Payout from rerolling a die until it shows above one

Question

You keep rolling a fair die for as long as it shows a $1$ . As soon as any other number appears you stop and are paid that number in dollars. What is your expected payout?

Short answer

$4. Conditional on not being $1$ , the final face is uniform on $\{2,3,4,5,6\}$ , so its mean is $\tfrac{2+3+4+5+6}{5}=4$ .

Adapted from Crack, Heard on the Street, Q4.11, p.38; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Conditional Expectation and Variance Revisited
Counting and Combinatorics
Expectation, Mean, and Variance

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