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Optional rerolls of a die

Expected valueMedium

You roll a fair $6$ -sided die and may reroll it up to $2$ times, seeing each result before you decide. The last roll is your payoff, so if you use both rerolls the third roll counts. Playing optimally, what is the expected payoff?

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Hint

Work backwards: find the value with $1$ roll left, then with $2$ rolls left, then with $3$ rolls left.

Solution

With $1$ roll left the value is $3.5$ .
With $2$ rolls left you keep faces above $3.5$ , which gives value $\frac{17}{4}$ .
With $3$ rolls left you keep faces above $\frac{17}{4}$ , that is $5$ or $6$ , with probability $\frac{1}{3}$ and average $5.5$ , contributing $\frac{11}{6}$ .
Otherwise, with probability $\frac{2}{3}$ , you reroll and get $\frac{17}{4}$ , contributing $\frac{17}{6}$ .
The total is $\frac{11}{6} + \frac{17}{6} = \frac{14}{3}$ .

Answer: $\frac{14}{3}$

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