# Probability of defective machines question

[solved]probability question

## Homework Statement

Three machines A, B, and C produce 50%, 30% and 20% respectively of the total
number of items at a factory. Each produces a number of defective: 3%, 4% and 5%
respectively.
(a) If an item produced by one of these machines is selected, find the probability that
it is defective.
(b) Now suppose that a manufactured item is selected at random and is found to be
defective. Find the probability that this item was produced by machine A.

N/A

## The Attempt at a Solution

is part (a)
Pr = (3%+4%+5%)/(50%*30%*20%)?

and no idea on part b

thanks a lot!

Last edited:

Use Bayes' theorem.

Let D be the event that a selected item is defective

Let A, B, and C be the events that that an item is produced by machine A, B, and C respectively

$$P(D) = P(D\cap A) + P(D \cap B) +P(D \cap C)$$

$$= P(D|A)P(A) + P(D|B)P(B) + P(D|C)P(C)$$

$$P(A|D) = \frac {P(A \cap D)}{P(D)}$$

$$= \frac {P(D|A)P(A)}{P(D|A)P(A) + P(D|B)P(B) + P(D|C)P(C)}$$

HallsofIvy