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A local PTA has decided to have a "poker night" as a fundraiser. The cost to play is $5. Instead of playing against others, each participant will be randomly dealt five cards from a full deck. The payout for each winning hand (and the number of ways that each hand can be drawn from a full deck) is given below:

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Answer:

Part a: The probability of the P(X=-995) is 0.0002558

Part b: The net profit for 1500 participants is 1161.46.

Explanation:

The solution is presented in the table attached with. Here are the key steps in identifying the solution.

First the ways to deal 5 cards out of 52 cards is given as

[tex]n=^{52}C_5\\n_{total}=2598960[/tex]

For the payout and number of ways, considering the table in the complete question referred in the comments section we have

Hand                     # Ways Payout X: Net profit to the PTA

Royal Flush                  4           1000         5-1000=-995

Straight Flush                 36          1000             5-1000=-995

Four of a Kind        624          1000         5-1000=-995

Full House                3744           100           5-100=-95

Flush                        5108           100           5-100=-95

Straight                      10200    50            5-50=-45

Three of a Kind      54912            20                   5-20=-15

Two Pairs             123552          10                     5-10=-5

One Pair                    1098240     6                      5-6=-1

[tex]n_{win}=[/tex]Total            1296420  

Number of ways in which the participant does not win is given as

[tex]n_{not win}=n_{total}-n_{win}\\n_{not win}=2598960-1296420\\n_{not win}=1302540[/tex]

Now for this the profit is 5.

Part a:

P(X=-995) is given as below

[tex]P(X=-995)=P_{Royal Flush}+P_{Straight Flush}+P_{Four of a kind}\\P(X=-995)=\frac{n_{royal}}{n_{total}}+\frac{n_{straight}}{n_{total}}+\frac{n_{four}}{n_{total}}\\P(X=-995)=\frac{n_{royal}+n_{straight}+n_{four}}{n_{total}}\\P(X=-995)=\frac{4+36+624}{2598960}\\P(X=-995)=0.00025548[/tex]

So the probability of the P(X=-995) is 0.0002558

Part b:

For this the individual probability for each value of X is calculated using the same method as in part a. the result is given as below

X                P(X)

-995 0.000255487

-95         0.003405978

-45        0.003924647

-15        0.021128451

-5       0.047539016

-1      0.422569028

5     0.501177394

For the last value the number of not wins is used.

Now as the total number of the participants is 1500, the profits are calculated as

X                 P(X)                    n                P*X*n

-995 0.000255487 1500 -381.3143475

-95         0.003405978 1500 -485.351865

-45           0.003924647 1500 -264.9136725

-15             0.021128451 1500 -475.3901475

-5           0.047539016 1500  -356.54262

-1           0.422569028 1500 -633.853542

5           0.501177394         1500 3758.830455

Total                                     1161.464261

So the net profit for 1500 participants is 1161.46.

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