Draw chips until a red one ends you

Question

You draw chips one at a time from a large well-mixed bag (each draw is replaced, so the odds stay fixed). With probability $\tfrac13$ a chip is red and wipes your entire pile to zero, ending the turn. Otherwise (probability $\tfrac23$ ) the chip is a cash chip that adds an amount averaging $6 to your pile. You may stop and bank at any point before drawing.

At what pile size should you stop, to maximize the expected banked amount?

Accepted Answer

Consider one more draw at pile size t: - With probability 	frac13 you draw red and lose t, a change of -t. - With probability 	frac23 you gain \6 on average, a change of +6. Expected change of one more draw: \frac{1}{3}(-t) + \frac{2}{3}(6) = -\frac{t}{3} + 4 = \frac{12 - t}{3}. This is positive while t < 12, so you keep drawing below \12 and stop once your pile reaches \12 or more. The red-chip risk here is steep (one in three), which keeps the stopping point low: even though each cash chip adds a healthy \6, the frequent wipeouts mean you should lock in early. \boxed{	ext{Stop as soon as…

Draw chips until a red one ends you

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