# Keno Odds

I was in Las Vegas at a computer industry trade show with a friend of mine, and we were eating lunch at a casino's cafe. On our table was a brochure, entitled "Keno," along with some handy playing tickets. This was the first that we had heard of the game. The brochure gives the returns at that casino, i.e., "bet this and win that," but does not provide any probabilities on the expectation of various winnings (so one can evaluate if they should bet at all). I took the brochure home with the hope of doing the analysis just for curiosity.

Keno is an entirely electronic game that is played every few minutes by a computer. Pretend that the computer has a basket of eighty balls, each sequentially numbered from one to eighty. You take a card with numbers from one to eighty and mark from one to fifteen "spots" (i.e., "lucky balls") that you expect to be drawn, and then place a bet. After the bets are closed, the computer pulls out twenty balls from the basket, and if you "catch" enough "spots," you'll get a return on your bet.

### Summary of Results

If following section's calculations are correct, along with my program, the fraction of your bet that you lose
on average per game *for this particular casino* depends on the
number of spots that you play on the card, as follows:

Spots bet | Percentage lost |
---|---|

1 | 25% |

4 | 26% |

3 | 26% |

5 | 27% |

8 | 27% |

7 | 28% |

10 | 28% |

2 | 28% |

11 | 28% |

9 | 28% |

12 | 28% |

6 | 28% |

13 | 29% |

14 | 29% |

15 | 29% |

So, from what I can tell, you're crazy to play Keno for any reason. If you can get your hands on the odds brochure from your favorite casino (or state lottery) and email them to me, I'll be happy to run my program for you, or you can try it yourself. The theory is so simple that you can write your own program in a few minutes, and even if you cannot program, all you need is a calculator to determine the odds yourself.

### How to Calculate the Odds for your Casino.

The probability of catching exactly r spots when you bet on N of them (where N >= r) is given by

P(N,r) = c(N,r) * c(80-N, 20-r) / c(80,20).

In other words, P(N,r) is the number of ways that you can pick r of the N spots times the number of ways that the computer can pick all of the spots that you didn't bet on divided by the ways that it can pick twenty spots. Note that c(N,r) is the famous "binomial coefficient", where

c(N,r) = f(N,r) / f(r,r),

and we can define f(n,r) according to the rules

f(n,0) = 1, otherwise f(n,r) = n * f(n-1,r-1).

#### Example

A casino says "play four spots, catch two and get 1:1, three and get 4:1, all four and get 115:1." Should we bet on four spots?

The probability of getting zero of four spots is p0 = P(4,0) = 97527 / 316316 = 0.3083. The probability of getting one of four spots is p1 = P(4,1)= 34220 / 79079 = 0.4327. The probability of getting two of four spots is p2 = P(4, 2) = 16815 / 79079 = 0.2126. The probability of getting three of four spots is p3 = P(4, 3) = 3420 / 79079 = 0.04324. The probability of getting all four spots is p4 = P(4, 4) = 969 / 316316 = 0.003063. (Note that the values of P can be found in the following table for your convenience.)

Note that all of these probabilities add to one: p0 + p1 + p2 + p3 + p4 = 1. Now you start by giving them one dollar, but you have a chance to win it back! You earn $1 with probability p2, $4 with probability p3, and $115 with probability p4, so your expected return per dollar bet is

=(what you put in) + (what you expect to get out)

= (-$1) + (($1)p2 + ($4)p3 + ($115)p4)

= (-$1) + (($0.213) + ($0.173) + (0.35229012))

= (-$1) + ($0.738)

= -$0.26.

In other words, you expect to loose about 0.26 cents per dollar that you bet on four spots, and this is horrible---pick a better game, like Baccarat.

#### A Specific Casino

For a dollar bet,

Spots | Catch | Win | Probability | Expected return |
---|---|---|---|---|

1 | 1 | $3.00 | 0.25 | $0.7500 |

2 | 2 | $12.00 | 0.060126584 | $0.7215 |

3 | 2 | $1.00 | 0.13875365 | $0.1388 |

3 | 3 | $43.00 | 0.013875365 | $0.5966 |

4 | 2 | $1.00 | 0.21263547 | $0.2126 |

4 | 3 | $4.00 | 0.04324789 | $0.1730 |

4 | 4 | $115.00 | 0.0030633924 | $0.3523 |

5 | 3 | $2.00 | 0.08393505 | $0.1679 |

5 | 4 | $20.00 | 0.012092338 | $0.2418 |

5 | 5 | $500.00 | 6.449247e-4 | $0.3225 |

6 | 3 | $1.00 | 0.12981954 | $0.1298 |

6 | 4 | $4.00 | 0.028537918 | $0.1142 |

6 | 5 | $90.00 | 0.0030956385 | $0.2786 |

6 | 6 | $1500.00 | 1.2898494e-4 | $0.1935 |

7 | 3 | $0.50 | 0.17499325 | $0.0875 |

7 | 4 | $1.50 | 0.052190967 | $0.0783 |

7 | 5 | $20.00 | 0.008638505 | $0.1728 |

7 | 6 | $360.00 | 7.320767e-4 | $0.2635 |

7 | 7 | $5000.00 | 2.4402556e-5 | $0.1220 |

8 | 5 | $9.00 | 0.018302586 | $0.1647 |

8 | 6 | $90.00 | 0.0023667137 | $0.2130 |

8 | 7 | $1500.00 | 1.6045517e-4 | $0.2407 |

8 | 8 | $25000.00 | 4.3456605e-6 | $0.1086 |

9 | 4 | $0.50 | 0.11410519 | $0.0571 |

9 | 5 | $3.00 | 0.03260148 | $0.0978 |

9 | 6 | $40.00 | 0.0057195583 | $0.2288 |

9 | 7 | $300.00 | 5.9167844e-4 | $0.1775 |

9 | 8 | $4000.00 | 3.2592455e-5 | $0.1304 |

9 | 9 | $37500.00 | 7.242768e-7 | $0.0272 |

10 | 5 | $2.00 | 0.05142769 | $0.1029 |

10 | 6 | $20.00 | 0.0114793945 | $0.2296 |

10 | 7 | $140.00 | 0.0016111432 | $0.2256 |

10 | 8 | $1000.00 | 1.3541937e-4 | $0.1354 |

10 | 9 | $4000.00 | 6.120649e-6 | $0.0245 |

10 | 10 | $50000.00 | 1.1221189e-7 | $0.0056 |

11 | 5 | $1.00 | 0.074080355 | $0.0741 |

11 | 6 | $8.00 | 0.020203736 | $0.1616 |

11 | 7 | $80.00 | 0.0036078098 | $0.2886 |

11 | 8 | $315.00 | 4.1141692e-4 | $0.1296 |

11 | 9 | $1800.00 | 2.837358e-5 | $0.0511 |

11 | 10 | $12500.00 | 1.05799794e-6 | $0.0132 |

11 | 11 | $65000.00 | 1.603027e-8 | $0.0010 |

12 | 5 | $0.50 | 0.09938732 | $0.0497 |

12 | 6 | $3.00 | 0.032208852 | $0.0966 |

12 | 7 | $35.00 | 0.0070273858 | $0.2460 |

12 | 8 | $260.00 | 0.0010195985 | $0.2651 |

12 | 9 | $500.00 | 9.5401025e-5 | $0.0477 |

12 | 10 | $1500.00 | 5.427989e-6 | $0.0081 |

12 | 11 | $20000.00 | 1.672724e-7 | $0.0033 |

12 | 12 | $70000.00 | 2.0909049e-9 | $0.0001 |

13 | 0 | $1.00 | 0.016395647 | $0.0164 |

13 | 6 | $1.00 | 0.047501296 | $0.0475 |

13 | 7 | $18.00 | 0.012315149 | $0.2217 |

13 | 8 | $80.00 | 0.0021831403 | $0.1747 |

13 | 9 | $700.00 | 2.5989765e-4 | $0.1819 |

13 | 10 | $3000.00 | 2.0062272e-5 | $0.0602 |

13 | 11 | $10000.00 | 9.433671e-7 | $0.0094 |

13 | 12 | $50000.00 | 2.3983909e-8 | $0.0012 |

13 | 13 | $75000.00 | 2.459888e-10 | $0.0000 |

14 | 0 | $1.00 | 0.011501424 | $0.0115 |

14 | 6 | $1.00 | 0.06575738 | $0.0658 |

14 | 7 | $10.00 | 0.019851286 | $0.1985 |

14 | 8 | $40.00 | 0.0041816365 | $0.1673 |

14 | 9 | $310.00 | 6.0823804e-4 | $0.1886 |

14 | 10 | $1100.00 | 5.973766e-5 | $0.0657 |

14 | 11 | $3100.00 | 3.8110152e-6 | $0.0118 |

14 | 12 | $25000.00 | 1.4784111e-7 | $0.0037 |

14 | 13 | $50000.00 | 3.084039e-9 | $0.0002 |

14 | 14 | $100000.00 | 2.5700322e-11 | $0.0000 |

15 | 0 | $1.00 | 0.008016144 | $0.0080 |

15 | 6 | $1.00 | 0.08634808 | $0.0863 |

15 | 7 | $5.00 | 0.02988972 | $0.1494 |

15 | 8 | $30.00 | 0.0073314407 | $0.2199 |

15 | 9 | $130.00 | 0.0012671626 | $0.1647 |

15 | 10 | $310.00 | 1.5205951e-4 | $0.0471 |

15 | 11 | $2500.00 | 1.2342492e-5 | $0.0309 |

15 | 12 | $7500.00 | 6.4960486e-7 | $0.0049 |

15 | 13 | $25000.00 | 2.0677078e-8 | $0.0005 |

15 | 14 | $50000.00 | 3.5045895e-10 | $0.0000 |

15 | 15 | $125000.00 | 2.336393e-12 | $0.0000 |

(I summed the values up in the table at the start of this page.)

*I want to extend my thanks to Rick Nungester for reading this
page and taking time to carefully point out that I didn't realize the
horror of what "2:1" odds really mean (for example) in Keno. I
incorrectly used to think that it meant, "we'll give you $2 plus your
dollar back," but this is NOT the case---they keep your dollar,
and give you $2 back. So, I corrected this serious mistake on my
part. Thanks, Rick!*

WARNING: READ SOME OTHER SOURCES ON THE MATHEMATICS OF KENO TO BE SURE THAT THIS IS RIGHT! It seems that mathematicians and statisticians have done lots of work to study casino games, but I have not yet had a chance to read any of their works.

*Kleanthes Koniaris, email.*