Calculate Expected Impressions and Probability for Users

Quick Overview

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.

Calculate Expected Impressions and Probability for Users

Company: Meta

Role: Data Scientist

Category: Statistics & Math

Difficulty: easy

Interview Round: Onsite

##### 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.

Overview: 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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Jul 12, 2025
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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?
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