Bank before you roll the bad face

Question

You grow a running total by rolling a fair die and adding each roll. This time, rolling a 6 wipes your entire pile to zero and ends the turn. You may stop and bank at any point before rolling.

At what running total should you stop, to maximize the expected banked amount?

Accepted Answer

Weigh one more roll at pile size t: - With probability 	frac16 you roll the killer 6 and lose your pile, a change of -t. - With probability 	frac56 you roll one of 1,2,3,4,5, whose average is 	frac{1+2+3+4+5}{5} = 3, a change of +3 on average. Expected change of one more roll: \frac{1}{6}(-t) + \frac{5}{6}(3) = \frac{15 - t}{6}. This is positive while t < 15, so you keep rolling below 15 and stop once your pile reaches 15 or more. Notice the break-even total dropped from 20 (when a 1 was the killer) to 15 here. That is because the surviving faces now average only 3 instead of 4: each safe …

Bank before you roll the bad face

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