Suppose that 30 percent of the bottles produced in a certain

Suppose that 30 percent of the bottles produced in a certain plant are deffective. If a bottle 1/1 effective, the probability is 0.9 that an inspector will notice it and remove it from the line. If a bottle is not deffective, the probability is 0.2 that the inspector will think it is deffective and remove it from the filling line. If a bottle is removed from the filling line, what is the probability that it is deffective?

Solution

Let

G = the bottle is good
B = the bottle is defective

I = the inspector think it is defective

Thus,

P(B|I) = P(B n I) / P(I)

As

P(B n I) = P(B) P(I|B) = 0.3*0.9 = 0.27

and

P(I) = P(B) P(I|B) + P(G) P(I|G) = 0.3*0.9 + (1-0.3)*0.2 = 0.41

Then,

P(B|I) = 0.27/0.41 = 0.658536585 [answer]