Paid only if the last roll wins

MediumExpected Value~7m

You roll a fair die three times. You are paid the value of the third roll in dollars, but only if it is strictly greater than both of the first two rolls; otherwise you get nothing. What is the expected payout? Give a decimal to two places.

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Condition on the third roll $c$ : you're paid $c$ only if both earlier rolls are below $c$ .
That has probability $\left(\tfrac{c-1}{6}\right)^2$ , so sum $c \cdot \tfrac16 \left(\tfrac{c-1}{6}\right)^2$ .

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1.3 (± 0.01)

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Paid only if the last roll wins

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