Die roll, forced re-roll on a one or two

Question

You roll a fair six-sided die. If it shows a 1 or a 2, you must re-roll once and keep that second roll no matter what. If it shows 3, 4, 5, or 6, you keep the first roll.

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

Accepted Answer

Split into the two cases based on the first roll, then blend by how likely each case is. Case A: you keep the first roll. This happens when the first roll is 3, 4, 5, or 6, which is 4 of the 6 faces, so probability 4/6. Given you kept it, the value is the average of 3, 4, 5, 6: \frac{3+4+5+6}{4} = \frac{18}{4} = 4.5. Case B: you re-roll. This happens when the first roll is 1 or 2, probability 2/6. The kept value is then just a fresh die, which averages 3.5. Blend the two cases by their probabilities: EV = \frac{4}{6}(4.5) + \frac{2}{6}(3.5) = \frac{4(4.5) + 2(3.5)}{6} = \frac{18 + 7}{6} = \fr…

Die roll, forced re-roll on a one or two

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