Two dice, keep the sum or re-roll both once
Asked at SIG, Optiver
You roll two fair six-sided dice and are paid their sum in dollars. You may keep the sum, or discard it and re-roll both dice once, taking whatever they then total.
What is the optimal strategy, and what is the fair value of the game?
Show a hint
The re-roll is worth the mean sum, . Keep sums above , but be careful: you need the conditional mean of the kept sums, not just their average.