Goldman Sachs·Software Engineer·Technical Phone Screen·Intermediate

IntermediatePrefer not to say

Apr 2026

Summary

Goldman Sachs data science interview threw a probability puzzle at me that looked deceptively simple. Spent more time second-guessing my intuition than actually working through the math.

Questions Asked (1)

You roll a standard six-sided die repeatedly, adding each result to a running total, and stop when the total hits 100 or more. Between the numbers 1 and 6, which do you expect to appear more frequently across all the rolls?

Algorithms & Data StructuresProduct Analytics & Metrics

Author's notes

My gut said 6 because higher numbers feel more dominant somehow, which is completely wrong.

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AI HintsAI Generated

Suggested Approach

Recognize that the stopping condition (total ≥ 100) introduces a bias: the final roll is more likely to be a small number because it must not overshoot too much, while larger numbers are more likely to appear earlier. Use renewal theory or simulation to show that 1 appears most frequently, and explain the intuition clearly.

Pro tip: Mention that the expected total after n rolls is 3.5n, so around 29 rolls are needed; the last roll is special and skews the distribution. Quantify the effect with a quick simulation or approximation to demonstrate rigor.

1. Clarify the process

Restate the problem: roll a fair six-sided die repeatedly, sum results, stop when total ≥ 100. Ask which number appears more often across all rolls.

2. Identify the bias

Explain that the stopping condition creates a bias: the final roll must be small enough to not overshoot 100 too much, so smaller numbers are more likely to be the last roll.

3. Analyze the final roll

Argue that the final roll is more likely to be 1 because it can occur when the previous total is 99, while larger numbers require the previous total to be in a narrower range (e.g., 6 requires previous total 94-99).

4. Consider all rolls

Note that non-final rolls are uniformly distributed, but the final roll's bias affects the overall frequency. Since there are many rolls (~29), the bias is diluted but still significant.

5. Conclude and quantify

Conclude that 1 appears most frequently. Optionally, provide a rough estimate or mention simulation to show the magnitude (e.g., 1 appears about 20% more often than 6).

Key Points to Mention

The stopping condition introduces a bias favoring smaller numbers on the final roll.
The final roll is more likely to be 1 because it has the widest range of previous totals that lead to stopping.
Non-final rolls are uniformly distributed, so the overall effect is a slight increase in frequency for smaller numbers.
Expected number of rolls is about 29 (since 100/3.5 ≈ 28.6), so the final roll is a small fraction but still noticeable.
Simulation or renewal theory can confirm that 1 appears most frequently.
The difference in frequencies between 1 and 6 is small but consistent.

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