Optimal Strategy for Rolling a 20-Sided Die with 100 Chances
You are playing a game with a fair 20-sided die (faces labeled
$ through
0$) and you have 00$ chances. Each chance, you roll the die and see the result. You can either **collect** the number (adding it to your running total) or **skip** it. Either way, one chance is used. Your goal is to maximize the expected total collected value over all
00$ chances.
What is the optimal strategy, and what is the expected total under that strategy?
As a follow-up: suppose instead that each chance you can either roll the die (using up one chance) or collect the currently showing number (without using a chance). Now your 100 chances are a budget for rolling only, and you want to maximize your total collected value. How does the optimal strategy change?
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