00145623       0.00376192          0.07635          0.97693
   63        0.00231503       0.00607694         0.020822          0.99775
   64        0.00364617       0.00972311        0.0021959          0.99994
   65        0.00569158        0.0154147      5.6922e-005                1
   66        0.00880839        0.0242231      1.7918e-007                1
   67         0.0135199        0.0377429      2.2415e-011                1
   68         0.0205871          0.05833                0                1
   69         0.0311093        0.0894393                0                1
   70          0.046664           0.1361                0                1
   71         0.0694996           0.2056                0                1
   72            0.1028           0.3084                0                1
   73           0.15106          0.45946                0                1
   74           0.22054             0.68                0                1
   75              0.32                1                0                1

Note: Numerical calculations involving large powers (e.g. 1,000) will tend to magnify precision errors.

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