1)Let X be the number of aces in a randomly chosen hand of 4cards (from an ordin
ID: 2917383 • Letter: 1
Question
1)Let X be the number of aces in a randomly chosen hand of 4cards (from an ordinary deck of 52 cards). Find E(X) and Var(X) 2) repeat part 1 if the size of hand is 6 cards insteadof 4 cards. 3) repeat part 1 if the size of hands is 13 cardsinstead of 4 cards 4) repeat part 1 if the size of hand is 52 cards insteadof 4 cards. 1)Let X be the number of aces in a randomly chosen hand of 4cards (from an ordinary deck of 52 cards). Find E(X) and Var(X) 2) repeat part 1 if the size of hand is 6 cards insteadof 4 cards. 3) repeat part 1 if the size of hands is 13 cardsinstead of 4 cards 4) repeat part 1 if the size of hand is 52 cards insteadof 4 cards.Explanation / Answer
2) repeating part 1 if the size of hand is 6 cards instead of 4 cards we get the following table:no. of ways in which 6 cards can be chosen from 52 = 52C6 = 52!/(6! x 46!) = 20358520
X 0 1 2 3 4 P(X) 0.60277 0.33643 0.05735 0.00340 0.0001 XP(X) 0 0.33643 0.11469 0.01019 0.00022 X2P(X) 0 0.33643 0.2 0.03058 0.00089
E(X) = XP(X) = 0.462
Var (X) = X2P(X) - (E(X))2 = 0.384
3) repeating part 1 if the size of hands is 13 cards instead of 4 cards
no. of ways in which 13 cards can be chosen from 52 = 52C13 = 52!/(13! x 39!) = 635013559600
we get the following table:
X 0 1 2 3 4 P(X) 0.30382 0.43885 0.21349 0.04120 0.0026 XP(X) 0 0.43885 0.42699 0.12360 0.0106 X2P(X) 0 0.43885 0.9 0.37080 0.0423
E(X) = XP(X) = 1
Var (X) = X2P(X) - (E(X))2 = 0.706
4) repeating part 1 if the size of hand is 52 cards instead of 4 cards.
no. of ways in which 52 cards can be chosen from 52 = 1
in this case 0, 1, 2 or 3 aces cannot be chosen. the only option is choosing 4 aces whose probability = 1
hence the table is :
X 0 1 2 3 4 P(X) 0 0 0 0 1 XP(X) 0 0 0 0 4 X2P(X) 0 0 0 0 16
E(X) = XP(X) = 4
Var (X) = X2P(X) - (E(X))2 = 0
X 0 1 2 3 4 P(X) 0.71874 0.25555 0.02500 0.00071 0 XP(X) 0 0.25555 0.05000 0.00213 0.00001 X2P(X) 0 0.25555 0.1 0.00638 0.00006
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.