Full house in a 32-card deck

Question

A short deck keeps only ranks 7, 8, 9, 10, J, Q, K, A, that is 8 ranks in 4 suits, for 32 cards. You are dealt 5 cards. What is the probability of a full house (three of one rank, two of another)?

Accepted Answer

Count full houses by construction, now with only 8 ranks: - Rank for the triple: 8 choices; its suits: \binom{4}{3} = 4. - Rank for the pair: 7 remaining choices; its suits: \binom{4}{2} = 6. \text{full houses} = 8 	imes 4 	imes 7 	imes 6 = 1{,}344. Total hands: \binom{32}{5} = \dfrac{32 	imes 31 	imes 30 	imes 29 	imes 28}{120} = 201{,}376. So P = \frac{1{,}344}{201{,}376} \approx 0.00667, about 1 in 150. Why more likely than in a full deck. In the standard 52-card deck a full house is about 1 in 694; here it is roughly 4.6 times more common. Two effects push the same way. The suit f…

Full house in a 32-card deck

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