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
`(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:

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)=((^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)=((^(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)
`=((^(20)_2)(^(60)_(2-2)))/((^(80)_2))`

Probability P ( Both number in the set 20)
`= (20*19)/(80*79)`

Probability P ( Both number in the set 20)
`=(19)/(316)`

Probability P ( Both number in the set 20)
`=(1)/(16.63)`

Thus; the payoff odd for
`=(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)=((^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)=((^(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.