One reroll

You roll a fair die and may reroll once, keeping the second result. With optimal play, what is the expected final value?

Hint

Compare the value of keeping a roll with the expected value of rerolling.

Solution

A reroll is worth $3.5$ on average, so keep any roll of 4 or more and reroll 1, 2, 3.

$E = \tfrac{1}{2}\cdot 5 + \tfrac{1}{2}\cdot 3.5 = 4.25$

Keeping 4, 5 or 6 averages 5 (half the time), and rerolling gives 3.5 (the other half).

Answer

$4.25$