Two interviewers both say hire

Question

Among applicants, 30\% are genuinely strong and 70\% are weak. Two interviewers assess a candidate independently. Each recommends 90\% of strong candidates and 40\% of weak ones. Both interviewers recommend hiring this candidate. What is the probability the candidate is genuinely strong?

Accepted Answer

Imagine 1{,}000 candidates: 300 strong, 700 weak. The two assessments are independent given the candidate's true quality, so multiply. - Strong, both recommend: 300 	imes 0.90 	imes 0.90 = 243. - Weak, both recommend: 700 	imes 0.40 	imes 0.40 = 112. Of the 243 + 112 = 355 double-recommendations, 243 are strong: P(	ext{strong} \mid 	ext{both yes}) = \frac{243}{355} \approx 0.685 = 68\%. A single yes gives about 49\%; the second independent yes tips the balance because a weak candidate impressing both interviewers is much rarer. Takeaway. Independent positive opinions compound, but noisy…

Two interviewers both say hire

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