A company that produces a particular machine component has 3 factories, one each in Buffalo, Detroit, and Pittsburgh. 39% of the components produced come from the Buffalo factory, 36% of the components come from the Detroit factory, and 25% of the components come from the Pittsburgh factory. It is known that 1.1% of the components from the Buffalo factory, 1.4% of the components from the Detroit factory, and 0.9% of the components from the Pittsburgh factory are defective. Given that a component is selected at random and is found to be defective, what is the probability that the component was made in Detroit?

Question

A company that produces a particular machine component has 3 factories, one each in Buffalo, Detroit, and Pittsburgh. 39% of the components produced come from the Buffalo factory, 36% of the components come from the Detroit factory, and 25% of the components come from the Pittsburgh factory. It is known that 1.1% of the components from the Buffalo factory, 1.4% of the components from the Detroit factory, and 0.9% of the components from the Pittsburgh factory are defective. Given that a component is selected at random and is found to be defective, what is the probability that the component was made in Detroit?

Answer 1

To find the probability that the defective component was made in Detroit, we can use Bayes' theorem.

Let's denote the events as follows:
A: The component is made in Buffalo.
D: The component is made in Detroit.
P: The component is made in Pittsburgh.
Def: The component is defective.

We want to find P(D|Def), the probability that the component was made in Detroit given that it is defective.

According to Bayes' theorem, we have:
P(D|Def) = (P(Def|D) * P(D)) / P(Def)

We are given:
P(A) = 0.39 (Probability that the component is made in Buffalo)
P(D) = 0.36 (Probability that the component is made in Detroit)
P(P) = 0.25 (Probability that the component is made in Pittsburgh)

We are also given:
P(Def|A) = 0.011 (Probability that a component made in Buffalo is defective)
P(Def|D) = 0.014 (Probability that a component made in Detroit is defective)
P(Def|P) = 0.009 (Probability that a component made in Pittsburgh is defective)

Now, we need to find P(Def), the probability that the component is defective. We can do this by using the law of total probability.

P(Def) = P(Def|A) * P(A) + P(Def|D) * P(D) + P(Def|P) * P(P)

Plugging in the given values, we get:
P(Def) = 0.011 * 0.39 + 0.014 * 0.36 + 0.009 * 0.25

Once we have P(Def), we can calculate P(D|Def) using Bayes' theorem.

P(D|Def) = (P(Def|D) * P(D)) / P(Def)

Plugging in the given values, we get:
P(D|Def) = (0.014 * 0.36) / P(Def)

Now, substitute the value of P(Def) into the equation to calculate P(D|Def).