Question

A certain firm has plants A, B, and C producing respectively 35%, 15%, and 50% of the total output. The probabilities of a non-defective product are, respectively, 0.75, 0.95, and 0.85. A customer receives a defective product. What is the probability that it came from plant C?

Answer 1

There is a 44.12% probability that the defective product came from C.

Explanation:

This can be formulated as the following problem:

What is the probability of B happening, knowing that A has happened.

It can be calculated by the following formula

`P = (P(B).P(A/B))/(P(A))`

Where P(B) is the probability of B happening, P(A/B) is the probability of A happening knowing that B happened and P(A) is the probability of A happening.

-In your problem, we have:

P(A) is the probability of the customer receiving a defective product. For this probability, we have:

`P(A) = P_(1) + P_(2) + P_(3)`

In which
`P_(1)` is the probability that the defective product was chosen from plant A(we have to consider the probability of plant A being chosen). So:

`P_(1) = 0.35*0.25 = 0.0875`

`P_(2)` is the probability that the defective product was chosen from plant B(we have to consider the probability of plant B being chosen). So:

`P_(2) = 0.15*0.05 = 0.0075`

`P_(3)` is the probability that the defective product was chosen from plant B(we have to consider the probability of plant B being chosen). So:

`P_(3) = 0.50*0.15 = 0.075`

So

`P(A) = 0.0875 + 0.0075 + 0.075 = 0.17`

P(B) is the probability the product chosen being C, that is 50% = 0.5.

P(A/B) is the probability of the product being defective, knowing that the plant chosen was C. So P(A/B) = 0.15.

So, the probability that the defective piece came from C is:

`P = (0.5*0.15)/(0.17) = 0.4412`

There is a 44.12% probability that the defective product came from C.