Parity of tosses until two matching in a row

Question

Paul tosses a fair coin until two consecutive tosses show the same face for the first time. What is the probability that the total number of tosses is even?

Short answer

$2/3$ . After the first toss, each subsequent toss either matches the previous one (stop) or not, each with probability $\tfrac12$ . So the stop happens at toss $k\ge2$ with probability $2^{-(k-1)}$ , and the even values sum to $\tfrac12+\tfrac18+\dots=\tfrac23$ .

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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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