Sanderson M. Smith

Home | About Sanderson Smith | Writings and Reflections | Algebra 2 | AP Statistics | Statistics/Finance | Forum

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:

• If none of your numbers are among the twenty, you win $500.
• If 1 of your numbers is among the twenty, you win $10
• If 2 of your numbers are among the twenty, you win $5.
• etc., etc., etc.
• If 19 or 20 of your numbers are among the twenty, you win $100,000.

  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 TI-83. 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 (20-L1))/(80 nCr 20).

This will put the probabilities in the appropriate cells. For example, the probability of matching 4 numbers is (₂₀C₄)(₆₀C₁₆)/(₈₀C₂₀) = .2050318987.

You must type in the appropriate payoffs in L3.

L4 is generated by highlighting L4 and typing L2*L3

The sum of the last column is your expected payoff for each $5. The sum $3.45 can be easily obtained by using 1-Var Stats L4 on your TI-83. 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.

RETURN TO WRITING HOME PAGE