Expected meetings in Room 1 after random assignment

Q: Expected meetings in Room 1 after random assignment

Evaluates conditional expectation in a balls-in-bins meeting-room assignment problem. Strong answers model Room 1 occupancy as binomial, compute non-empty probability, and derive expected meetings given Room 1 is non-empty.

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This is a medium difficulty Analytics & Experimentation question, commonly asked during Onsite rounds at Meta.

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This question is commonly asked for Data Scientist candidates at Meta during technical interviews.

Question

Expected Meetings in Room 1 Conditional on Being Non-empty

There are N rooms and k meetings. Each meeting independently chooses a room uniformly at random, and multiple meetings can be assigned to the same room. Let X1 be the number of meetings assigned to Room 1.

Given that Room 1 is non-empty, compute E[X1 | X1 >= 1].

Constraints & Assumptions

Each meeting assignment is independent.
Each room is equally likely for each meeting.
N and k are positive integers.
Use conditional expectation and linearity of expectation.

Clarifying Questions to Ask Guidance

Can multiple meetings be assigned to the same room?
Is Room 1 fixed in advance?
Are all rooms equally likely?
Are we conditioning on Room 1 specifically being non-empty or some room being non-empty?

What a Strong Answer Covers Guidance

Models X1 as Binomial(k, 1/N).
Computes E[X1] = k/N.
Computes P(X1 >= 1) = 1 - (1 - 1/N)^k.
Uses E[X | X > 0] = E[X] / P(X > 0) for nonnegative X.
Gives the exact expression (k/N) / [1 - (1 - 1/N)^k].
Checks edge cases such as k = 1 or large k.

Follow-up Questions Guidance

What is the expected number of non-empty rooms?
How would the result change if meetings choose rooms with unequal probabilities?
What is the probability Room 1 has exactly m meetings?

Expected meetings in Room 1 after random assignment

Quick Overview

Expected meetings in Room 1 after random assignment

Expected Meetings in Room 1 Conditional on Being Non-empty

Constraints & Assumptions

Clarifying Questions to Ask Guidance

What a Strong Answer Covers Guidance

Follow-up Questions Guidance

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