In this problem we have a deck of 52 cards and we shuffle th

In this problem, we have a deck of 52 cards and we shuffle them randomly and deal the whole deck out to 4 people, so that each player has 13 cards. As usual, when dealing cards, it is without replacement and the \"hand\" that each player has is a set.

(a) What is the probability that each player will receive exactly three face cards?

(b) What is the probability that each player will receive all cards of one suit (i.e., one gets all clubs, another all hearts, another all diamonds, and another all spades)?

Solution

a)

There are 52!/(13!13!13!13!) = 5.36447*10^28 ways to distribute the cards.

Now, there are 12!/(3!3!3!3!) = 369600 ways to distrbute the face cards, and 40!/(10!10!10!10!) = 4.70536*10^21 ways to distrbute the remaining 40 cards.

Thus, there are 369600*4.70536*10^21 = 1.7391*10^27 ways to do the job.

Thus, its probability is

P = (1.7391*10^27) / (5.36447*10^28) = 0.03241886 [answer]

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b)

There is only 1 way to do this.

Thus, the probability of this is

P = 1/(5.36447*10^28) = 1.86412*10^-29 [ANSWER]