Odds of a four-of-a-kind poker hand

Question

Five cards are dealt from a well-shuffled standard 52-card deck. What is the probability that the hand contains four cards of the same rank?

Short answer

$1/4165\approx 0.00024$ . Choose the rank (13 ways) and the fifth card from the other 48, giving $624$ hands out of $\binom{52}{5}=2{,}598{,}960$ .

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Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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