Alternating biased coins, first head wins

Question

Annie and Bob alternate tossing their own coins, Annie first. Annie's coin shows heads with probability $\tfrac13$ and Bob's with probability $\tfrac25$ . Whoever first tosses a head wins. What is Annie's probability of winning?

Short answer

$\tfrac59$ . Annie wins immediately with probability $\tfrac13$ ; otherwise, if Bob also misses (probability $\tfrac23\cdot\tfrac35=\tfrac25$ ), the game restarts. So $p=\tfrac13+\tfrac25p$ , which gives $p=\tfrac59$ .

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Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Maximum Likelihood Estimation: Score and Log-Likelihood
Bias, Variance, and Mean Squared Error

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