A release-checked easy problem for training Expected Value, Conditional Expectation, Games.
Question
Alice rolls a fair 6-sided die with the values $1-6$ on the sides. She sees that value showing up and then is allowed to decide whether or not she wants to roll again. The re-roll costs $\$1$. If she decides to stop, Alice receives a payout equal to the upface of the die. If she rolls again, she receives a payout equal to the upfaces of the second roll. Assuming Alice plays optimally and can only re-roll at most once, what is her expected payout on this game?
Practice focus
This Probability problem is tagged Expected Value, Conditional Expectation, Games. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.