Roll at least a five

Question

You roll a fair die and win if your final face is at least $5$ (that is, a $5$ or a $6$ ). After the first roll you may keep it, or reroll once and keep the new face.

What is your optimal rule, and what is your probability of winning?

Accepted Answer

The target is fixed: you need a 5 or 6. So keep the first roll only if it already clears the bar, and reroll otherwise (a 1 through 4 is a guaranteed loss if kept). Keep a 5 or 6, reroll 1 through 4. The chance of clearing the bar on any single roll is P(\ge 5) = 	frac{2}{6} = 	frac13. With probability 	frac13 your first roll already wins. With probability 	frac23 it fails and you reroll, winning that reroll with probability 	frac13: P(	ext{win}) = \frac13 + \frac23 \cdot \frac13 = \frac13 + \frac29 = \frac{3}{9} + \frac{2}{9} = \frac{5}{9} \approx 0.556. Takeaway. Against a fixed targe…

Roll at least a five

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