Beating the dealer with one card each

Question

From a well-shuffled standard $52$ -card deck, you and the dealer each receive one card. You win only if your card's rank is strictly higher than the dealer's; ties and lower ranks lose. What is your probability of winning?

Short answer

$8/17$ . Given your card, the dealer's card shares its rank with probability $\tfrac{3}{51}=\tfrac1{17}$ . Otherwise higher and lower are equally likely by symmetry, so you win with probability $\tfrac12\cdot\tfrac{16}{17}=\tfrac{8}{17}$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Conditional Probability, Total Probability, and Bayes' Rule
Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating
Sample Spaces, Events, and the Probability Axioms

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