Answer to Question #154563 in Statistics and Probability for 75ed86

Question

 Each item coming off a given production line is inpected by either inspector 1 or inspector 2. Inspector 1 inspects about 60% of the production items, while inspector 2 the rest. Inspector 1, who has been at her present job for some time, will not find 1% of the defective items she inspects. Inspector 2, who newer on the job, misses about 5% of the defective items he inspects. If an item that has passed an ispector is found to be defective, what is the probability that it was inspected by inspector 1?

Answer 1

we will count how many defective products both inspectors will not notice, for all products

$0.6*0.01=6*10^{-3}$ - the proportion of defects that were not found by the first inspector

$0.4*0.05=20*10^{-3}$- the proportion of defects that were not found by the second inspector

$20*10^{-3}+6*10^{-3}$ - the proportion of defects that were not found by both inspectors

Then the probability that defect was inspected by inspector 1 is:

$\frac{6*10^{-3}}{20*10^{-3}+6*10^{-3}} = \frac{6}{20+6}\thickapprox 0.23077$