Difficulty: Hard · Puzzle 6 of 6 · Topic: Probability Puzzles
A silicon issue occurs in 1% of chips. A screening test has 99% sensitivity and 95% specificity. If a chip tests positive, what is P(issue is real)?
Use Bayes with prevalence, not just test accuracy.
1) Let D be real defect, + be positive test.
2) Given: P(D)=0.01, P(+|D)=0.99, P(+|~D)=0.05 (from 95% specificity).
3) Bayes: P(D|+) = P(+|D)P(D) / [P(+|D)P(D) + P(+|~D)P(~D)].
4) Numerator = 0.99*0.01 = 0.0099.
5) Denominator = 0.0099 + 0.05*0.99 = 0.0099 + 0.0495 = 0.0594.
6) P(D|+) = 0.0099/0.0594 = 1/6 approx 0.1667.Answer: Only about 16.67% of positive results are true defects in this setup.
Confusing P(+|D)=99% with P(D|+).
If prevalence rises to 10%, what does P(D|+) become?