Question

. A factory manufactures widgets using three machines, A, B, and C. Of the total output, machine A is responsible for 30%, machine B for 20%, and machine C for the rest. It is known from previous experience with the machines that 10% of the output from machine A is defective, 5% from machine B, and 3% 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 machine A? Round your final answers to three decimal places.

Answer 1

0.545 = 54.5% probability that it came from machine A

Explanation:

Bayes Theorem:

Two events, A and B.

`P(B|A) = (P(B)*P(A|B))/(P(A))`

In which P(B|A) is the probability of B happening when A has happened and P(A|B) is the probability of A happening when B has happened.

In this question:

Event A: Defective.

Event B: Coming from machine A.

Machine A is responsible for 30%

This means that
`P(B) = 0.3`

10% of the output from machine A is defective

This means that
`P(B|A) = 0.1`

Probability of being defective:

Machine A is responsible for 30%. Of those, 10% are defective.

Machine B is responsible for 20%. Of those, 5% are defective.

Machine C is responsible for 100 - (30+20) = 50%. Of those, 3% are defective. Then

`P(A) = 0.3*0.1 + 0.2*0.05 + 0.5*0.03 = 0.055`

Finally:

`P(B|A) = (P(B)*P(A|B))/(P(A)) = (0.3*0.1)/(0.055) = 0.545`

0.545 = 54.5% probability that it came from machine A