medium Games

Distinct and Gamey

A release-checked medium problem for training Expected Value/LOTUS, Dice.

Question

A standard fair 6-sided die is repeatedly rolled. After each roll, as long as the value showing isn't $1$, you have the option to roll again or cash out an amount (in dollars) equal to the number of distinct faces you've observed thus far. If you roll a $1$, you receive no payout. Under the optimal strategy, what is your expected payout?

Practice focus

This Games problem is tagged 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.