Question

A particular electronic component is produced at two plants for an electronics manufacturer. Plant A produces 70% of the components used and the remainder are produced by plant B. Among the components produced at plant A, the proportion of defective components is 1%. Among the components produced at plant B, the proportion of defective components is 2%. If a component received by the manufacturer is defective, the probability that it was produced at plant A is

Answer 1

If a component received by the manufacturer is defective, the probability that it was produced at plant A is 0.5385 = 53.85%.

Explanation:

We use the Bayes Theorem to solve this question.

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 component

Event B: Produced at plant A.

Plant A produces 70% of the components used

This means that
`P(B) = 0.7`

Among the components produced at plant A, the proportion of defective components is 1%.

This means that
`P(A|B) = 0.01`

Probability of a defective component:

1% of 70%(defective at plant A)

2% of 100 - 70 = 30%(defective at plant B). So

`P(A) = 0.01*0.7 + 0.02*0.3 = 0.013`

If a component received by the manufacturer is defective, the probability that it was produced at plant A is

`P(B|A) = (0.7*0.01)/(0.013) = 0.5385`

If a component received by the manufacturer is defective, the probability that it was produced at plant A is 0.5385 = 53.85%.