Bayes' Theorem Statement:

Bayes' theorem provides a way to update probabilities based on new evidence. It states that:

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

where:

• $P(E_i | A)$ = Probability that event $E_i$ occurred given that $A$ has occurred.
• $P(A | E_i)$ = Probability of event $A$ occurring given that $E_i$ has occurred.
• $P(E_i)$ = Prior probability of event $E_i$.
• $P(A)$ = Total probability of event $A$ occurring.

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Step 1: Define Given Data

Let:

• $E_1, E_2, E_3$ be the events that a bolt is manufactured by missions $E_1, E_2, E_3$, respectively.
• $A$ be the event that a randomly drawn bolt is defective.

From the problem:

• $P(E_1) = 20\% = 0.2$
• $P(E_2) = 30\% = 0.3$
• $P(E_3) = 50\% = 0.5$

The probability of producing a defective bolt:

• $P(A | E_1) = 6\% = 0.06$
• $P(A | E_2) = 3\% = 0.03$
• $P(A | E_3) = 2\% = 0.02$

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Step 2: Compute Total Probability of Defective Bolt, $P(A)$

Using the Law of Total Probability:

$P(A) = P(A | E_1) P(E_1) + P(A | E_2) P(E_2) + P(A | E_3) P(E_3)$

$P(A) = (0.06 \times 0.2) + (0.03 \times 0.3) + (0.02 \times 0.5)$

$P(A) = 0.012 + 0.009 + 0.01 = 0.031$

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Step 3: Compute $P(E_1 | A)$ Using Bayes’ Theorem

$P(E_1 | A) = \frac{P(A | E_1) P(E_1)}{P(A)}$

$P(E_1 | A) = \frac{(0.06 \times 0.2)}{0.031}$

$P(E_1 | A) = \frac{0.012}{0.031} = 0.387$

Thus, the probability that the defective bolt was manufactured by mission $E_1$ is 0.387 or 38.7%.

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Conclusion

There is a 38.7% probability that a randomly selected defective bolt was manufactured by mission E1.

Do you need further details or clarifications?

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Related Questions:

1. How can Bayes' theorem be applied in medical diagnosis?
2. What is the Law of Total Probability, and how does it work?
3. How do you calculate conditional probability using Bayes’ theorem?
4. Can you solve a similar problem with different probabilities?
5. What are real-world applications of Bayes' theorem in machine learning?

Tip:

Bayes' theorem is widely used in spam filtering, medical diagnosis, and AI-based predictions.