Calculate Expected Meetings in Randomly Assigned Rooms

Quick Overview

Meta probability prompt on meetings assigned to rooms, covering binomial conditional expectation, conditioning on non-empty rooms, two-room occupancy states, total probability, and Bayes' theorem.

Calculate Expected Meetings in Randomly Assigned Rooms

Company: Meta

Role: Data Scientist

Category: Statistics & Math

Difficulty: medium

Interview Round: Onsite

##### Scenario Meetings randomly assigned to conference rooms ##### Question N rooms are available and K meetings are scheduled uniformly at random. Given that room 1 is not empty, what is the expected total number of meetings in room 1? Now assume two rooms where the trio of states {both occupied, exactly one occupied, both empty} are equally likely. Before entering, what is the probability the room you pick is occupied? If you enter and find it occupied, what is the probability the other room is also occupied? ##### Hints Apply Bayes’ theorem and conditional expectations; the first part is a binomial conditioned on non-zero, the second uses the law of total probability.

Overview: Meta probability prompt on meetings assigned to rooms, covering binomial conditional expectation, conditioning on non-empty rooms, two-room occupancy states, total probability, and Bayes' theorem.

|Home/Statistics & Math/Meta
Meta logo
Meta
Jul 12, 2025
mediumData ScientistOnsiteStatistics & Math
86
0

Expected Meetings in Randomly Assigned Rooms

You are solving two probability questions about room occupancy.

Constraints & Assumptions

  • In the first question, each of K meetings independently chooses one of N rooms uniformly at random.
  • In the second question, the three macro-states {both occupied, exactly one occupied, both empty} are equally likely before you choose a room.
  • State the conditioning event carefully.
  • Use Bayes' theorem or conditional expectation where appropriate.

Clarifying Questions to Ask Guidance

  • Are meetings assigned independently?
  • Is room 1 conditioned to have at least one meeting, or exactly one?
  • In the two-room question, are the three macro-states really equally likely, or are they induced by a meeting-assignment process?
  • Do we pick a room uniformly at random before observing occupancy?

What a Strong Answer Covers Guidance

  • For room 1, define X ~ Binomial(K, 1/N) .
  • Use E[X | X > 0] = E[X] / P(X > 0) for nonnegative X .
  • Compute E[X | X > 0] = (K/N) / [1 - (1 - 1/N)^K] .
  • For two rooms, compute P(picked room occupied) using total probability over the three macro-states.
  • After observing the chosen room is occupied, compute the posterior probability that both rooms are occupied using Bayes' theorem.
  • Explain why the equal-macro-state assumption is different from assigning meetings independently.

Follow-up Questions Guidance

  • What is the Poisson approximation when K/N is moderate?
  • How would the answer change if every room were equally likely to be empty or occupied independently?
  • Why does conditioning on "not empty" increase the expected count?
Loading comments...

Browse More Questions

More Statistics & Math•More Meta•More Data Scientist•Meta Data Scientist•Meta Statistics & Math•Data Scientist Statistics & Math