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

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- feach player has one card from each suit? (c) What is the probability of {each player has all four cards of the same value? (d) * Compute PB(A) and PA(B). Are A and B independent? (e) What is the probabiility that players one, two, and three have been dealt only aces, kings, or queens. (f) (**) What is the probability that one player has one card from each suit but that nobody As usual, explain. else has cards from more than one suit?

Explanation / Answer

Solution

(a) There are 52C4 ways to choose the cards that Player 1 gets, and for each of these ways there are 48C4 ways to choose the cards that Player 2 gets, and so on, for a total of 52C4*48C4*44C44C4 ways.

This number simplifies to 52!/(4!)13 All these ways are equally likely.

(b) For P(A), 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

134*124*114=13!4

(c) For P(B), there are 13 possible hands for player 1, 12 possible hands for player 2, and so on. Thus, the total is

13!

(d) Note that in order for a player to have four of the same value, they must be different suits. Thus,

PB(A)=1

and

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

It is also clear to see that the events are not independent

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