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 need to use Bayes' theorem. Bayes' theorem allows us to calculate the probability of an event given some prior information.

Let's define our events:
A: The component was made in Buffalo
D: The component was made in Detroit
P: The component was made in Pittsburgh
Def: The component is defective

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

Bayes' theorem states:
P(D|Def) = P(Def|D) * P(D) / P(Def)

We have the following information:
P(B) = 0.39 (probability that a component comes from Buffalo)
P(D) = 0.36 (probability that a component comes from Detroit)
P(P) = 0.25 (probability that a component comes from Pittsburgh)

P(Def|B) = 0.011 (probability that a component from Buffalo is defective)
P(Def|D) = 0.014 (probability that a component from Detroit is defective)
P(Def|P) = 0.009 (probability that a component from Pittsburgh is defective)

Now, let's find P(Def), the overall probability that a component is defective:
P(Def) = P(Def|B) * P(B) + P(Def|D) * P(D) + P(Def|P) * P(P)

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

Next, we can substitute all the values into Bayes' theorem to find P(D|Def):
P(D|Def) = P(Def|D) * P(D) / P(Def)

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

To find P(Def), plug in the values we calculated earlier:
P(D|Def) = 0.014 * 0.36 / (0.011 * 0.39 + 0.014 * 0.36 + 0.009 * 0.25)

Calculating this expression, we find that the probability that the component was made in Detroit given that it is defective is approximately 0.447 or 44.7%.