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Jane Street·Software Engineer·Technical Phone Screen·Intermediate

Intermediate
May 2026

Summary

Jane Street SWE interview with a probability puzzle that looks clean on the surface but has a recursive structure that'll trip you up if you rush it.

Questions Asked (1)

Q1

Three players A, B, and C roll a fair six-sided die. B and C have already rolled and both got 4. You (player A) roll once. If your result beats 4 you win outright, if it's below 4 you lose outright, and if you also roll a 4 then all three reroll simultaneously until one player has a strictly higher result than the others. What is your probability of ultimately winning?

Algorithms & Data Structures
Author's notes

I set up the base cases fine.

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AI HintsAI Generated

Suggested Approach

Break the problem into two stages: the initial roll and the tie-breaking reroll phase. Compute the probability of winning outright on the first roll, then compute the probability of entering the tie and winning the subsequent fair game, and combine them.

Pro tip: In the tie-breaking phase, all players are symmetric, so each has an equal chance of winning. This symmetry argument simplifies the calculation and avoids unnecessary complexity.

1. Analyze the initial roll

Determine the outcomes of A's first roll: win outright if >4 (5 or 6), lose outright if <4 (1,2,3), and tie if =4.

2. Compute probabilities for initial outcomes

Calculate P(win outright) = 2/6 = 1/3, P(lose outright) = 3/6 = 1/2, P(tie) = 1/6.

3. Analyze the tie-breaking phase

Recognize that when all three reroll simultaneously until one has a strictly higher result, the game is symmetric among the three players, so each has an equal probability of winning.

4. Compute probability of winning from tie

Since the game is symmetric and must eventually produce a winner, the probability that A wins given a tie is 1/3.

5. Combine probabilities

Use the law of total probability: P(win) = P(win outright) + P(tie) * P(win | tie) = 1/3 + (1/6)*(1/3) = 1/3 + 1/18 = 7/18.

Key Points to Mention

  • Law of total probability: breaking the problem into disjoint cases (win outright, lose outright, tie).
  • Symmetry argument: in the tie-breaking phase, all players have equal chances.
  • Calculation of probabilities for each outcome of a fair six-sided die.
  • Combining probabilities correctly: P(win) = P(win outright) + P(tie) * P(win | tie).
  • Final answer: 7/18 (approximately 0.3889).
  • Justification that the tie-breaking game must eventually end (probability of infinite ties is zero).

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.