Meta·Data Scientist·Technical Phone Screen·Intermediate

Intermediate

Jul 2026

Summary

Meta data scientist interview with a probability question that felt deceptively simple on the surface. Conditional expectation with a non-empty room constraint, the kind of thing that trips you up if you just pattern-match to the unconditional case.

Questions Asked (1)

k meetings are randomly assigned across N rooms. Given that Room 1 already has at least one meeting, what is the expected number of meetings in Room 1?

Product Analytics & MetricsAlgorithms & Data Structures

Author's notes

My first instinct was just k/N and I almost said it out loud.

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AI HintsAI Generated

Suggested Approach

Recognize that the number of meetings in Room 1 follows a Binomial(k, 1/N) distribution. Use the conditional expectation formula E[X | X ≥ 1] = E[X] / P(X ≥ 1) = (k/N) / (1 - (1 - 1/N)^k).

Pro tip: Clearly state the distributional assumption (independent, uniformly random assignment) and then apply the conditional expectation formula. This shows you can translate a word problem into a probabilistic model and handle conditioning rigorously.

1. Define the random variable

Let X be the number of meetings assigned to Room 1. Assume each meeting is independently assigned to one of N rooms with equal probability 1/N.

2. Identify the distribution

X follows a Binomial distribution with parameters k (number of trials) and p = 1/N (probability of success). So X ~ Binomial(k, 1/N).

3. Compute unconditional expectation

The expected number of meetings in Room 1 without any condition is E[X] = k * (1/N) = k/N.

4. Compute the conditioning probability

The probability that Room 1 has at least one meeting is P(X ≥ 1) = 1 - P(X = 0) = 1 - (1 - 1/N)^k.

5. Apply conditional expectation formula

Using E[X | X ≥ 1] = E[X] / P(X ≥ 1), the expected number of meetings in Room 1 given at least one is (k/N) / (1 - (1 - 1/N)^k).

Key Points to Mention

Independence and uniform random assignment of meetings to rooms.
Binomial distribution for the count in a specific room.
Conditional expectation formula: E[X | A] = E[X * I_A] / P(A).
Complement rule for calculating P(X ≥ 1).
Simplification and interpretation of the final expression.
Edge cases: when N=1 or k=0, and behavior as k or N grows.

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