6. 4 cards are randomly dealt to each of 13 players. (a) Describe the sample spa

Question

6. 4 cards are randomly dealt to each of 13 players.

(a) Describe the sample space and write down its size.

(b) What is the probability of A = {each player has one card from each suit}?

(c) What is the probability of B = {each player has all four cards of the same value}?

(d) (*) Compute PB(A) and PA(B). Are A and B independent? 1

(e) What is the probabiility that players one, two, and three have been dealt only aces, kings, or queens. As usual, explain.

(f) (**) What is the probability that one player has one card from each suit but that nobody else has cards from more than one suit?

Explanation / Answer

a)
4 cards are dealt to each of 13 players
Sample space has all possible combinations of 52 cards among 13 players.

there are 52! ways to deal out the cards, however the order within a hand does not matter hence divide by (4!)^12
total possible outcomes in sample space are 52!/(4!)^13

b)
there are 13 cards of each suit for player 1, 12 cards of each suit for player 2, and so on. Thus, the total is
13^4 * 12^4 * ... = (13!)^4
P(A) = (13!)^4 / 52!/(4!)^13

c)
there are 13 possible hands for player 1, 12 possible hands for player 2, and so on. Thus, the total is 13!
P(B) = 13! / 52!/(4!)^13

d)
PB(A) = 1 (in order for a player to have four of the same value, they must be different suits)

PA(B) = P(B)/P(A) = 1/(13!)^3

The events are not independent

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