Answer:
Part a: <em>The probability of the P(X=-995) is 0.0002558</em>
Part b:<em> </em><em>The net profit for 1500 participants is 1161.46.</em>
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
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
Total 1296420
Number of ways in which the participant does not win is given as
Now for this the profit is 5.
<em>Part a:</em>
P(X=-995) is given as below
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<em>So the probability of the P(X=-995) is 0.0002558</em>
<em>Part b:</em>
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
<em>X P(X)</em>
<em>-995 0.000255487</em>
<em>-95 0.003405978</em>
<em>-45 0.003924647</em>
<em>-15 0.021128451</em>
<em>-5 0.047539016</em>
<em>-1 0.422569028</em>
<em>5 0.501177394</em>
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.