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Reroll before a duel

Expected valueHard

Player A rolls a fair $6$ -sided die and may reroll once, in which case the second result is kept. Then player B rolls one fair $6$ -sided die with no reroll, independently. A wins only if the final face of A is strictly greater than the face of B; ties and lower faces are not wins. If A maximizes the probability of winning, what is that probability?

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Hint

Find the winning probability for each face of A, then compare it with the average over a fresh roll.

Solution

If A holds face $a$ , the winning probability is $\frac{a - 1}{6}$ .
A fresh roll has average winning probability $\frac{15}{36} = \frac{5}{12}$ .
A keeps $a$ exactly when $\frac{a - 1}{6} > \frac{5}{12}$ , that is $a \geq 4$ .
Keeping $4$ , $5$ or $6$ contributes $\frac{3 + 4 + 5}{36} = \frac{1}{3}$ .
Rerolling happens with probability $\frac{1}{2}$ and contributes $\frac{1}{2} \times \frac{5}{12} = \frac{5}{24}$ .
The total is $\frac{1}{3} + \frac{5}{24} = \frac{13}{24}$ .

Answer: $\frac{13}{24}$

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