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Derive expected meetings given nonempty room | Meta Interview Question

Derive expected meetings given nonempty room

Last updated: Mar 29, 2026

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.

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].

Solution

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