This problem can be solved using Bayes' Theorem.

Let:

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

Given Information:

• $P(A) = 0.40$ (40% of the output is produced by machine A)
• $P(B) = 0.60$ (60% of the output is produced by machine B)
• Defective rates:
  • $P(D | A) = \frac{9}{1000} = 0.009$ (the probability of an item being defective if it is produced by machine A)
  • $P(D | B) = \frac{1}{250} = 0.004$ (the probability of an item being defective if it is produced by machine B)

We are tasked with finding:

• $P(A | D)$: the probability that the item was produced by machine A given that it is defective.
• $P(B | D)$: the probability that the item was produced by machine B given that it is defective.

Step 1: Find the total probability that the item is defective, $P(D)$, using the Law of Total Probability:

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

Substitute the given values:

$P(D) = (0.009 \times 0.40) + (0.004 \times 0.60)$ $P(D) = 0.0036 + 0.0024 = 0.006$

Step 2: Use Bayes' Theorem to find the conditional probabilities.

(a) Probability that the defective item was produced by machine A:

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

Substitute the known values:

$P(A | D) = \frac{0.009 \times 0.40}{0.006} = \frac{0.0036}{0.006} = 0.6$

(b) Probability that the defective item was produced by machine B:

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

Substitute the known values:

$P(B | D) = \frac{0.004 \times 0.60}{0.006} = \frac{0.0024}{0.006} = 0.4$

Final Answer:

• The probability that the defective item was produced by machine A is $P(A | D) = 0.6$.
• The probability that the defective item was produced by machine B is $P(B | D) = 0.4$.

Would you like further details on Bayes' Theorem or the Law of Total Probability?

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5 Relative Questions:

1. How does Bayes' Theorem apply in real-world problems?
2. What if machine C was added with a different defective rate, how would that affect the calculation?
3. Can you calculate the overall defect rate if the defect rate for machine A changes?
4. How can the Law of Total Probability be extended to more machines?
5. What is the importance of conditional probability in decision making?

Tip:

Always ensure to correctly identify the "prior probabilities" (like $P(A)$ and $P(B)$) and the "likelihoods" (like $P(D | A)$ and $P(D | B)$) when applying Bayes' Theorem.