# Probability of a full house

Please help with the following problems.

4. In one variety of poker, players are dealt five cards from an ordinary deck of 52 cards.

(a) A full house in poker is a hand of five cards of one denomination and 3 cards of another denomination. Find the probability of a player being dealt a full house.

(b) Two pairs is a poker hand containing 2 cards of one denomination, another 2 cards of a second denomination, and a fifth cards of a third dimension. Find the probability of a player being dealt two pairs.

5. Given n marbles where k of the marbles are black and the rest are white, a game is played as follows. You choose two marbles at random, and you win the game if the marbles are of different colors. Prove that for a fixed n, you have the highest probability of winning when k = n/2 (for even values of n) and when k = (n (plus or minus) 1)/2 (for odd values of n).

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#### Solution Preview

Please see the attached document for step by step calculations for both of the questions.

Probability of a full house:

13(■(4@3))12(■(4@2))/((■(52@5)) ...

#### Solution Summary

In this solution the probability of a full house is carefully depicted in this solution. Step by step calculations are given, along with explanations.