Probability of a full house

Question

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

Accepted Answer

Count full houses by construction: - Rank for the triple: 13 choices; its suits: \binom{4}{3} = 4. - Rank for the pair: 12 remaining choices; its suits: \binom{4}{2} = 6. \text{full houses} = 13 \cdot 4 \cdot 12 \cdot 6 = 3744. Total hands: \binom{52}{5} = 2{,}598{,}960. So P = \frac{3744}{2598960} = \frac{6}{4165} \approx 0.00144, about 1 in 694 hands. Common pitfalls. (1) The triple-rank and pair-rank choices are ordered, "aces full of kings" differs from "kings full of aces", so 13 	imes 12 is correct, not \binom{13}{2}. (2) Don't multiply by orderings of the five cards: both numerator a…

Probability of a full house

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