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 betPercentage lost
125%
426%
326%
527%
827%
728%
1028%
228%
1128%
928%
1228%
628%
1329%
1429%
1529%

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,

SpotsCatchWinProbabilityExpected return
11$3.000.25$0.7500
22$12.000.060126584$0.7215
32$1.000.13875365$0.1388
33$43.000.013875365$0.5966
42$1.000.21263547$0.2126
43$4.000.04324789$0.1730
44$115.000.0030633924$0.3523
53$2.000.08393505$0.1679
54$20.000.012092338$0.2418
55$500.006.449247e-4$0.3225
63$1.000.12981954$0.1298
64$4.000.028537918$0.1142
65$90.000.0030956385$0.2786
66$1500.001.2898494e-4$0.1935
73$0.500.17499325$0.0875
74$1.500.052190967$0.0783
75$20.000.008638505$0.1728
76$360.007.320767e-4$0.2635
77$5000.002.4402556e-5$0.1220
85$9.000.018302586$0.1647
86$90.000.0023667137$0.2130
87$1500.001.6045517e-4$0.2407
88$25000.004.3456605e-6$0.1086
94$0.500.11410519$0.0571
95$3.000.03260148$0.0978
96$40.000.0057195583$0.2288
97$300.005.9167844e-4$0.1775
98$4000.003.2592455e-5$0.1304
99$37500.007.242768e-7$0.0272
105$2.000.05142769$0.1029
106$20.000.0114793945$0.2296
107$140.000.0016111432$0.2256
108$1000.001.3541937e-4$0.1354
109$4000.006.120649e-6$0.0245
1010$50000.001.1221189e-7$0.0056
115$1.000.074080355$0.0741
116$8.000.020203736$0.1616
117$80.000.0036078098$0.2886
118$315.004.1141692e-4$0.1296
119$1800.002.837358e-5$0.0511
1110$12500.001.05799794e-6$0.0132
1111$65000.001.603027e-8$0.0010
125$0.500.09938732$0.0497
126$3.000.032208852$0.0966
127$35.000.0070273858$0.2460
128$260.000.0010195985$0.2651
129$500.009.5401025e-5$0.0477
1210$1500.005.427989e-6$0.0081
1211$20000.001.672724e-7$0.0033
1212$70000.002.0909049e-9$0.0001
130$1.000.016395647$0.0164
136$1.000.047501296$0.0475
137$18.000.012315149$0.2217
138$80.000.0021831403$0.1747
139$700.002.5989765e-4$0.1819
1310$3000.002.0062272e-5$0.0602
1311$10000.009.433671e-7$0.0094
1312$50000.002.3983909e-8$0.0012
1313$75000.002.459888e-10$0.0000
140$1.000.011501424$0.0115
146$1.000.06575738$0.0658
147$10.000.019851286$0.1985
148$40.000.0041816365$0.1673
149$310.006.0823804e-4$0.1886
1410$1100.005.973766e-5$0.0657
1411$3100.003.8110152e-6$0.0118
1412$25000.001.4784111e-7$0.0037
1413$50000.003.084039e-9$0.0002
1414$100000.002.5700322e-11$0.0000
150$1.000.008016144$0.0080
156$1.000.08634808$0.0863
157$5.000.02988972$0.1494
158$30.000.0073314407$0.2199
159$130.000.0012671626$0.1647
1510$310.001.5205951e-4$0.0471
1511$2500.001.2342492e-5$0.0309
1512$7500.006.4960486e-7$0.0049
1513$25000.002.0677078e-8$0.0005
1514$50000.003.5045895e-10$0.0000
1515$125000.002.336393e-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.