Derive expected meetings given nonempty room

Quick Overview

This question evaluates a candidate's understanding of conditional probability, zero-truncated binomial distributions, and the calculation of conditional expectation and variance in finite-sample random assignment settings.

Derive expected meetings given nonempty room

Company: Meta

Role: Data Scientist

Category: Statistics & Math

Difficulty: medium

Interview Round: Onsite

There are N rooms labeled 1..N and K meetings. Each meeting independently chooses a room uniformly at random. Let K1 be the number of meetings scheduled in Room 1. Condition on the event {K1 > 0}. (a) Derive E[K1 | K1 > 0] in closed form as a function of N and K. (b) Derive Var(K1 | K1 > 0). (c) Briefly explain your approach (e.g., via Bayes’ rule or conditioning identities).

Quick Answer: This question evaluates a candidate's understanding of conditional probability, zero-truncated binomial distributions, and the calculation of conditional expectation and variance in finite-sample random assignment settings.

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Oct 13, 2025, 9:49 PM
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Zero-Truncated Binomial: Random Room Assignment

Setup

  • There are N rooms labeled 1, 2, ..., N.
  • K meetings are scheduled; each meeting independently chooses a room uniformly at random.
  • Let K1 be the number of meetings scheduled in Room 1. Then K1 ~ Binomial(K, p = 1/N).
  • Condition on the event {K1 > 0} (i.e., at least one meeting is in Room 1).

Assume N ≥ 1 and K ≥ 1 (otherwise P(K1 > 0) = 0 when K = 0).

Tasks

(a) Derive E[K1 | K1 > 0] in closed form as a function of N and K.

(b) Derive Var(K1 | K1 > 0].

(c) Briefly explain the approach (e.g., via Bayes’ rule or conditioning identities).

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