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Single optional reroll

Expected valueEasy

You roll a fair $6$ -sided die once and see the face. You may keep it, or reroll once and then you must keep the second result. Your payoff equals the face you keep, and you play to maximize the expected payoff. What is the expected payoff?

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Hint

Compare the first face with the expected value of a fresh roll.

Solution

A fresh fair roll has expected value $3.5$ , so that is what a reroll is worth.
Reroll exactly when the first face is below $3.5$ , that is on $1$ , $2$ or $3$ .
Keeping happens with probability $\frac{1}{2}$ and the kept face averages $5$ , contributing $2.5$ .
Rerolling happens with probability $\frac{1}{2}$ and is worth $3.5$ , contributing $1.75$ .
Adding the parts gives $\frac{17}{4}$ .

Answer: $\frac{17}{4}$

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