Questions: 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 notices that it is defective. If you want to know the probability that the bolt was made by the first machine (event C), given a defective bolt (event A), which form of Bayes' rule would you use? a.) P(A C) = (P(C A) P(A)) / P(C) b.) P(C A) = (P(A C) P(C)) / P(A) c.) P(A C) = (P(A C) P(A)) / P(C) d.) P(C A) = (P(A C) P(A)) / P(C)

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 notices that it is defective. If you want to know the probability that the bolt was made by the first machine (event C), given a defective bolt (event A), which form of Bayes' rule would you use? a.) P(A C) = (P(C A) P(A)) / P(C) b.) P(C A) = (P(A C) P(C)) / P(A) c.) P(A C) = (P(A C) P(A)) / P(C) d.) P(C A) = (P(A C) P(A)) / P(C)

Transcript text: 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 notices that it is defective. If you want to know the probability that the bolt was made by the first machine (event C), given a defective bolt (event A), which form of Bayes' rule would you use? a.) $P(A \mid C)=\frac{P(C \mid A) P(A)}{P(C)}$ b.) $P(C \mid A)=\frac{P(A \mid C) P(C)}{P(A)}$ c.) $P(A \mid C)=\frac{P(A \mid C) P(A)}{P(C)}$ d.) $P(C \mid A)=\frac{P(A \mid C) P(A)}{P(C)}$

Solution

Solution Steps

To determine the probability that the bolt was made by the first machine given that it is defective, we need to use Bayes' rule. The correct form of Bayes' rule for this scenario is:

\[ P(C \mid A) = \frac{P(A \mid C) P(C)}{P(A)} \]

where:

$ P(C \mid A) $ is the probability that the bolt was made by the first machine given that it is defective.
$ P(A \mid C) $ is the probability that the bolt is defective given that it was made by the first machine.
$ P(C) $ is the probability that the bolt was made by the first machine.
$ P(A) $ is the overall probability that the bolt is defective.

Solution Approach

Identify the given probabilities:
- $ P(C) = 0.70 $ (probability that the bolt was made by the first machine)
- $ P(A \mid C) = 0.03 $ (probability that the bolt is defective given that it was made by the first machine)
- $ P(\neg C) = 0.30 $ (probability that the bolt was made by the second machine)
- $ P(A \mid \neg C) = 0.05 $ (probability that the bolt is defective given that it was made by the second machine)
Calculate the overall probability of a defective bolt, $ P(A) $: \[ P(A) = P(A \mid C)P(C) + P(A \mid \neg C)P(\neg C) \]
Use Bayes' rule to find $ P(C \mid A) $: \[ P(C \mid A) = \frac{P(A \mid C) P(C)}{P(A)} \]

Step 1: Given Probabilities

We are given the following probabilities:

$ P(C) = 0.70 $ (the probability that the bolt was made by the first machine)
$ P(A \mid C) = 0.03 $ (the probability that the bolt is defective given that it was made by the first machine)
$ P(\neg C) = 0.30 $ (the probability that the bolt was made by the second machine)
$ P(A \mid \neg C) = 0.05 $ (the probability that the bolt is defective given that it was made by the second machine)

Step 2: Calculate Overall Probability of a Defective Bolt

We calculate the overall probability of a defective bolt, $ P(A) $, using the law of total probability: \[ P(A) = P(A \mid C) P(C) + P(A \mid \neg C) P(\neg C) \] Substituting the values: \[ P(A) = (0.03 \times 0.70) + (0.05 \times 0.30) = 0.021 + 0.015 = 0.036 \]

Step 3: Apply Bayes' Rule

Now, we apply Bayes' rule to find the probability that the bolt was made by the first machine given that it is defective, $ P(C \mid A) $: \[ P(C \mid A) = \frac{P(A \mid C) P(C)}{P(A)} \] Substituting the values: \[ P(C \mid A) = \frac{0.03 \times 0.70}{0.036} = \frac{0.021}{0.036} \approx 0.5833 \]