Reroll on a twenty-sided die

Question

You roll a fair twenty-sided die (faces $1$ through $20$ ) and are paid its value in dollars. Before being paid, you may reroll once; if you do, you must accept the second roll.

What is the fair value of this game under optimal play?

Accepted Answer

A fresh d20 is worth its mean: E[	ext{d20}] = \frac{1 + 20}{2} = 10.5. So keep any first roll worth more than 10.5, that is 11 through 20, and reroll on 1 through 10. Now value the game at the start: - With probability 	frac{10}{20} = 	frac12, the first roll is 11 to 20, averaging \dfrac{11+20}{2} = 15.5. - With probability 	frac12, you reroll, worth 10.5. E = 	frac12(15.5) + 	frac12(10.5) = 7.75 + 5.25 = 13. The game is worth \13.00. Takeaway. The reroll rule is scale-free: keep the first roll if it beats the mean of the distribution, then average the "keep" region against the continua…

Reroll on a twenty-sided die

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