4.34 Keno is a favorite game in casinos, and similar games are popular with the
states that operate lotteries. Balls numbered 1 to 80 are tumbled in a machine
as the bets are placed, then 20 of the balls are chosen at random. Players select
numbers by marking a card. The simplest of the many wagers available is marking
1 number." Your payoff is $3 on a $1 bet if the number you select is one of those
chosen. Because 20 of 80 numbers are chosen, your probability of winning is 20/80,
or 0.25.
(a) What is the probability distribution (the outcomes and their probabilities) of
the payoff X on a single play?
(b) What is the mean payoff X?
(c) In the long run, how much does the casino keep from each dollar bet?
In the simple KENO game where one number is played.
a) There is a 25% chance of winning $3 and a 75% chance of winning zero.
b) the mean payoff is .25*3 + .75*0. = 0.75 or 75 cents on each dollar bet.
c) from b, the casino keeps 25 cents on each dollar bet.
(a) The probability distribution of the payoff X on a single play is as follows:
X = 3 if the number chosen is one of the 20 chosen balls
X = 0 if the number chosen is not one of the 20 chosen balls
The probabilities of each outcome can be calculated as:
P(X = 3) = 0.25 (since the probability of selecting a chosen ball is 20/80 = 0.25)
P(X = 0) = 0.75 (since the probability of not selecting a chosen ball is 1 - P(X = 3) = 1 - 0.25 = 0.75)
(b) The mean payoff X can be calculated as:
Mean payoff, E(X) = (Value of X1 * P(X = X1)) + (Value of X2 * P(X = X2)) + ...
E(X) = (3 * 0.25) + (0 * 0.75)
E(X) = 0.75
Therefore, the mean payoff X is $0.75.
(c) In the long run, the casino keeps the amount that players lose on average per dollar bet. This is equal to the expected value of the loss for each dollar bet. In this case, the expected value of the loss is the opposite of the mean payoff, so the casino keeps $0.75 from each dollar bet.
To determine the probability distribution of the payoff X, we need to consider the possible outcomes and their corresponding probabilities.
(a) The possible outcomes for the payoff X are:
- If your selected number is chosen among the 20 balls, you win $3.
- If your selected number is not chosen among the 20 balls, you lose $1.
The corresponding probabilities for these outcomes can be calculated as follows:
- Probability of winning: Since 20 out of the 80 balls are chosen, the probability of your selected number being among them is 20/80 = 0.25.
- Probability of losing: Since there are 80 balls in total, and only 20 are chosen, the probability of your selected number not being among them is 60/80 = 0.75.
Therefore, the probability distribution of the payoff X is:
- X = $3 with a probability of 0.25.
- X = -$1 with a probability of 0.75.
(b) To calculate the mean payoff X, we multiply each outcome by its corresponding probability and sum them up:
Mean payoff X = ($3 * 0.25) + (-$1 * 0.75)
= $0.75 - $0.75
= $0
The mean payoff X is $0, indicating that on average, you neither win nor lose money in the long run.
(c) To determine how much the casino keeps from each dollar bet in the long run, we subtract the mean payoff from 1:
Casino's profit per dollar bet = 1 - mean payoff
= 1 - $0
= $1
Therefore, in the long run, the casino keeps the entire dollar bet as profit.