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).
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.
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