Chance of a non-flush straight in five cards

Question

Call a five-card poker hand a straight if its ranks are five in a row (an ace may be high or low) and the cards are not all of one suit. What is the probability that a random five-card hand from a standard deck is a straight?

Short answer

$\dfrac{10(4^5-4)}{\binom{52}{5}}=\dfrac{10200}{2598960}\approx0.0039$ . There are 10 possible rank sequences (A–5 up to 10–A). Each card can be any of 4 suits, giving $4^5$ suit choices, minus the 4 single-suit (straight flush) cases.

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Lessons behind this question

Counting and Combinatorics
Matrix Algebra: Operations, Rules, and the Transpose
Kernel, Image, and the Rank-Nullity Theorem

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