Expected tosses until heads lead by 10 or trail by 7

Question

You toss a fair coin repeatedly and track (number of heads) minus (number of tails). What is the expected number of tosses until this difference first reaches $+10$ or $-7$ ?

Short answer

$70$ . The difference is a simple symmetric random walk from $0$ . Its expected exit time from the interval $(-a,b)$ is $ab$ , here $7\cdot10=70$ .

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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