326.Roll-or-Take on a d20
★★★optimal stoppingCh. 6 — Random Variables and Expectation444+ Problems in Probability
You have a fair 20-sided die and 100 actions. Each action is either a roll (re-roll the current face) or a take (cash out the current face in dollars) — and you must roll again before each take. You fix a threshold in advance and take whenever the face is at least . With the optimal threshold, what is your expected payout, rounded to the nearest whole dollar?
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With threshold , a success costs on average rolls plus 1 take, where , and pays the conditional mean . Balance value per action.
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