Keep your die roll or reroll once, played optimally

Question

You roll a fair six-sided die. You may keep its value, or reroll once and must take the second roll no matter what. Playing to maximize your expected score, what score should you expect?

Accepted Answer

First decide the best rule, then compute the expected score by conditioning on the first roll (the tower rule). Step 1, the reroll's worth. If you reroll, you take a fresh fair die, worth E[	ext{reroll}] = 3.5 on average. So you should keep the first roll only when it beats 3.5, that is when it is 4, 5, or 6, and reroll when it is 1, 2, or 3. Step 2, condition on the first roll. Split into the keep branch and the reroll branch: - First roll is 4, 5, or 6 (probability 	frac12): you keep it, averaging \frac{4+5+6}{3} = 5. - First roll is 1, 2, or 3 (probability 	frac12): you reroll, averagin…

Keep your die roll or reroll once, played optimally

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