Beating the dealer with one card each

Question

You and a dealer each receive one card from a shuffled 52-card deck. You win only if your card's rank is strictly higher than the dealer's. What is your probability of winning?

Short answer

$8/17$ . Given your card, the dealer's card has the same rank with probability $\tfrac{3}{51}=\tfrac1{17}$ . Otherwise either card is equally likely to be higher, so you win with probability $\tfrac12\left(1-\tfrac1{17}\right)=\tfrac8{17}$ .

Adapted from OpenQuant; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

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

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