# Bingo Odds

## Introduction

Calculating bingo odds is similar to calculating keno odds. From a probability standpoint, bingo is basically a game of keno where you don't win anything for partial matches. Of course, I am solely talking about the probability of matching a particular pattern as opposed to winning a game of bingo because the probability of winning a bingo game is dependent on how many other people you are competing against.

## Calculating bingo odds

Here is the equation for calculating the probability (p) of matching a particular pattern made up of (x) numbers when (y) numbers were drawn out of (z) total numbers (i.e. - "What is the probability of hitting a pattern made up of 4 particular numbers (e.g. "corners") when 20 numbers are drawn out of 75?").

p(x,n) = (((combin(x,(x-x)))*combin(z-x,y-x))/combin(z,y))

where:       x = 4
y = 4
z = 75

To calculate different bingo odds, you can download my odds spreadsheet by right-clicking and selecting "Save target as" to calculate the casino's edge of the keno game you play.

#### Bingo Odds Table

 # of Calls Cover All Four corners "X" pattern 1 - - - 2 - - - 3 - - - 4 - 0.00008% - 5 - 0.00041% - 6 - 0.00123% - 7 - 0.00288% - 8 - 0.00576% 0.000000006% 9 - 0.01037% 0.000000053% 10 - 0.01728% 0.000000267% 11 - 0.02715% 0.000000978% 12 - 0.04073% 0.000002934% 13 - 0.05883% 0.000007628% 14 - 0.08236% 0.000017800% 15 - 0.11230% 0.000038142% 16 - 0.14974% 0.000076285% 17 - 0.19581% 0.000144093% 18 - 0.25176% 0.000259367% 19 - 0.31889% 0.000447998% 20 - 0.39862% 0.000746663% 21 - 0.49241% 0.001206149% 22 - 0.60183% 0.001895377% 23 - 0.72854% 0.002906244% 24 0.000000000000000004% 0.87424% 0.004359366% 25 0.000000000000000097% 1.04077% 0.006410833% 26 0.000000000000001261% 1.23000% 0.009260091% 27 0.000000000000011347% 1.44391% 0.013159077% 28 0.000000000000079426% 1.68456% 0.018422708% 29 0.000000000000460671% 1.95409% 0.025440883% 30 0.000000000002303355% 2.25472% 0.034692113% 31 0.000000000010200573% 2.58875% 0.046758935% 32 0.000000000040802291% 2.95858% 0.062345246% 33 0.000000000149608400% 3.36665% 0.082295725% 34 0.000000000508668560% 3.81554% 0.107617487% 35 0.000000001618490874% 4.30787% 0.139504149% 36 0.000000004855472621% 4.84635% 0.179362478% 37 0.000000013819422074% 5.43379% 0.228841782% 38 0.000000037509859915% 6.07306% 0.289866257% 39 0.000000097525635779% 6.76712% 0.364670453% 40 0.000000243814089448% 7.51903% 0.455838066% 41 0.000000588022215727% 8.33189% 0.566344264% 42 0.000001372051836696% 9.20893% 0.699601738% 43 0.000003105169946207% 10.15344% 0.859510706% 44 0.000006831373881656% 11.16879% 1.050513085% 45 0.000014638658317834% 12.25842% 1.277651050% 46 0.000030608103755472% 13.42589% 1.546630218% 47 0.000062546994630747% 14.67481% 1.863887699% 48 0.000125093989261493% 16.00889% 2.236665238% 49 0.000245184218952526% 17.43190% 2.673087724% 50 0.000471508113370243% 18.94771% 3.182247290% 51 0.000890626436366014% 20.56029% 3.774293298% 52 0.001654020524679740% 22.27364% 4.460528443% 53 0.003022865096828490% 24.09190% 5.253511277% 54 0.005441157174291290% 26.01925% 6.167165412% 55 0.009653665954387760% 28.05998% 7.216895695% 56 0.016893915420178600% 30.21844% 8.419711644% 57 0.029180399362126600% 32.49907% 9.794358443% 58 0.049778328323627800% 34.90641% 11.361455794% 59 0.083912039174115500% 37.44506% 13.143644939% 60 0.139853398623526000% 40.11971% 15.165744160% 61 0.230569116649597000% 42.93513% 17.454913090% 62 0.376191716638816000% 45.89617% 20.040826140% 63 0.607694311493471000% 49.00777% 22.955855397% 64 0.972310898389554000% 52.27496% 26.235263310% 65 1.541468497446850000% 55.70283% 29.917405529% 66 2.422307638845060000% 59.29656% 34.043944223% 67 3.774293297735320000% 63.06142% 38.660072253% 68 5.832998732863680000% 67.00276% 43.814748554% 69 8.943931390390970000% 71.12600% 49.560945086% 70 13.610330376681900000% 75.43667% 55.955905742% 71 20.560286313711000000% 79.94035% 63.061417582% 72 30.840429470566500000% 84.64272% 70.944094780% 73 45.945945945945900000% 89.54955% 79.675675676% 74 68.000000000000000000% 94.66667% 89.333333333% 75 100.000000000000000000% 100.00000% 100.000000000%

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Hey! Thanks for all this great info! I'm a fan.
I have a question, what would be the equation if I needed to make it dynamic in a way where the odds react to the number of real players in the room and the # of bingo cards they each have keeping in mind that I want a balance between AI odds of winning and player odds.

dandan on September 7, 2016

How would this formula change if the number chosen was put back into the hopper instead of kept out (ie with replacement)

DaveG44131 on September 11, 2014

Bingo Odds

HPG ADMIN on February 27, 2013