Question

n a certain assembly plant, three machines, B1, B2, and B3, make 30%, 45%, and 25%, respectively,

of the products. It is known from past experience that 2%, 3%, and 2% of the products made by each

machine, respectively, are defective. Now, suppose that a finished product is randomly selected.

(a) What is the probability that it is defective?

(b) if a product was chosen randomly and found to be defective, what is the probability that it was

made by machine B3?

Answer 1

Answer:

Consider the following events

A: The product is defective

B₁: The product is made by machine B₁

B₂: The product is made by machine B2

B3 The product is made by machine B3

Using the theorem of total probability

P(A) = P(B₁).P(A|B₁) + P(B₂).P(A|B₂) +

But We have:

P(B₁) = 0.30, P(A/B₁)

0.02

P(B₂) = 0.45, P(A|B₂) = 0.03

P(B3) = 0.25, P(A/B3)

Hence

P(B₁).P(A B₁) - 0.3 × 0.02 = 0.006

P(B₂).P(A|B2) = 0.45 × 0.03 = 0.0135

P(B3).P(A/B3) 0.25 × 0.02 0.005

So

0.02

P(A)

=

P(B₁).P(A|B₁) + P(B₂).P(A|B₂) +

P(A) 0.006 + 0.0135 + 0.005

P(A) = 0.0245

0.0245 is the probability of defective