##### Scenario Randomly allocating Y ad impressions to X users ##### Question Given X users and Y impressions assigned uniformly at random, what is the expected number of impressions a particular user receives? What is the probability that a specific user sees at least one impression? What is the expected number of users who see at least one impression? ##### Hints Model each assignment as Bernoulli with p = 1⁄X, then apply binomial expectation, complement rule, and linearity.

Meta probability question on random ad-impression allocation, including binomial expectation, probability a user sees at least one impression, expected unique reach, and large-population approximations.

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Calculate Expected Impressions and Probability for Users

Expected Impressions From Random Ad Allocation

There are X distinct users and Y ad impressions. Each impression is assigned independently and uniformly at random to one of the X users. A user may receive multiple impressions.

Constraints & Assumptions

Model each impression assignment as an independent trial.
Keep answers symbolic in terms of X and Y .
State when the binomial model and complement rule are being used.
You may include large- X approximations if helpful.

Clarifying Questions to Ask Guidance

Are impressions assigned independently with replacement across users?
Are users equally likely to receive each impression?
Are we counting total impressions, unique reached users, or both?

What a Strong Answer Covers Guidance

For a fixed user, the impression count follows Binomial(Y, 1/X) .
Expected impressions for one user are Y / X .
Probability a specific user sees at least one impression is 1 - (1 - 1/X)^Y .
Expected number of users reached at least once is X * [1 - (1 - 1/X)^Y] by linearity of expectation.
For large X , with lambda = Y/X , use (1 - 1/X)^Y approx exp(-lambda) as a useful approximation.
Edge cases such as Y = 0 , very small X , and non-uniform allocation.

Follow-up Questions Guidance

What is the variance of impressions for a particular user?
How would the answer change if impressions were capped at one per user?
How would non-uniform user eligibility affect expected reach?

Expected Impressions From Random Ad Allocation

There are X distinct users and Y ad impressions. Each impression is assigned independently and uniformly at random to one of the X users. A user may receive multiple impressions.

Constraints & Assumptions

Model each impression assignment as an independent trial.

Keep answers symbolic in terms of X and Y .

State when the binomial model and complement rule are being used.

You may include large- X approximations if helpful.

Clarifying Questions to Ask Guidance

Are impressions assigned independently with replacement across users?

Are users equally likely to receive each impression?

Are we counting total impressions, unique reached users, or both?

What a Strong Answer Covers Guidance

For a fixed user, the impression count follows Binomial(Y, 1/X) .

Expected impressions for one user are Y / X .

Probability a specific user sees at least one impression is 1 - (1 - 1/X)^Y .

Expected number of users reached at least once is X * [1 - (1 - 1/X)^Y] by linearity of expectation.

For large X , with lambda = Y/X , use (1 - 1/X)^Y approx exp(-lambda) as a useful approximation.

Edge cases such as Y = 0 , very small X , and non-uniform allocation.

Follow-up Questions Guidance

What is the variance of impressions for a particular user?

How would the answer change if impressions were capped at one per user?

How would non-uniform user eligibility affect expected reach?

Calculate Expected Impressions and Probability for Users

Quick Overview

Calculate Expected Impressions and Probability for Users

Expected Impressions From Random Ad Allocation

Constraints & Assumptions

Clarifying Questions to Ask Guidance

What a Strong Answer Covers Guidance

Follow-up Questions Guidance

Write your answer

Calculate Expected Impressions and Probability for Users

Quick Overview

Calculate Expected Impressions and Probability for Users

Expected Impressions From Random Ad Allocation

Constraints & Assumptions

Clarifying Questions to Ask Guidance

What a Strong Answer Covers Guidance

Follow-up Questions Guidance

Write your answer