Jane Street·Software Engineer·Technical Phone Screen·Intermediate

Intermediate

Jul 2026

Summary

Jane Street software engineer interview with a probability brain teaser. The kind of question where you either see the structure quickly or spend five minutes going in circles.

Questions Asked (1)

In a three-player dice game where each person rolls a standard 6-sided die and the highest roll wins (with ties broken by re-rolling among tied players), your two friends have both rolled a 4. You roll next. What is the probability that you eventually win?

Algorithms & Data Structures

Author's notes

I stared at this longer than I'd like to admit.

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AI HintsAI Generated

Suggested Approach

Break the problem into cases based on your first roll: if you roll 1-3, you lose immediately; if you roll 5 or 6, you win immediately; if you roll 4, you enter a symmetric tie with the two friends. For the tie case, use symmetry to determine your probability of eventually winning, then combine all cases using the law of total probability.

Pro tip: When explaining, emphasize the symmetry argument: in a three-way tie, each player has an equal chance to win, so your probability is 1/3. This shows you can simplify complex recursive problems elegantly.

1. Identify immediate outcomes

List the possible results of your first roll and their immediate consequences: rolling 1, 2, or 3 means you lose; rolling 5 or 6 means you win; rolling 4 leads to a tie.

2. Assign probabilities to immediate outcomes

Since the die is fair, each outcome has probability 1/6. So P(lose immediately) = 3/6 = 1/2, P(win immediately) = 2/6 = 1/3, and P(tie) = 1/6.

3. Analyze the tie scenario

If you roll a 4, all three players are tied. By symmetry, each player has an equal chance to eventually win, so your probability of winning from the tie is 1/3.

4. Combine probabilities using total probability

Compute overall win probability: P(win) = P(win immediately) + P(tie) * P(win | tie) = 1/3 + (1/6)*(1/3) = 1/3 + 1/18 = 7/18.

5. Verify and present the answer

Check that the result is reasonable (between 1/3 and 1/2) and clearly state the final probability as 7/18.

Key Points to Mention

Law of total probability: breaking the problem into mutually exclusive cases.
Symmetry argument: in a fair tie, each player has equal chance to win.
Independence of dice rolls: each roll is independent, so past rolls do not affect future rolls.
Recursive nature of the tie: if a tie occurs, the process repeats until a winner emerges.
The final answer 7/18 is approximately 0.3889, which is between 1/3 and 1/2 as expected.
Clear communication: explaining each step logically, as expected in a Jane Street interview.

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