A release-checked easy problem for training Conditional Expectation, Expected Value/LOTUS, Dice.
Question
You roll a fair standard 6-sided die. You have two options: keep or roll again. If you decide to keep, you receive a payout equal to the upface (in dollars). Otherwise, you can roll again. If the sum of your two rolls is at least $7$, you receive nothing. If the sum of the two rolls is less than $7$, you receive the upface of your second roll (in dollars). Assuming optimal play, what is your expected payout?
Practice focus
This Games problem is tagged Conditional Expectation, Expected Value/LOTUS, Dice. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.