Dice Reroll Game: Puzzle and Step-by-Step Solution
Probability
Easy
Reported at SIG, Optiver, Akuna, Belvedere, IMC
The puzzle
You roll a fair six-sided die and are paid its face value in dollars. After the first roll you may choose to roll once more, in which case you are paid the second roll instead. What is the game worth, and what is your strategy? What if you are allowed two rerolls?
Short answer: With one reroll the game is worth 4.25: keep 4, 5 or 6 and reroll 1, 2 or 3. With two rerolls it is worth 14/3, about 4.67.
Step 1: The value of a single roll
A fair die shows 1 to 6 with equal probability, so its expected value is (1 + 2 + 3 + 4 + 5 + 6) / 6 = 21 / 6 = 3.5. If you have no reroll left, the game is worth 3.5.
This number is the threshold for every earlier decision. It is what you give up when you keep a roll, and what you get when you throw one away.
Step 2: One reroll: keep what beats 3.5
After the first roll, keeping it pays its face value and rerolling pays 3.5 on average. So keep any roll above 3.5, which means 4, 5 or 6, and reroll 1, 2 or 3.
With probability 1/2 you keep, and the average kept roll is (4 + 5 + 6) / 3 = 5. With probability 1/2 you reroll and get 3.5 on average. The game is worth 1/2 × 5 + 1/2 × 3.5 = 2.5 + 1.75 = 4.25.
Step 3: Two rerolls: the threshold rises
Work backward. With one reroll left the continuation is worth 4.25, from the step above. So on the first of three rolls, keep only what beats 4.25: a 5 or a 6. Reroll 1 to 4.
With probability 2/6 you keep, averaging 5.5. With probability 4/6 you move on to a game worth 4.25. The value is (2/6) × 5.5 + (4/6) × 4.25 = 11/6 + 17/6 = 28/6 = 14/3, about 4.67.
Step 4: The pattern and the pricing question
This is optimal stopping by backward induction: each stage's value becomes the next stage's threshold. With more rerolls the threshold keeps rising toward 6, and the value approaches 6.
The common follow-up is a price: 'I will let you play the one-reroll game for $4. Do you?' At a fair value of 4.25 you buy, and your edge is 25 cents a game. Being able to turn the value into a trade decision is what a trading interview is really asking for.
Why interviewers ask it
State the threshold rule before you compute. Trading interviewers usually follow up by quoting a price and asking whether you would play, so have the fair value ready as a number.
Knowing the proof is half of it. Work Dice Reroll Game interactively, then take it under a clock with the rest of the Outcry simulators.