##### 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.

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.

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Calculate Expected Meetings in Randomly Assigned Rooms

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?

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?

Calculate Expected Meetings in Randomly Assigned Rooms

Quick Overview

Calculate Expected Meetings in Randomly Assigned Rooms

Expected Meetings in Randomly Assigned Rooms

Constraints & Assumptions

Clarifying Questions to Ask Guidance

What a Strong Answer Covers Guidance

Follow-up Questions Guidance

Write your answer

Calculate Expected Meetings in Randomly Assigned Rooms

Quick Overview

Calculate Expected Meetings in Randomly Assigned Rooms

Expected Meetings in Randomly Assigned Rooms

Constraints & Assumptions

Clarifying Questions to Ask Guidance

What a Strong Answer Covers Guidance

Follow-up Questions Guidance

Write your answer