The scanner beeped, is the part really defective?

Question

On a production line, 10\% of parts are truly defective and 90\% are good. An automatic scanner flags 90\% of defective parts, but it also mistakenly flags 5\% of good parts. A part just triggered the scanner. What is the probability the part is truly defective?

Accepted Answer

Consider 1{,}000 parts: 100 defective and 900 good. Count the flags. - Defective flagged: 100 	imes 0.90 = 90. - Good flagged: 900 	imes 0.05 = 45. Of the 90 + 45 = 135 flagged parts, 90 are truly defective: P(	ext{defective} \mid 	ext{flag}) = \frac{90}{135} = \frac{2}{3} \approx 0.667. With a moderate 10\% defect rate and a low 5\% false-alarm rate, a flag is fairly informative, raising belief from 10\% to about 67\%. Takeaway. A lower false-positive rate and a not-too-rare base rate together make a positive flag much more trustworthy.

The scanner beeped, is the part really defective?

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