To solve this problem, we use Bayes' Theorem and calculate the probability that the defective bolt was manufactured by machine 3. Let's define the problem mathematically:

Definitions:

• $A_1, A_2, A_3$: Events that the bolt is manufactured by Machine 1, Machine 2, and Machine 3 respectively.
• $D$: Event that the bolt is defective.

Given Data:

1. $P(A_1) = 0.25, P(A_2) = 0.35, P(A_3) = 0.40$ (probabilities of bolts being manufactured by each machine).
2. Defective bolt ratios:
   • $P(D|A_1) = 0.03$ (Machine 1),
   • $P(D|A_2) = 0.04$ (Machine 2),
   • $P(D|A_3) = 0.05$ (Machine 3).

Goal:

Find $P(A_3|D)$: the probability that the defective bolt was manufactured by Machine 3.

Step-by-Step Solution:

1. Apply Bayes' Theorem:

$P(A_3|D) = \frac{P(D|A_3)P(A_3)}{P(D)}$

2. Find $P(D)$ (total probability of defectiveness):

Using the law of total probability: $P(D) = P(D|A_1)P(A_1) + P(D|A_2)P(A_2) + P(D|A_3)P(A_3)$ Substitute the values: $P(D) = (0.03 \times 0.25) + (0.04 \times 0.35) + (0.05 \times 0.40)$ $P(D) = 0.0075 + 0.014 + 0.02 = 0.0415$

3. Calculate $P(A_3|D)$:

Substitute into Bayes' formula: $P(A_3|D) = \frac{P(D|A_3)P(A_3)}{P(D)}$ $P(A_3|D) = \frac{0.05 \times 0.40}{0.0415}$ $P(A_3|D) = \frac{0.02}{0.0415} \approx 0.482$

Final Answer:

The probability that the defective bolt was manufactured by Machine 3 is approximately 0.482 (48.2%).

Would you like a more detailed explanation or clarification?

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Related Questions:

1. What is the probability that the defective bolt was manufactured by Machine 1?
2. How can the law of total probability help in solving similar problems?
3. What is the role of Bayes' Theorem in conditional probability scenarios?
4. If the defect rates for Machine 2 were halved, how would the result change?
5. What is the total probability of defectiveness for all three machines combined?

Tip: Bayes' Theorem is particularly useful when you have multiple conditions leading to a single outcome,