Waiting time for two consecutive heads

Question

Danielle keeps tossing a fair coin and stops the first time she gets heads on two tosses in a row. On average, how many tosses does she make?

Short answer

$6$ . Let $E$ be the expected count from scratch. A tail costs one toss and restarts, heads then tails costs two and restarts, and two heads ends after two: $E=\tfrac12(1+E)+\tfrac14(2+E)+\tfrac14\cdot 2$ , so $E=6$ .

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

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Lessons behind this question

Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited
Sample Spaces, Events, and the Probability Axioms

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