Chapter 4, Problem 4.80P

A game popular in Nevada gambling casinos is Keno, which is played as follows: Twenty numbers are selected at random by the casino from the set of numbers 1 through 80. A player can select from 1 to 15 numbers; a win occurs if some fraction of the player’s chosen subset matches any of the 20 numbers drawn by the house. The payoff is a function of the number of elements in the player’s selection and the number of matches. For instance, if the player selects only I number, then he or she wins if this number is among the set of 20, and the payoff is $2.20 won for every dollar bet. (As the player’s probability of winning in this case is $\frac{1}{4}$, it is clear that the “fair” payoff should be $3 won for every $1 bet.) When the player selects 2 numbers, a payoff (of odds) of $12 won for every $1 bet is made when both numbers are among the 20.

a. What would be the fair payoff in this case?

Let $p_{n,k}$, denote the probability that exactly of the n, numbers chosen by the player are among the 20 selected by the house.

b. Compute $p_{n,k}$

c. The most typical wager at Keno consists of selecting 10 numbers. For such a bet, the casino pays off as shown in the following table. Compute the expected payoff: