## 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

## Keno Odds Calculator

### Odds Calculator for Keno

#### Results

This is a free online Keno odds calculator which will determine the return, house advantage and jackpot break-even point (if any) for a given Keno pay table.

Enter the data into the relevant fields. Below is a description of what each field represents. When this page first loads it has data entered for 8 numbers selected with a jackpot awarded for matching all 8 numbers.

**Game Details** * Numbers Selected*: The number of selections made by the player. Select an integer between 1 and 15.

*: The bet made by the player. Enter as a floating point number between 0.01 and 10. Do not use a dollar sign.*

**Bet Amount***: The size of the jackpot if on offer. Enter as a floating point number between 0 and 10000000. Do not use a dollar sign.*

**Jackpot Amount****Pay Table** * Match*: The number of player selected numbers that match the 20 numbers drawn by the house.

*: The pay out in dollars or percentage of the jackpot depending on*

**Collect***Type*selected (see below) for each

*Match*. If type selected is dollar amount ($) enter as a floating point number between 0 and 10000000. If type selected is percentage of the jackpot (%) enter as a floating point number between 0 and 100. Do not use any symbols (i.e. $ or %).

*: The type of collect, either a dollar amount ($) or percentage of the jackpot (%).*

**Type**Once the data is correctly entered press the “Calculate” button to analyze. Below is a description of what each result represents.

**Summary** * Return on $x bet*: The average amount the player will collect each time a game of Keno with the given data is played.

*: The percentage of the total amount bet the the house will win on average. A negative number represents a player advantage.*

**House Advantage***: The amount the jackpot needs to reach for there to be zero house advantage.*

**Jackpot Break-even Point**A free online calculator to determine odds for a variety of Keno pay tables.