Half of a set of the parts are manufactured by machine A and half by machine B. Four (.04) of all the parts are defective. Six percent (6%) of the parts manufacture on machine A are defective. Find the probability that a part was manufactured on machine A, given the part is defective."

"Half of a set of the parts" and "Six percent (6%) of the parts manufacture on machine A are defective."

.5 * .06 = ?

To find the probability that a part was manufactured on machine A, given the part is defective, we can use Bayes' theorem.

Let's define the following events:
A: Part is manufactured on machine A
B: Part is manufactured on machine B
D: Part is defective

We are given the following probabilities:
P(A) = 0.5 (since half of the parts are manufactured by machine A)
P(B) = 0.5 (since half of the parts are manufactured by machine B)
P(D|A) = 0.06 (6% of the parts manufactured on machine A are defective)
P(D) = 0.04 (4% of all the parts are defective)

We want to find P(A|D), the probability that a part was manufactured on machine A, given the part is defective. Using Bayes' theorem:

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

Let's calculate:

P(D) = P(D|A) * P(A) + P(D|B) * P(B)
P(D) = 0.06 * 0.5 + P(D|B) * 0.5
0.04 = 0.03 + P(D|B) * 0.5
P(D|B) * 0.5 = 0.04 - 0.03
P(D|B) = 0.01 / 0.5
P(D|B) = 0.02

Now we can calculate P(A|D):

P(A|D) = (P(D|A) * P(A)) / P(D)
P(A|D) = (0.06 * 0.5) / 0.04
P(A|D) = 0.03 / 0.04
P(A|D) = 0.75

Therefore, the probability that a part was manufactured on machine A, given the part is defective, is 0.75 or 75%.

To find the probability that a part was manufactured on machine A, given that the part is defective, we can use Bayes' theorem. Bayes' theorem states:

P(A|B) = (P(B|A) * P(A)) / P(B)

where:
P(A|B) is the probability of event A occurring given that event B has occurred.
P(B|A) is the probability of event B occurring given that event A has occurred.
P(A) is the probability of event A occurring.
P(B) is the probability of event B occurring.

In this case:
Event A: Part is manufactured on machine A.
Event B: Part is defective.

We are given the following information:
P(defective) = 0.04 (probability that a part is defective)
P(defective|A) = 0.06 (probability that a part is defective given that it was manufactured on machine A)

We need to find P(A|defective), the probability that a part was manufactured on machine A given that it is defective.

First, let's find P(A), the probability that a part is manufactured on machine A.
Since half of the parts are manufactured by machine A and half by machine B, we have:
P(A) = 0.5 (probability that a part is manufactured on machine A)

Now, let's substitute the values into Bayes' theorem.

P(A|defective) = (P(defective|A) * P(A)) / P(defective)

P(A|defective) = (0.06 * 0.5) / 0.04
= 0.03 / 0.04
= 0.75

Therefore, the probability that a part was manufactured on machine A, given that the part is defective, is 0.75 or 75%.