A game popular in Nevada gambling casinos is Keno, which is...

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 1 number, then he or she wins if this number is among the set of $20,$ and the payoff is $\$ 2.2$ 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 $k$ 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: $$\begin{array}{cc} \hline \multicolumn{2}{c} {\text { Keno Payoffs in 10 Number Bets }} \\ \hline \text { Number of matches } & \text { Dollars won for each \$1 bet } \\ \hline 0-4 & -1 \\ 5 & 1 \\ 6 & 17 \\ 7 & 179 \\ 8 & 1,299 \\ 9 & 2,599 \\ 10 & 24,999 \\ \hline \end{array}$$

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 1 number, then he or she wins if this number is among the set of $20,$ and the payoff is $\$ 2.2$ 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
$k$ 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:
$$\begin{array}{cc}
\hline \multicolumn{2}{c} {\text { Keno Payoffs in 10 Number Bets }} \\
\hline \text { Number of matches } & \text { Dollars won for each \$1 bet } \\
\hline 0-4 & -1 \\
5 & 1 \\
6 & 17 \\
7 & 179 \\
8 & 1,299 \\
9 & 2,599 \\
10 & 24,999 \\
\hline
\end{array}$$

Key Concepts

Hypergeometric Distribution

This concept describes the probability of drawing a specific number of successes in a finite number of draws without replacement from a finite population. In the context of the Keno game, the hypergeometric distribution is used to compute the probability that exactly k of a player’s n selected numbers are among the 20 numbers drawn by the house from the set of 80 numbers. It is a fundamental tool when events are dependent and are sampled without replacement.

Combinatorial Analysis

Combinatorial analysis involves calculating combinations, often represented by binomial coefficients, which count the number of ways in which a set of items can be selected. This is crucial for setting up the probability expression in the hypergeometric scenario, where both the number of ways to choose k successes from n picks and the number of ways to choose the remaining items from the rest of the population must be computed and compared to the total number of outcomes.

Expected Value Calculation

Expected value is a central concept in probability and statistics that measures the average outcome of a random event over many trials. In the gambling setting described in the problem, the expected value is computed by weighting each outcome (number of matches) by its corresponding probability and payoff. This calculation helps determine whether a wager is beneficial, fair, or unfavorable in the long run.

Fair Betting and Payoff Determination

The notion of a 'fair' bet in gambling is defined as one in which the expected value of the wager is zero, meaning that the player neither gains nor loses money on average. When determining the fair payoff, one typically sets the payoff equal to the reciprocal of the probability of winning, ensuring that the average return over many plays balances the cost of the bet. This concept is critical for designing games that are equitable in a theoretical sense.

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