This problem involves using Bayes' Theorem to find the probability that a defective bolt came from Machine C.

We are given:

• The probabilities that a bolt comes from each machine:

  • $P(A) = 0.20$ (Machine A manufactures 20% of bolts)
  • $P(B) = 0.30$ (Machine B manufactures 30% of bolts)
  • $P(C) = 0.50$ (Machine C manufactures 50% of bolts)

• The probabilities that a bolt from each machine is defective:

  • $P(D|A) = 0.06$ (6% of bolts from Machine A are defective)
  • $P(D|B) = 0.03$ (3% of bolts from Machine B are defective)
  • $P(D|C) = 0.02$ (2% of bolts from Machine C are defective)

We are asked to find $P(C|D)$, the probability that the bolt came from Machine C given that it is defective.

Bayes' Theorem

Bayes' Theorem states:

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

Where:

• $P(C|D)$ is the probability the bolt came from Machine C, given that it is defective.
• $P(D|C)$ is the probability that the bolt is defective given it came from Machine C.
• $P(C)$ is the probability that the bolt came from Machine C.
• $P(D)$ is the total probability that a randomly chosen bolt is defective.

To calculate $P(D)$, we use the law of total probability:

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

Step-by-Step Calculation

1. Calculate $P(D)$: $P(D) = (0.06)(0.20) + (0.03)(0.30) + (0.02)(0.50)$ $P(D) = 0.012 + 0.009 + 0.01 = 0.031$

2. Apply Bayes' Theorem:

$P(C|D) = \frac{(0.02)(0.50)}{0.031}$ $P(C|D) = \frac{0.01}{0.031} \approx 0.3226$

Thus, the probability that the defective bolt came from Machine C is approximately 0.3226, or 32.26%.

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Do you need further clarification or details on the calculation? Here are some follow-up questions that might interest you:

1. What is the total probability that a bolt is defective?
2. How does Bayes' Theorem apply to situations other than defective products?
3. What happens if the defective rates for each machine were all the same?
4. How would the probability change if Machine C produced fewer bolts?
5. Can this method be used for multiple outcomes (i.e., more than three machines)?

Tip: Understanding Bayes' Theorem is essential when dealing with conditional probabilities. It helps to reverse the direction of probability to find the cause (like the machine) when the effect (defect) is known.