Mathematical Analysis of Keno.

Let Pn(k) be the probability that “k” out of “n” numbers chosen by a player will occur in the twenty (20) numbers chosen by the computer

Lets see now.. what exactly is this Pn(k) anyway?

1) The number of possible outcomes is equal to the number of the combination of the eighty (80) numbers taken twenty (20) at a time.

*(SamplingArea / Number of all possible cases)*

2) The number of ways in which “k” out of “n” chosen numbers occur in the twenty (20) given by the computer, is equal to the number of ways in which “k” numbers can be chosen from a set of “n” numbers.

*(Number of favorable Cases)*

3) The number of ways in which the rest of the numbers shall not occur in the twenty (20) numbers chosen by the machine, is given by the number of ways in which the 20-k numbers can be chosen from a set of 80-n numbers

*(Number of the rest of numbers that did not occur)*

To put it simple, we’re talking about a HyperGeometric Distribution.

As we all (players) know, N = 80, and r = 20 in Keno. So with a simple substitution we get this formula:

For example, if we want to calculate the probability of us chosing 12 numbers, 11 out of which occur, we get:

## Expected Payout

Lets see now. What is the expected payout, and how do we calculate it?

f the player participates in the n-spot game and ends up matching k of the twenty numbers selected, we will refer to that payout as: Wn(k).

The expected payout for the n-spot game can be determined by summing, over all values of i from one to n (from zero to n if the game pays out in the case of zero numbers matched), the product of the payout for that result and the probability of occurrence of that result

In other words, it is calculate by this formula:

Which could alternatively be represented as the inner product of the vector of probabilities and the vector of payouts.

## Probabilities:

You may use the “Probability Form” of the program to calculate any Probability you want to. Some probabilities have already been calculated for you, and are presented below.

Probability of 1 out of 1:

P1(1) = 0.25 = **25%**

Probability of 2 out of 2:

P2(2) = 0.0601265822784810126582278481 = 6.01265822784810126582278481% = **6%**

Probability of 2 out of 3:

P3(2) = 0.1387536514118792599805258033 = 13.87536514118792599805258033% = **13.9%**

Probability of 3 out of 3:

P3(3) = 0.0138753651411879259980525803 = 1.38753651411879259980525803% = **1.4%**

Probability of 4 out of 4:

P4(4) = 0.0030633923038986330125570632 = 0.30633923038986330125570632% = **0.3%**

Probability of 4 out of 5:

P5(4) = 0.0120923380417051303127252494 = 1.20923380417051303127252494% = **1.2%**

Probability of 5 out of 5:

P5(5) = 0.0006449246955576069500120133 = 0.06449246955576069500120133% = **0.06%**

Probability of 6 out of 6:

P6(6) = 0.0001289849391115213900024027 = 0.01289849391115213900024027% = **0.01%**

Probability of 7 out of 7:

P7(7) = 0.0000244025560481256683788329 = 0.00244025560481256683788329% = **0.002%**

Probability of 8 out of 8:

P8(8) = 0.000004345660666104571081162 = 0.0004345660666104571081162% = **0.0004%**

Probability of 9 out of 9:

P9(9) = 0.0000007242767776840951801937 = 0.00007242767776840951801937% = **0.00007%**

Probability of 10 out of 10:

P10(10) = 0.0000001122118951341555912976 = 0.00001122118951341555912976% = **0.00001%**

Probability of 11 out of 11:

P11(11) = 0.0000000160302707334507987568 = 0.00000160302707334507987568% = **0.000002%**

Probability of 12 out of 12:

P12(12) = 0.0000000020909048782761911422 = 0.00000020909048782761911422% = **0.0000002%**

Kino Statistics, By Giannis Mamalikidis ©N1h1l1sT

Mathematical Analysis of Keno. Let Pn(k) be the probability that “k” out of “n” numbers chosen by a player will occur in the twenty (20) numbers chosen by the computer Lets see now.. what## Calculating Keno Probabilities and Odds – Chances to Win Keno

### Odds and Probabilities in Keno

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 80 C_{20} ways for the house to draw 20 from 80 numbers.

Probability(n, k) = | n C_{k} x (80-n) C_{(20-k)} |

———————– | |

80 C_{20} |

2) The following formula uses the total number of combinations (i.e. denominator) from the players point of view. There are 80 C_{n} ways for the player to draw n from 80 numbers.

Probability(n, k) = | 20 C_{k} x 60 C_{(n-k)} |

———————– | |

80 C_{n} |

3) Both of the above equations can be reduced to the same equation involving factorials given below.

a C_{b} = |
a! |

————— | |

b! x (a – b)! |

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 the House Advantage in Keno

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 |

#### Related Pages

- Keno Probabilities and Odds Tables
- Keno Odds Calculator

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A description of calculating probabilities and odds for the game of Keno. ]]>