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))
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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Casino & Poker
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.