Question

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 $2.20 won for every dollar bet. (As the player’s probability of winning in this case is , 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, 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 Pn, 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.

Answer 1

The missing part in the question;

and the payoff is $2.20 won for every dollar bet. (As the player’s probability of winning in this case is $\dfrac{1}{4}$........

Also:

For such a bet, the casino pays off as shown in the following table.

The table can be shown as:

Keno Payoffs in 10 Number bets

Number of matches        Dollars won for each $1 bet

0  -   4                                        -1

5                                                  1

6                                                  17

7                                                  179

8                                                 1299

9                                                 2599

10                                               24999

Answer:

Step-by-step explanation:

Given that:

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

Let assume X to represent the numbers of player chooses which are in the Casino-selected-set of 20.

Let assume the random variable X has a hypergeometric distribution with parameters  N= 80 and m =20.

Then, the probability mass function of a hypergeometric distribution can be defined as:

$P(X=k)=\dfrac{(^m_k)(^{N-m}_{n-k})}{(^N_n)}, k =1,2,3 ... n$

Now; the probability that i out of  n numbers chosen by the player among 20  can be expressed as:

$P(X=k)=\dfrac{(^{20}_k)(^{60}_{n-k})}{(^{80}_n)}, k =1,2,3 ... n$

Also; given that ; 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

So; n= 2; k= 2

Then :

Probability P ( Both number in the set 20)  $=\dfrac{(^{20}_2)(^{60}_{2-2})}{(^{80}_2)}$

Probability P ( Both number in the set 20) $= \dfrac{20*19}{80*79}$

Probability P ( Both number in the set 20) $=\dfrac{19}{316}$

Probability P ( Both number in the set 20) $=\dfrac{1}{16.63}$

Thus; the payoff odd for $=\dfrac{1}{16.63}$ is 16.63:1 ,as such fair payoff in this case is $16.63

Again;

Let assume X to represent the numbers of player chooses which are in the Casino-selected-set of 20.

Let assume the random variable X has a hypergeometric distribution with parameters  N= 80 and m =20.

The probability mass function of the hypergeometric distribution can be defined as :

$P(X=k)=\dfrac{(^m_k)(^{N-m}_{n-k})}{(^N_n)}, k =1,2,3 ... n$

Now; the probability that i out of  n numbers chosen by the player among 20  can be expressed as:

$P(n,k)=\dfrac{(^{20}_k)(^{60}_{n-k})}{(^{80}_n)}, k =1,2,3 ... n$

From the table able ; the expected payoff can be computed as shown in the attached diagram below. Thanks.