Question

A factory production line is manufacturing bolts using three machines, A, B and C. Of the total output, machine A is responsible for $25$%, machine B for $35$% and machine C for the rest. It is known from previous experience with the machines that $5$% of the output from machine A is defective, $4$% from machine B and $2$% from machine C. A bolt is chosen at random from the production line and found to be defective. What is the probability that it came from (a) machine A

• A. $0.632$
• B. $0.362$
• C. $0.487$
• D. none of these

Answer 1

The correct option is B $0.362$

Let

A be bolt manufactured from machine A

B be bolt manufactured from machine B

C be bolt manufactured from machine C

D be bolt is defective

Given:

$P\left(A\right)=25\%=0.25,P\left(B\right)=35\%=0.35,P\left(C\right)=40\%=0.4$

$P\left(D/A\right)=5\%=0.05,P\left(D/B\right)=4\%=0.04,P\left(D/C\right)=2\%=0.02$

To find:

The probability that the bolt is manufactured by machine A if it is defective.

i.e., $P\left(A/D\right)=\frac{P\left(A\right).P\left(D/A\right)}{P\left(A\right).P\left(D/A\right)+P\left(B\right).P\left(D/B\right)+P\left(C\right).P\left(D/C\right)}⟹=\frac{0.25\times 0.05}{0.25\times 0.05+0.35\times 0.04+0.4\times 0.02}⟹=\frac{0.0125}{0.0345}⟹P\left(A/D\right)=0.362$