Oct 08, 2026 by Offer

IMC Optimal Stopping 2027: Dice Re-Roll Strategy

The IMC Trading optimal stopping dice question — commonly reported by candidates as "You roll a die and may accept the value or re-roll once (must accept the second). What is your expected value?" — solves to 4.25. The strategy: accept 4, 5, or 6; re-roll 1, 2, or 3.

What This IMC Trading Optimal Stopping Dice Question Assesses

This is the purest optimal-stopping problem in the interview canon, and IMC asks it to test threshold thinking: do you compare the bird in hand against the expected value of the bush? Traders make this comparison constantly — hold a position or exit, fill now or wait. The candidates who shine state the threshold rule before computing anything.

How to Answer This IMC Trading Optimal Stopping Dice Question

Derive the strategy, then the value:

Step 1 — Value the option. A forced re-roll is worth the plain die expectation: 3.5. This is your walk-away value — the number every first roll gets compared against.
Step 2 — Set the threshold. Accept the first roll if it exceeds 3.5; re-roll otherwise. With integer faces: keep 4, 5, 6 — re-roll 1, 2, 3. State this as a rule, not a feeling.
Step 3 — Compute the expectation. Half the time you keep, contributing (4+5+6)/6 = 2.5; half the time you re-roll for 3.5, contributing 1.75. Total: 4.25.

Example line: "The re-roll is worth 3.5, so I keep anything above it — 4, 5, 6 — giving 2.5 plus half of 3.5, which is 4.25."

Note the "must accept the second" clause doesn't change the math — it just confirms the re-roll's value is exactly 3.5 with no further decisions.

Common Mistakes

Threshold errors. Re-rolling a 4 or keeping a 3 both destroy value — the cutoff must be exactly at the re-roll's expectation.
Answering 3.5. Ignoring the option value entirely is the classic failure; the question exists to test whether you price the choice.
Overcomplicating. Some candidates try to model the second roll strategically — but "must accept" means no decision remains, so 3.5 is exact.

Candidates commonly report extensions: a cost per re-roll, multiple re-rolls, or a different die. The method never changes — threshold equals the expected value of continuing — so nail the base case and the variants solve themselves.

Keep Reading

FAQ

What if the re-roll cost $1? The re-roll's net value becomes 2.5, so keep anything above 2.5 — accept 3 through 6, re-roll 1 and 2.

What with two re-rolls allowed? Work backwards: the final re-roll is worth 3.5, so with one re-roll remaining the game is worth 4.25 — meaning you'd keep 5+ at the first roll. Thresholds rise with more options.

Does "must accept" ever matter? Only if you were tempted to strategize on the second roll — the clause removes that, making 3.5 exact. It's a simplification, not a complication.

Why do trading firms love this question? Optimal stopping is the skeleton of execution decisions — when to cross the spread versus wait. They're testing the instinct in its simplest form.

Preparing for IMC Trading's interview? Our 2027 IMC Trading Assessment BrainsFirst Games NeurOlympics Exact Questions and Answers has practice questions and answers — $79 one-time, instant download.