One reroll on a twenty-sided die, played optimally

Question

You roll a fair twenty-sided die (faces 1 through 20). You may keep its value, or reroll once and must take the second roll. Playing to maximize your expected score, what score should you expect?

Accepted Answer

Set the rule, then condition on the first roll (the tower rule). Step 1, the reroll's worth. A fresh twenty-sided die averages E[	ext{reroll}] = \frac{1 + 20}{2} = 10.5. So keep the first roll only when it beats 10.5, that is when it is 11 through 20, and reroll on 1 through 10. Step 2, condition on the first roll. - First roll is 11 through 20 (probability 	frac12): keep it, averaging \frac{11 + 20}{2} = 15.5. - First roll is 1 through 10 (probability 	frac12): reroll, averaging 10.5. Step 3, combine: E[	ext{score}] = 	frac12 	imes 15.5 + 	frac12 	imes 10.5 = 7.75 + 5.25 = 13. Your e…

One reroll on a twenty-sided die, played optimally

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