A manufacturing process produces microprocessors using an entirely new technolog

Question

A manufacturing process produces microprocessors using an entirely new technology. Historical results suggest that 25% of the final product is damaged and therefore not usable. In the past, the manufacturer has used an extremely expensive (but extremely reliable) diagnostic procedure to test if a microprocessor is damaged. In order to cut costs, management is considering a very inexpensive test that is not as reliable. Preliminary data suggest that if a microprocessor is damaged, then the probability that the new test would be negative (i.e., the test declares that the microprocessor is damaged), is 85%. If the microprocessor is working properly, the probability that the test is negative is 10%.

Clearly there are two types of errors that can occur with the new test. The first type of error is that a damaged microprocessor might test positive, and so be shipped to a customer. The second type of error is that a perfectly good microprocessor might test negative and so be unnecessarily discarded. Please figure out the probabilities that each of these two errors might occur, that is,

a) The probability that a microprocessor is damaged given that it tests positive;

b) The probability that a microprocessor is working properly given that it tests negative.

Explanation / Answer

a)probability of test positive =P(damaged and test positive+not damaged and test positive)

=0.25*(1-0.85)+(1-0.25)*(1-0.10)=0.7125

hence probability that a microprocessor is damaged given that it tests positive

=P(damaged and test positive)/P(test positive) ==0.25*(1-0.85)/0.7125=0.0526

b) P(test negative) =1-P(test positive) =1-0.7125=0.2875

hence probability that a microprocessor is working properly given that it tests negative

=P(not damaged and test negative)/P(negative) =(1-0.25)*0.1/0.2875=0.26087

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