Probability of a flush

Question

You are dealt 5 cards from a well-shuffled standard 52-card deck. What is the probability of a flush (all five cards of the same suit)?

Accepted Answer

Count five-same-suit hands by construction: - Choose the suit: 4 ways. - Choose 5 of the 13 cards in that suit: \binom{13}{5} = 1287. \text{(all same suit)} = 4 	imes 1287 = 5148. Total hands: \binom{52}{5} = 2{,}598{,}960. So the probability of five cards all in one suit is P = \frac{5148}{2{,}598{,}960} \approx 0.00198, about 1 in 505. The subtlety poker players insist on. In poker, a "flush" usually excludes straight flushes (five in a row, same suit), which rank higher. There are 10 straight sequences per suit (from A-2-3-4-5 up through 10-J-Q-K-A) and 4 suits, so 40 straight flushes. S…

Probability of a flush

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