Sanderson M. Smith

KENO...A MONEY-MAKING GAME FOR THE CASINO

(AND AN OPPORTUNITY TO ILLUSTRATE PROBABILITY CONCEPTS)

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"If you believe in miracles, head for the Keno lounge"

Jimmy the Greek

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.

S = {1, 2, 3,....,78, 79, 80}

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.) If a good portion of the numbers you chose are among the 20 numbers, you win some money.

Let's consider 8 SPOT KENO.

In this game, you pay to have the opportunity to choose eight numbers from the set S. After you do this, the casino then generates its list of 20 numbers. In most casinos, you are a winner in 8 SPOT KENO if the set of 20 numbers contains 5 or more of your numbers.

At John Ascuaga's Nugget in Reno, if you pay \$1 to play 8 SPOT KENO, and you match

• 5 numbers, you win \$9 (net gain of \$8, since you paid \$1 to play)
• 6 numbers, you win \$90 (net gain of \$89)
• 7 numbers, you win \$1,480 (net gain of \$1,479)
• 8 numbers you win \$20,000 (net gain of \$19,999)

The number of different sets of 20 numbers that can be chosen from a set of 80 numbers is 80C20, which is approximately 3.535x1018. In 8 SPOT KENO, the probability that you would match x numbers, where x is 0, 1, 2, 3, 4, 5, 6, 7, or 8, is

(8Cx)(72C(20-x))/(80C20).

In 8 SPOT KENO, you have 8 goodies (the numbers you chose), and 72 baddies (the numbers you didn't choose). How many ways could you match five numbers. Well, from the 8 goodies, you must choose 5. And, these would be combined with 15 baddies chosen from the set of 72 baddies. Hence, the probability that you match exactly 5 numbers is

(8C5)(72C15)/(80C20) = .018302586, or about 1.83%.

The probability that you would match all 8 numbers is

(8C8)(72C12)/(80C20) = .00000434566

Obviously your chances of matching all 8 are quite slim. But, how slim? If x is not zero, we can use the algebraic identity x = 1/(1/x) and represent .00000434566 as 1/230114.61. In other words, we would expect to match all 8 number once in 230,115 games.

The following table lists probabilities associated with 8 SPOT KENO.

 # matches Probability G = Player's gain 0 .088266, which is approximately 1/11 -\$1 = \$(0-1) 1 .266464, which is approximately 1/4 -\$1 = \$(0-1) 2 .328146, which is appoximately 1/3 -\$1 = \$(0-1) 3 .214786, which is approximately 1/5 -\$1 = \$(0-1) 4 .081504, which is approximately 1/12 -\$1 = \$(0-1) 5 .018303, which is approximately 1/55 \$8 = \$(9-1) 6 .002367, which is approximately 1/423 \$89=\$(90-1) 7 .000160, which is approximately 1/6232 \$1479=\$(1480-1) 8 .000004, which is approximately 1/230155 \$19999=\$(20000-1)

So, if a player spends \$1 to play 8 SPOT KENO, the player's dollar expectation is

mG = (-1)(.088266) + (-1)(.266464) + (-1)(.328146) + (-1)(.214786) + (-1)(.081504) + (8)(.018303) +(89)(.002367) + (1479)(.000160) + (19999)(.000004) = -.305443.

In other words, for each dollar spent playing 8 SPOT KENO, John Ascuaga's Nugget expects to gain about 30 cents. This is equivalent to the player handing over a dollar bill to the casino and getting 70 cents in return. Not a bad deal for the casino! Not a great deal for the player!

All versions of KENO at John Ascuaga's Nugget result in a casino gain of approximately 30 cents for each dollar spent playing the game.

But, let's not forget that you could win in the short run. That is, if you spent just a few dollars, it is quite possible to end up ahead for the evening. And, it is possible to win \$50,000 if you pay \$1 to play 20 SPOT KENO at John Ascuaga's Nuggett. Basically, you would pick 20 numbers, and your numbers would have to match the 20 randomly chosen by the casino. The probability of this is

(20C20)(60C0)/(80C20) = 2.8286x10-19, or approximately 1/3500000000000000000.

If you played one game of 20 SPOT KENO per second, you would have to play for approximately 1,121,000,000 centuries to get one match of 20 numbers.

Again, KENO is a fun and relaxing numbers game if you can contain yourself and realize that you are not likely to get rich by gambling on the numbers. As with any gambling game of pure chance, it is a definite money-maker for the casino.