You roll a fair six-sided die and receive a payoff equal to the number shown. a) What is the expected payoff? b) If you are given the option to re-roll after seeing the result, under what conditions should you choose to re-roll? What is your expected payoff with this re-roll option?

Free worked solution · Quant Blueprint

The initial expected payoff is 3.5; reroll 1, 2 or 3, keep 4, 5 or 6, and obtain an expected payoff of 4.25.

1. Assume the reroll is independent, free, and replaces the first result; you must keep the second roll. Its expected payoff is 3.5.
2. After observing x, compare keeping x with the continuation value 3.5. Reroll exactly when x < 3.5.
3. Average the optimal conditional values: (3.5+3.5+3.5+4+5+6)/6 = 25.5/6 = 4.25.

Common mistake

This is not the maximum of two dice: you cannot recover the first result after choosing to reroll.

Follow-up

If a reroll costs c, its continuation value is 3.5 − c. Reroll only when x < 3.5 − c; at equality either action has the same expected payoff.

Original educational explanation for this practice question. Reviewed 2026-10-13. Firm-style practice; the label describes the collection’s preparation context.