Calculate Probabilities for Mixed Reviewer Types

Q: Calculate Probabilities for Mixed Reviewer Types

Meta probability prompt on a mixture of lazy and careful reviewers, covering total probability, Bayes theorem, repeated-review likelihoods, expected good reviews, and posterior convergence.

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Q: What difficulty level is this interview question?

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

Q: What role is this question designed for?

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

Question

Probabilities for Mixed Reviewer Types

Two types of reviewers exist in a marketplace:

Lazy reviewers are 20% of reviewers and always give good reviews.
Careful reviewers are 80% of reviewers and give good reviews 60% of the time.

Assume reviews from a given reviewer are independent conditional on reviewer type.

Constraints & Assumptions

Use the law of total probability and Bayes' theorem.
Distinguish a random review from repeated reviews by the same reviewer.
For repeated reviews, condition on the same reviewer type throughout the sequence.
State what happens as the number of consecutive good reviews grows.

Clarifying Questions to Ask Guidance

Is each review drawn from a new random reviewer, or are multiple reviews from the same reviewer?
Are the 20% and 80% type proportions priors over reviewers?
Is "good" the only observed label, or do we know the true quality of reviewed items?

What a Strong Answer Covers Guidance

Probability a random review is good: 1 * 0.2 + 0.6 * 0.8 = 0.68 .
Probability a negative review came from a lazy reviewer is 0 because lazy reviewers never give negative reviews.
Expected good reviews in 100 random reviews is 100 * 0.68 = 68 by linearity.
After three consecutive good reviews from the same reviewer, use Bayes' theorem with likelihoods 1^3 and 0.6^3 .
General posterior after N good reviews: 0.2 / (0.2 + 0.8 * 0.6^N) .
As N approaches infinity, the posterior probability of lazy approaches 1.
Mention that variance depends on whether reviews come from many reviewers or one reviewer.

Follow-up Questions Guidance

How would the answer change if lazy reviewers gave good reviews 90% of the time?
What if you observed one negative after several good reviews?
How many consecutive good reviews are needed for the lazy posterior to exceed 90%?

Calculate Probabilities for Mixed Reviewer Types

Quick Overview

Calculate Probabilities for Mixed Reviewer Types

Probabilities for Mixed Reviewer Types

Constraints & Assumptions

Clarifying Questions to Ask Guidance

What a Strong Answer Covers Guidance

Follow-up Questions Guidance

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