Question
You are given a fair 20-sided die and 100 actions in a game. The die starts with upface $1$. The two options you can perform are to roll and to take. Performing a roll re-rolls the current upface of the die. Performing a take allows you to cash out the current upface of the die. Note that the game does not end when you perform a take and that you do not have to roll between takes. Therefore, for example, you can just perform $100$ takes on the initial $\$1$ upface and walk away with $\$100$ guaranteed. Your strategy is to cash out the upface when you roll at least some threshold $n$ for the first time. You fix this $n$ at the beginning of the game. Assuming rational strategy in selecting $n$, what is your expected payout on this game?