A company has two factories that manufacture tires. Suppose 65% of the tires are manufactured in factory 1 and 35% are manufactured in factory 2. In factory 1, 5% of the tires are defective. In factory 2, 2% of the tires are defective.

a) Determine the probability that a defective tire is produced

b) A defective tire is discovered. What is the probability the defective tire came from factory 2?

Please show your work.

To find the probability of a defective tire being produced or coming from a specific factory, we can use conditional probability. Let's solve the problem step by step:

a) To determine the probability that a defective tire is produced, we need to consider both factories' defective tire rates.

Let's denote the events as follows:
D1 = A tire is defective and produced in factory 1.
D2 = A tire is defective and produced in factory 2.

We are given the following information:
P(D1) = 0.05 (5% of tires from factory 1 are defective)
P(D2) = 0.02 (2% of tires from factory 2 are defective)

Now, we can calculate the probability of a defective tire being produced:
P(D) = P(D1) + P(D2)
= 0.05 + 0.02
= 0.07

Therefore, the probability that a defective tire is produced is 0.07 or 7%.

b) To find the probability that a defective tire came from factory 2, we need to calculate the conditional probability using Bayes' Theorem.

Let's denote the events as follows:
F1 = A tire is produced in factory 1.
F2 = A tire is produced in factory 2.

We want to find P(F2|D), which means the probability that the tire came from factory 2 given that it is defective.

According to Bayes' Theorem:
P(F2|D) = (P(D|F2) * P(F2)) / P(D)

We are given the following information:
P(D|F2) = 0.02
P(F2) = 0.35
P(D) = 0.07 (from part a)

Now, we can substitute these values into Bayes' Theorem:
P(F2|D) = (0.02 * 0.35) / 0.07
= 0.007 / 0.07
= 0.1

Therefore, the probability that the defective tire came from factory 2 is 0.1 or 10%.