Die roll, single re-roll only when you roll a one

Question

You roll a fair six-sided die. If it shows a 1, you must re-roll once and keep that second roll. Otherwise (2 through 6) you keep the first roll.

On average, what is the value of the roll you keep, in dollars?

Accepted Answer

Two cases, blended by their probabilities. Case A: you keep the first roll. This happens for 2, 3, 4, 5, or 6, which is 5 of the 6 faces, probability 5/6. Given you kept it, the value averages: \frac{2+3+4+5+6}{5} = \frac{20}{5} = 4. Case B: you re-roll. This happens only on a 1, probability 1/6. The kept value is a fresh die, averaging 3.5. Blend by probability: EV = \frac{5}{6}(4) + \frac{1}{6}(3.5) = \frac{5(4) + 3.5}{6} = \frac{20 + 3.5}{6} = \frac{23.5}{6} = \frac{47}{12} \approx 3.92. Re-rolling only the single worst face barely moves the average, since you almost always keep the first …

Die roll, single re-roll only when you roll a one

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