Probability of exactly one pair

Question

You are dealt 5 cards from a well-shuffled standard 52-card deck. What is the probability of exactly one pair (two cards of one rank, and three other cards of three different ranks)?

Accepted Answer

Build the hand by construction: - Rank of the pair: 13 choices; its two suits: \binom{4}{2} = 6. - Ranks of the three singletons: choose 3 distinct ranks from the remaining 12, which is \binom{12}{3} = 220 (unordered, because the three kickers play no particular role). - Suit of each singleton: 4 each, so 4^3 = 64. \text{(one pair)} = 13 	imes 6 	imes 220 	imes 64 = 1{,}098{,}240. Total hands: \binom{52}{5} = 2{,}598{,}960. So P = \frac{1{,}098{,}240}{2{,}598{,}960} \approx 0.4226, about 42.3\%, by far the most common "made" category. The pitfall. The three singleton ranks must be chosen …

Probability of exactly one pair

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