In a factory that manufactures bolts, the first machine manufactures 70% of the bolts and a second
machine manufactures the remaining 30%. The percentage of defective bolts is 3% and 5%, respectively.
An employee picks a bolt off a shelf at random and it is from the first machine (event A).
If you want to know the probability that it is a defective bolt (event C), which form of Bayes' rule would
you use?

a.) P(C|A)= P(A|C)P(C)/P(A)
b.) P(A|C)= P(A|C)P(A)/P(C)
c.) P(A|C)= P(C|A)P(A)/P(C)

To determine the probability that a bolt from the first machine is defective, we need to use Bayes' rule. Bayes' rule is used to calculate the probability of an event given another event has occurred.

In this scenario, we want to know the probability that a bolt from the first machine (event A) is defective (event C).

The correct form of Bayes' rule to use in this case is:

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

Let's break down the components of this equation:

- P(C|A): This represents the probability of event C (defective bolt) given event A (bolt from the first machine). This is what we want to calculate.

- P(A|C): This represents the probability of event A (bolt from the first machine) given event C (defective bolt). In this scenario, it is not provided, and we don't need it to calculate the probability we are interested in.

- P(C): This represents the probability of event C (defective bolt) occurring. In this scenario, it is given as 3% for the bolts from the first machine.

- P(A): This represents the probability of event A (bolt from the first machine) occurring. In this scenario, it is given as 70% for the bolts from the first machine.

By substituting the given values into the equation, we have:

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

P(C|A) = (1 * 0.03) / 0.70

P(C|A) = 0.03 / 0.70

Therefore, the correct form of Bayes' rule to use in this scenario is:

a.) P(C|A) = P(A|C) * P(C) / P(A)