Calculate Probabilities for Mixed Reviewer Types

Q: Calculate Probabilities for Mixed Reviewer Types

This question evaluates proficiency in probability theory and Bayesian inference, focusing on mixture models, conditional independence, posterior updating, expected value calculation, and asymptotic behavior of beliefs.

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Question

Scenario

Two types of reviewers exist in a marketplace:

Lazy reviewers (20%) always give good reviews.
Careful reviewers (80%) give good reviews 60% of the time.

Assume that, for a given reviewer, their reviews are independent conditional on their type.

Questions

(a) What is the probability that a random review is good?

(b) If a review is negative, what is the probability it came from a lazy reviewer?

(c) What is the expected number of good reviews in 100 reviews?

(d) After a reviewer gives three consecutive good reviews, what is the probability the reviewer is lazy?

(e) How does this probability change as the number of consecutive good reviews N → ∞?

Calculate Probabilities for Mixed Reviewer Types

Scenario

Questions

Solution

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