Probability of two pair

Question

You are dealt 5 cards from a well-shuffled standard 52-card deck. What is the probability of two pair (two cards of one rank, two of another rank, and a fifth card of a third rank)?

Accepted Answer

Build the hand: - The two pair-ranks: choose 2 of the 13 ranks, \binom{13}{2} = 78. These are unordered, because "kings and queens" is the same two-pair as "queens and kings," the two pairs have equal status. - Suits for each pair: \binom{4}{2} = 6 per pair, so 6 	imes 6 = 36. - The fifth card: any card of a different rank, which is 52 - 8 = 44 cards (all ranks except the two already used). \text{(two pair)} = 78 	imes 36 	imes 44 = 123{,}552. Total hands: \binom{52}{5} = 2{,}598{,}960. So P = \frac{123{,}552}{2{,}598{,}960} \approx 0.0475, about 4.75\%, or roughly 1 in 21. The instructiv…

Probability of two pair

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