Calculating Keno Probabilities and Odds
There are 2 equivalent formulas using combinations to calculate probabilities for the game of Keno.
The two variables involved in the calculation are:
- n = Number of selections made by player
- k = How many of the player's selections match the drawn numbers.
1) The following formula uses the total number of combinations (i.e. denominator) from the houses point of view. There are 80C20 ways for the house to draw 20 from 80 numbers.
| Probability(n, k) = | nCk x (80-n)C(20-k) |
| ----------------------- |
| 80C20 |
2) The following formula uses the total number of combinations (i.e. denominator) from the players point of view. There are 80Cn ways for the player to draw n from 80 numbers.
| Probability(n, k) = | 20Ck x 60C(n-k) |
| ----------------------- |
| 80Cn |
3) Both of the above equations can be reduced to the same equation involving factorials given below.
| aCb = | a! |
| --------------- |
| b! x (a - b)! |
Therefore:
| Probability(n, k) = | 60! x 20 ! x n! x (80 - n)! |
| ---------------------------------------------------- |
| 80! x k! x (20 - k)! x (n - k)! x (60 - n + k)! |
Calculating House Advantage
To calculate the return to the player sum the returns (equal to probability multiplied by the pay out) for each possible outcome. The house advantage is determined by subtracting the return to the player from the bet amount and is usually expressed as a percentage. The table below calculates the player's return using formula 2 above for the Spot 9 pay outs given in the example section of the Standard Rules page. The house advantage in this case is 100 x ($1 - $0.669665) = 33.03%.
| Result Details |
|---|
| Match | Collect ($) | Frequency | Probability (%) | Return |
|---|
| 0 | 0.00 | 14,783,142,660 | 6.374783835335 | 0.0000000000 | | 1 | 0.00 | 51,172,416,900 | 22.066559430007 | 0.0000000000 | | 2 | 0.00 | 73,379,314,800 | 31.642613522274 | 0.0000000000 | | 3 | 0.00 | 57,072,800,400 | 24.610921628435 | 0.0000000000 | | 4 | 1.00 | 26,461,025,640 | 11.410518209547 | 0.1141051821 | | 5 | 5.00 | 7,560,293,040 | 3.260148059871 | 0.1630074030 | | 6 | 20.00 | 1,326,367,200 | 0.571955799977 | 0.1143911600 | | 7 | 210.00 | 137,210,400 | 0.059167841377 | 0.1242524669 | | 8 | 2,500.00 | 7,558,200 | 0.003259245500 | 0.0814811375 | | 9 | 100,000.00 | 167,960 | 0.000072427678 | 0.0724276778 | | Total | | 231,900,297,200 | 100.000000000000 | 0.669665027234 |
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