Dice Questions in Quant Interviews

Dice games are how trading firms ask "can you price an option?" without saying the word option: compute a fair value under uncertainty, often with an embedded choice. Three structures cover most of what appears.

Structure 1: rerolls = embedded options

"Roll a die, take the value or reroll once" — fair value 4.25: keep 4/5/6 (average 5), reroll 1/2/3 for expected 3.5, so $\frac{1}{2}(5) + \frac{1}{2}(3.5) = 4.25$. With two rerolls: now your threshold rises (keep ≥5 first roll, since a roll is worth 4.25), giving $\frac{1}{3}(5.5) + \frac{2}{3}(4.25) \approx 4.67$. Each reroll is an option, and its value comes from optimal exercise — say those words and the interviewer hears a trader. Backward induction from the last roll is the universal method — the same logic as the optimal stopping family.

Structure 2: sums, records, and counting

Distribution questions — probability two dice sum to 7 (the most likely sum, 1/6), expected number of rolls to see all six faces (coupon collector: $6H_6 \approx 14.7$), expected maximum of $n$ rolls ($6 - \sum_{k=1}^{5}(k/6)^n$ — or for one die, the max-of-two shortcut $E[\max] = 161/36 \approx 4.47$). Clean setups, testing whether you reach for indicator variables and linearity of expectation before brute force.

Structure 3: adversarial dice

Non-transitive dice (A beats B beats C beats A) test the same lesson as Penney's game: pairwise comparisons don't order things. Betting games on dice outcomes fold in bet sizing and game theory — "I roll, you can bet on high/low at these odds" is a market-making round in miniature.

Practice with solutions

• Three-roll dice game — THE reroll classic, fully worked.
• Repeat-or-stop dice game — pricing with an indefinite option.
• D20 vs 3d6 — expectation and spread comparisons across dice structures.
• Dice with all pairwise sums — the combinatorial end of the family.