An inspection by a quality officer found that in a large shipment of electronic parts, 0.025...

An inspection by a quality officer found that in a large shipment of electronic parts, 0.025 are bad. The parts are tested using a machine that correctly identifies bad parts as defective with a probability of 0.96 and correctly identifies good parts as non-defective with a probability of 0.92.

(a) If a part is randomly sampled from the shipment and tested, calculate the probability that the testing machine identifies the part as defective.

(b) Given that the test indicates a non-defective part, calculate the probability that the part is truly good?

Solutions

Expert Solution

P[ electronic parts are bad ] = 0.025

P[ electronic parts are good ] = 1 - P[ electronic parts are bad ] = 1 - 0.025 = 0.975

P[ machine identifies part as bad | electronic parts are bad ] = 0.96

P[ machine identifies part as good | electronic parts are good ] = 0.92

P[ machine identifies part as bad | electronic parts are good ] = 1 - P[ machine identifies part as good | electronic parts are good ] = 1 - 0.92 = 0.08

(a) If a part is randomly sampled from the shipment and tested, calculate the probability that the testing machine identifies the part as defective.

P[ machine identifies part as bad ] = P[ machine identifies part as bad | electronic parts are good ]*P[ electronic parts are good ] + P[ machine identifies part as bad | electronic parts are bad ]*P[ electronic parts are bad ]

P[ machine identifies part as bad ] = 0.08*0.975 + 0.96*0.025

P[ machine identifies part as bad ] = 0.078 + 0.024

P[ machine identifies part as bad ] = 0.102

(b) Given that the test indicates a non-defective part, calculate the probability that the part is truly good?

P[ electronic parts are good | machine identifies part as good ] = P[ electronic parts are good ]*P[ machine identifies part as good | electronic parts are good ] / P[ machine identifies part as good ]

P[ machine identifies part as good ] = 1 - P[ machine identifies part as bad ]

P[ machine identifies part as good ] = 1 - 0.102

P[ machine identifies part as good ] = 0.898

P[ electronic parts are good | machine identifies part as good ] = 0.92*0.975 / 0.898

P[ electronic parts are good | machine identifies part as good ] = 0.897/0.898

P[ electronic parts are good | machine identifies part as good ] = 0.9989

orchestra answered 1 year ago

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