Sanderson M. Smith
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KENO is an interesting numbers game. You can relax in a chair, be served free drinks (as long as you are playing), and even watch people who are involved in other casino games. I enjoy playing KENO every now and then, although I know that my chances of "coming out ahead" are extremely small. It's possible to win big in KENO, but it is definitely not the game to play if you are attempting to make money playing games of pure chance. Games of pure chance such as roulette and craps provide opportunities to lose money at a slower pace.
KENO is a relatively simple game. The set, S, of the first 80 positive integers, is involved in all versions of KENO.
After you choose some of these numbers, the casino will randomly select 20 number from S. (In all versions of KENO, the casino generates a list of 20 randomly selected numbers from S.)
Let's consider 20 SPOT KENO as played at John Ascuaga's Nugget in Reno.
In this game, you pay to have the opportunity to choose twenty numbers from the set S. It costs $5 to play. After paying the $5 and picking your twenty numbers, the casino then generates its list of 20 numbers. The table below provides the payouts for certain numbers of matches. Some examples:
Note carefully that you will at least break even unless you match 4. 5, or 6 numbers. This seems to make the game almost too good to be true. Let's check it out.
The table below can be generated on the TI83. If we assume the four columns represent lists L1, L2, L3, and L4, then
L1 can be generated by highlighting the L1 and typing seq(x,x,0,20,1).
L2 can be generated by highlighting L2 and typing (20 nCr L1)*(60 nCr (20L1))/(80 nCr 20).
This will put the probabilities in the appropriate cells. For example, the probability of matching 4 numbers is (_{20}C_{4})(_{60}C_{16})/(_{80}C_{20}) = .2050318987.
You must type in the appropriate payoffs in L3.
L4 is generated by highlighting L4 and typing L2*L3
M = # Matches 
p = Probability 
x =payoff for $5 
Product xp 
0 
.00119 
500 
.59285 
1 
.01157 
10 
.11568 
2 
.04971 
5 
.24857 
3 
.12486 
5 
.62432 
4 
.20503 
0 
0 
5 
.23328 
0 
0 
6 
.19017 
0 
0 
7 
.11330 
5 
.56648 
8 
.04986 
10 
.49862 
9 
.01628 
20 
.32563 
10 
.00394 
50 
.19700 
11 
7E4 
200 
.14047 
12 
9.1E5 
1000 
.09117 
13 
8.5E6 
5000 
.04234 
14 
5.5E7 
12500 
.00686 
15 
2.4E8 
25000 
6E4 
16 
7E10 
37500 
2.5E5 
17 
1E11 
50000 
5.5E7 
18 
1E13 
75000 
7.1E9 
19 
3E16 
100000 
3E11 
20 
3E19 
100000 
3E14 
TOTALS 


3.4506 
The sum of the last column is your expected payoff for each $5. The sum $3.45 can be easily obtained by using 1Var Stats L4 on your TI83. For each $5 "invested," you can expect a return of $3.45. Hence, you expect to lose a total of $1.55 for each $5 you spend on 20 SPOT KENO. You expected loss per dollar spent is this $.31, which puts the version of KENO on a par with other versions.
Note that the only losing numbers of matches are 4, 5, and 6. But the probability that you will end up matching one of these numbers is .20503 + .23328 + .19017 = .62848. And, 2, 3, and 7 are "break even" events. The sum of these probabilities is .04971 + .12486 + .11330 = .28787. Hence, the probability that you will not "come out ahead" in a game is 1  .62848  .28787 = 1  .91635 = .08365.
KENO can be fun and relaxing. Just don't expect to win much playing the game.
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