When to bank in a bust-or-build dice game

Question

You build up a running total by rolling a fair die and adding each roll to your pile. But there is a catch: if you ever roll a 1, your entire pile is wiped to zero and the turn ends. At any point before rolling, you may instead stop and bank whatever you have.

At what running total should you stop rolling? (Your goal is to maximize the expected amount you bank.)

Accepted Answer

Compare, at any pile size t, what one more roll does to you on average. A roll has two cases: - With probability 	frac16 you roll a 1 and lose your whole pile, a change of -t. - With probability 	frac56 you roll one of 2,3,4,5,6, whose average is 	frac{2+3+4+5+6}{5} = 4, a change of +4 on average. So the expected change from one more roll is \frac{1}{6}(-t) + \frac{5}{6}(4) = \frac{-t + 20}{6} = \frac{20 - t}{6}. This is positive while t < 20 and negative once t 20. So you should keep rolling as long as your pile is below 20 and stop once it reaches 20 or more. Why is this simple "one step…

When to bank in a bust-or-build dice game

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