This problem is a typical example of Bayes' Theorem applied to conditional probability.

Here is the given information:

• Machine A produces 30% of the items ($P(A) = 0.30$).
• Machine B produces 70% of the items ($P(B) = 0.70$).
• 4% of the items produced by machine A are defective ($P(D|A) = 0.04$).
• 2% of the items produced by machine B are defective ($P(D|B) = 0.02$).

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

Bayes' Theorem formula:

$P(B|D) = \frac{P(D|B) P(B)}{P(D)}$

Where $P(D)$ is the total probability of an item being defective, which can be found using the law of total probability:

$P(D) = P(D|A)P(A) + P(D|B)P(B)$

Now, let's compute each part step-by-step.

Step 1: Find $P(D)$

$P(D) = P(D|A)P(A) + P(D|B)P(B)$ $P(D) = (0.04)(0.30) + (0.02)(0.70)$ $P(D) = 0.012 + 0.014 = 0.026$

Step 2: Apply Bayes' Theorem

$P(B|D) = \frac{P(D|B) P(B)}{P(D)} = \frac{(0.02)(0.70)}{0.026}$ $P(B|D) = \frac{0.014}{0.026} \approx 0.5385$

So, the probability that a defective item was produced by machine B is approximately 0.5385 or 53.85%.

Let me know if you need further clarifications or more details about the steps!

Related questions:

1. How do we calculate conditional probabilities using Bayes' Theorem?
2. What is the law of total probability, and how is it applied in this context?
3. Can Bayes' Theorem be used for more than two machines or events?
4. How would the result change if the defect rates of the machines were reversed?
5. What are other real-world examples where Bayes' Theorem can be used?

Tip:

Understanding Bayes' Theorem is very powerful when dealing with scenarios where we have partial information and need to update our beliefs based on new evidence.