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. However, you must roll the die again before doing another take. Your strategy is to accept any number that is at least some threshold n. This n must be decided in advance and is fixed for the entire game. Assuming rational play in selecting n, find your expected payout.
