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Sig·Software Engineer·Technical Phone Screen·Intermediate

Intermediate
May 2026

Summary

Quant Engineer interview at SIG, probability and statistics heavy. The dice problem sounds textbook until you're actually sitting there trying to remember how to derive variance from scratch under pressure.

Questions Asked (1)

Q1

Given two fair six-sided dice, what is the probability that their sum equals some value k? Give a general formula, and walk through the expected value and variance of the sum.

Algorithms & Data StructuresTechnical Trade-offs
Author's notes

The counting part was fine, I drew out the triangle distribution in my head and got through the formula okay.

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AI HintsAI Generated

Suggested Approach

Start by defining the sample space of 36 equally likely outcomes for two fair six-sided dice. Derive the probability mass function for the sum k by counting the number of outcomes that yield k, then use linearity of expectation and the definition of variance (or properties of independent variables) to compute the expected value and variance. Present the general formula, then verify with a quick example like k=7.

Pro tip: Mention that the sum of two dice is a classic example of a triangular distribution, and that the expected value and variance can be derived without enumerating all outcomes by using linearity and independence—this shows you understand the underlying probability theory and can generalize to n dice.

1. Define the sample space and random variables

State that each die is a discrete uniform random variable on {1,2,3,4,5,6}, independent of the other. There are 36 equally likely ordered pairs (i, j).

2. Derive the probability mass function for the sum

For a given sum k (ranging from 2 to 12), count the number of pairs (i, j) such that i + j = k. The count is 6 - |k - 7| for k in 2..12, so P(S = k) = (6 - |k - 7|)/36.

3. Compute the expected value

Use linearity of expectation: E[S] = E[X] + E[Y] = 3.5 + 3.5 = 7. Alternatively, sum k * P(S=k) over k=2..12 to confirm.

4. Compute the variance

Since X and Y are independent, Var(S) = Var(X) + Var(Y). For a fair die, Var(X) = (6^2 - 1)/12 = 35/12 ≈ 2.9167, so Var(S) = 35/6 ≈ 5.8333.

5. Summarize and check with an example

State the general formula for P(S=k), E[S]=7, Var(S)=35/6. Optionally, verify for k=7: P(S=7)=6/36=1/6, which matches the formula.

Key Points to Mention

  • The sample space consists of 36 equally likely outcomes.
  • The probability mass function for the sum is triangular: P(S=k) = (6 - |k - 7|)/36 for k=2,...,12.
  • Expected value of a single fair die is 3.5, so the sum has expected value 7 by linearity.
  • Variance of a single fair die is 35/12, and variances add for independent variables, giving 35/6 for the sum.
  • The distribution is symmetric around 7, which is the most likely sum.
  • The approach generalizes to n dice: mean = 3.5n, variance = (35/12)n, and the distribution approaches normal.

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