Die payouts that continue only on high rolls

Question

You roll a fair die and receive the number shown in dollars. If the roll was $4$ , $5$ or $6$ you roll again (and are paid again); after a $1$ , $2$ or $3$ the game ends, though you still collect that roll. What is the expected total amount you receive?

Short answer

$7. Let $E$ be the expected total. Each roll pays $3.5$ on average and with probability $\tfrac12$ the game restarts, so $E=3.5+\tfrac12E$ , giving $E=7$ .

Adapted from Zhou, A Practical Guide to Quantitative Finance Interviews, sec. 4.5 Expected Value, Variance & Covariance, 'Dice game (roll again on 4-6)', p.93; question and answer rewritten by SelfMadeQuant.com.

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