This problem involves conditional probability, so let's break it down systematically.

We have the following information:

• Machine A produces 30% of the items, and machine B produces 70%.
• Of the items produced by machine A, 4% are defective.
• Of the items produced by machine B, 2% are defective.

We need to find the probability that a randomly selected defective item was produced by machine B.

Step 1: Define the events

Let:

• $A$ be the event that the item was produced by machine A.
• $B$ be the event that the item was produced by machine B.
• $D$ be the event that the item is defective.

Step 2: Known probabilities

From the problem:

• $P(A) = 0.30$, $P(B) = 0.70$
• $P(D|A) = 0.04$ (probability of being defective given that it was produced by machine A)
• $P(D|B) = 0.02$ (probability of being defective given that it was produced by machine B)

We are asked to find the probability that a randomly selected defective item was produced by machine B, i.e., $P(B|D)$.

Step 3: Apply Bayes' Theorem

Bayes' Theorem states that: $P(B|D) = \frac{P(D|B)P(B)}{P(D)}$ Where $P(D)$ is the total probability that an item is defective. We can compute $P(D)$ using the law of total probability: $P(D) = P(D|A)P(A) + P(D|B)P(B)$ Substitute the known values: $P(D) = (0.04)(0.30) + (0.02)(0.70)$ Now calculate $P(D)$.

Step 4: Compute the probability

After computing $P(D)$, we can use Bayes' Theorem to find $P(B|D)$.

Let me compute this step for you. Given values P_A = 0.30 # Probability that item was produced by A P_B = 0.70 # Probability that item was produced by B P_D_given_A = 0.04 # Probability that item is defective given it was produced by A P_D_given_B = 0.02 # Probability that item is defective given it was produced by B

Total probability of defect (P(D))

P_D = (P_D_given_A * P_A) + (P_D_given_B * P_B)

Calculate P(B|D) using Bayes' Theorem

P_B_given_D = (P_D_given_B * P_B) / P_D P_B_given_DThe probability that a randomly selected defective item was produced by machine B is approximately $0.538$ or 53.85%.

Would you like a more detailed breakdown of any part of this solution? Let me know if you need further clarifications.

Here are some related questions for you to explore:

1. How does the probability change if the defect rate of machine A increases?
2. What happens to the probability if the production rates of machine A and B are swapped?
3. How does Bayes' Theorem apply in situations with more than two machines or categories?
4. What is the importance of the prior probabilities $P(A)$ and $P(B)$ in Bayesian analysis?
5. How can we visualize this probability using a Venn diagram?

Tip: When working with Bayes' Theorem, always ensure that the sum of all prior probabilities equals 1 (i.e., $P(A) + P(B) = 1$) for a complete and consistent solution.