Respuesta :
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 Â
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 Â Â Â Â Â Â Â Â P(X)
-995 0.000255487
-95 Â Â Â Â 0.003405978
-45 Â Â Â Â 0.003924647
-15 Â Â Â Â 0.021128451
-5 Â Â Â 0.047539016
-1 Â Â Â 0.422569028
5 Â Â 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 Â Â Â Â 0.003405978 1500 -485.351865
-45 Â Â Â Â Â 0.003924647 1500 -264.9136725
-15 Â Â Â Â Â Â 0.021128451 1500 -475.3901475
-5 Â Â Â Â Â 0.047539016 1500 Â -356.54262
-1 Â Â Â Â Â 0.422569028 1500 -633.853542
5 Â Â Â Â Â 0.501177394 Â Â Â Â 1500 3758.830455
Total                   1161.464261
So the net profit for 1500 participants is 1161.46.
