**Poker Digest Vol. 2, No. 22, October 22 - November 4, 1999**

Keno is the next game we examine based on expectation. About
six months ago a message was posted on rec.gambling.poker, by someone
whose name I've
forgotten (just as well), in which the calculation for the probability of
getting six numbers out of six was wrong by a large factor of 10. Thus,
we start by showing how to make this kind of calculation. In fact,
let's use a ticket with six numbers selected as our example. First, the
total number of possible draws is *C*(80,20) since we are drawing 20
numbers from a total of 80 numbers. In order to determine the
probability of having all six numbers drawn, we simply count how many
draws have our six numbers included. Since our six numbers are to
be included in the draw, we need to count how many ways the remaining
14 numbers may appear. These 14 numbers are being chosen
from seventy-four numbers because essentially we have set aside our six
selected numbers to appear in the draw. This means there are *C*(74,14)draws which will include our six numbers. Therefore, the probability
of all six of our selected numbers appearing among the 20 numbers
drawn is *C*(74,14) divided by *C*(80,20). If one carries out the simple
calculation, the probability obtained is .000129 which is about 1/7,753.

Let's determine the probability of getting exactly five of our six
numbers drawn. There are
*C*(6,5) = 6 ways of choosing which set of five
numbers will be in the draw. The remaining 15 numbers must then be
chosen from the 74 numbers not on our ticket because we must
exclude the possibility of all six numbers appearing. Therefore, the number
of draws with exactly five of our
numbers is 6*C*(74,15). Dividing this by *C*(80,20) gives us a
probability of .00310 which is about 1/323.

Similarly, to determine the probability of getting four of the six
numbers, we observe there are
*C*(6,4) = 15 ways of choosing four of the
six numbers and there are *C*(74,16) ways to choose the remaining sixteen
numbers. Hence, there are
15*C*(74,16) draws in which precisely
four of our numbers appear. Dividing by *C*(80,20) gives us a probability
of .02854 that precisely four numbers appear in the draw. This is
about 1/35.

Finally, for precisely three numbers in the draw, there are
*C*(6,3) = 20 ways to choose the three numbers and *C*(74,17)
ways to choose the remaining
17 numbers. Multiplying these two numbers and dividing by *C*(80,20)
yields a probability of .1298 that exactly three numbers appear in the draw.
This is slightly better than 1/8.

The payouts for keno are not the same for all casinos. However, the
differences are insufficient to make much difference in the expected values.
I am going to use the payouts I found at a major Las Vegas casino in
August 1999. For all six numbers you receive $2,000; for five numbers
$80; for four numbers $4; and for three numbers $1. Be aware these
amounts are not your winnings since you have already paid $1 for the
ticket. You must subtract one dollar from each to get your winnings.
Multiplying the winnings by the various probabilities gives an expectation
of

1999(.000129) + 79(.00310) + 3(.02854) - 1(.83842) = -.25

which means the casino is taking about twenty-five cents from every dollar bet on a six-spot ticket in keno.

The expectation just worked out is very bad for the player and very good for the casino. In spite of this many people play keno. This column deals with mathematics and poker (or more generally mathematics and gambling). I have no intention of discussing psychology and gambling but the results just obtained and the fact the game is popular certainly indicate the psychology of gambling is an important topic. Somehow the possibility of a large payout for a small wager makes people willing to play a game with a terrible expectation.

The table below gives the probabilities and expectations for various choices of numbers played for keno. The expectations shown are the expectations for playing that many spots and not just the expectation for the line on which it is written.

Pick | Catch | Payout | Probability | Expectation |

1 | 1 | $3 | .25 | -.25 |

2 | 2 | $12 | .0601 | -.2788 |

3 | 2 | $1 | .1388 | |

3 | $45 | .01388 | -.2366 | |

4 | 2 | $1 | .2126 | |

3 | $2 | .04325 | ||

4 | $160 | .00306 | -.2113 | |

5 | 3 | $1 | .08393 | |

4 | $15 | .01209 | ||

5 | $750 | .000645 | -.2510 | |

6 | 3 | $1 | .12982 | |

4 | $4 | .02854 | ||

5 | $80 | .00310 | ||

6 | $2,000 | .000129 | -.25 | |

7 | 4 | $1 | .05219 | |

5 | $15 | .008639 | ||

6 | $400 | .000732 | ||

7 | $10,000 | .0000244 | -.2814 | |

8 | 5 | $5 | .0183 | |

6 | $75 | .002367 | ||

7 | $1,500 | .00016 | ||

8 | $50,000 | .00000435 | -.2735 | |

9 | 5 | $6 | .0326 | |

6 | $30 | .00572 | ||

7 | $300 | .000592 | ||

8 | $5,000 | .000033 | ||

9 | $50,000 | .0000007 | -.2561 | |

10 | 5 | $1 | .05143 | |

6 | $25 | .01148 | ||

7 | $125 | .00161 | ||

8 | $1,000 | .000135 | ||

9 | $10,000 | .0000061 | ||

10 | $100,000 | .00000011 | -.2529 |

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