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Conditional reroll payoff

Market makingMedium

A fair $6$ -sided die is rolled. If it shows $1$ to $4$ , the contract pays that number in dollars. If it shows $5$ or $6$ , the die is rolled once more and the contract pays the sum of both rolls in dollars. What is its fair value in dollars? Give an exact fraction.

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Hint

Split the payoff into the first roll plus a bonus roll that happens only sometimes.

Solution

The first roll always counts and has expectation $3.5$ .
The second roll happens with probability $\frac{2}{6} = \frac{1}{3}$ and has expectation $3.5$ , independent of the first roll.
The fair value is $3.5 + \frac{3.5}{3} = \frac{14}{3}$ .

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

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