Expected tosses for the winner of a first-head race

Question

Two people each toss their own fair coin until they get a head. Let $X$ and $Y$ be how many tosses each needs. Given that $X<Y$ (strictly), what is the expected value of $X$ ?

Short answer

$\tfrac43$ . $P(X=k,\,Y>k)=2^{-k}\cdot2^{-k}=4^{-k}$ , so conditional on $X<Y$ , $X$ is geometric with success probability $\tfrac34$ (total probability $\tfrac13$ ). Its mean is $\tfrac1{3/4}=\tfrac43$ .

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

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Lessons behind this question

Conditional Expectation and Variance Revisited
Conditional Probability, Total Probability, and Bayes' Rule
Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating

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