Fair value of a die game with one reroll

Question

You roll a fair six-sided die and get paid its face value in dollars. Before being paid, you may choose to reroll once; if you do, you must accept the second roll.

What is the fair value of this game (assuming you play optimally)?

Accepted Answer

Work backwards from the reroll. If you reroll, you receive a fresh fair die roll worth E[	ext{second roll}] = \frac{1+2+3+4+5+6}{6} = 3.5. So the optimal strategy after the first roll is a threshold rule: keep any first roll worth more than 3.5, i.e. keep 4, 5, 6 and reroll on 1, 2, 3. Now value the game at the start: - With probability 	frac{3}{6}, the first roll is 4, 5, or 6, worth on average \frac{4+5+6}{3} = 5. - With probability 	frac{3}{6}, you reroll, worth 3.5. E = \frac{1}{2}(5) + \frac{1}{2}(3.5) = 4.25 = \frac{17}{4}. The game is worth \4.25. Takeaway. Optionality is valued by …

Fair value of a die game with one reroll

Your answer

More Probability questions