Problem 4. Poker hands Each hand in Poker consists of 5-cards from a deck of 52

Question

Problem 4. Poker hands Each hand in Poker consists of 5-cards from a deck of 52 cards (4 suits and 13 ranks). The most common hands are one-pair and two-pair: One-pair: Two cards of one rank, plus three cards which are not of this rank nor the same as each other, such as 4,4e, Ke ,10*,5 Two-pair: Two cards of the same rank, plus two cards of another rank (that match each other but not the first pair), plus any card not of either rank, such asJ,4,4,9 (a) (5 points) How many 5-card poker hands are there? b)(10 points) What is the probability of getting one-pair in a 5-card poker hand? ( (10 points) What is the probability of getting two-pair in a 5-card poker hand?

Explanation / Answer

a)

52C5 = 52!/[(52-5)!*5!] = [52*51*50*49*48]/[1*2*3*4*5] = 2598960

b) 1098240

There are (13C1) ways to pick the kind we have a pair in. For each such way, we have (4C2)ways to pick the actual cards.

For each of these ways, there are (12C3) ways to pick the kinds we will have one each of. For each of these kinds, there are (4C1) ways to pick the actual cards, for a total of (13C1)(4C2)(12C3)(4C1)^3 ways.

c)

123552

This hand has the pattern AABBC where A, B, and C are from distinct kinds. The number of such hands is (13-choose-2)(4-choose-2)(4-choose-2)(11-choose-1)(4-choose-1)

There are (13C1) ways to pick the kind we have a pair in. For each such way, we have (4C2)ways to pick the actual cards.

For each of these ways, there are (12C3) ways to pick the kinds we will have one each of. For each of these kinds, there are (4C1) ways to pick the actual cards, for a total of (13C1)(4C2)(12C3)(4C1)^3 ways.

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