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Push-your-luck dice

Expected valueHard

You pay an entry fee of $3$ dollars. Then you roll a fair $6$ -sided die repeatedly: a $1$ ends the game with a payout of $0$ , otherwise the roll is added to your running total. At any moment before a roll you may stop and receive the running total in dollars. With optimal play, what is your expected net result, rounded to $2$ decimal places?

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Hint

Find the running total at which one more roll stops paying off.

Solution

From a total of $t$ , one more roll is worth $\frac{5}{6} \times (t + 4)$ , since the safe rolls $2$ to $6$ average $4$ and a $1$ loses everything.
That exceeds $t$ exactly when $t < 20$ , so the optimal rule is to roll until the total is at least $20$ .
Backward recursion over all totals from $0$ gives an expected gross payout of about $8.1418$ .
Subtracting the entry fee, the expected net result is $8.1418 - 3 \approx 5.14$ .

Answer: $\approx 5.14$

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