Cash out or bust

MediumProbability Expected Value~6m

You repeatedly roll a fair die. On a $1$ , $2$ , or $3$ you bank $1 and roll again. On a $4$ or $5$ you cash out and keep everything banked so far. On a $6$ you go bust and leave with nothing. What is the fair value of this game (the most you would pay to play)?

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Let $V(g)$ be the expected payout with $g banked: $V(g)=\tfrac12 V(g+1)+\tfrac13 g$ .
Guess $V(g)=ag+b$ ; you get $a=b=\tfrac23$ , so $V(0)=\tfrac23$ .

Answer

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0.6667 (± 0.005)

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Cash out or bust

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