1.	In a bolt factory, machine A, B and C produce respectively 25%, 35% and 40% of total. Of their output, 5%, 4% and 2% are defective. A bolt is drawn at random from the product and is found to be defective. What is the probability that it was manufactured by machine A, machine B or machine C?

To solve this problem, we will use Bayes' theorem, which allows us to find the probability of an event based on prior knowledge of conditions related to the event.

First, let's define the events:

- \( A_1 \): The bolt was produced by machine A.
- \( A_2 \): The bolt was produced by machine B.
- \( A_3 \): The bolt was produced by machine C.
- \( D \): The bolt is defective.

Given:
- \( P(A_1) = 0.25 \)
- \( P(A_2) = 0.35 \)
- \( P(A_3) = 0.40 \)
- \( P(D|A_1) = 0.05 \)
- \( P(D|A_2) = 0.04 \)
- \( P(D|A_3) = 0.02 \)

We want to find \( P(A_1|D) \), \( P(A_2|D) \), and \( P(A_3|D) \).

Using Bayes' theorem:
\[
P(A_i|D) = \frac{P(D|A_i)P(A_i)}{P(D)}
\]

First, we need to find \( P(D) \), the total probability of a defective bolt:
\[
P(D) = P(D|A_1)P(A_1) + P(D|A_2)P(A_2) + P(D|A_3)P(A_3)
\]
\[
P(D) = (0.05 \times 0.25) + (0.04 \times 0.35) + (0.02 \times 0.40)
\]
\[
P(D) = 0.0125 + 0.014 + 0.008
\]
\[
P(D) = 0.0345
\]

Now we can find each conditional probability:

\[
P(A_1|D) = \frac{P(D|A_1)P(A_1)}{P(D)} = \frac{0.05 \times 0.25}{0.0345} = \frac{0.0125}{0.0345} \approx 0.3623
\]

\[
P(A_2|D) = \frac{P(D|A_2)P(A_2)}{P(D)} = \frac{0.04 \times 0.35}{0.0345} = \frac{0.014}{0.0345} \approx 0.4058
\]

\[
P(A_3|D) = \frac{P(D|A_3)P(A_3)}{P(D)} = \frac{0.02 \times 0.40}{0.0345} = \frac{0.008}{0.0345} \approx 0.2319
\]

Thus, the probabilities are:
- \( P(A_1|D) \approx 0.3623 \)
- \( P(A_2|D) \approx 0.4058 \)
- \( P(A_3|D) \approx 0.2319 \)

Would you like further details or have any questions?

Here are five additional questions you might find interesting:
1. How do you find the mean and variance of a probability distribution?
2. What is the difference between permutations and combinations?
3. How do you solve a system of linear equations using matrix methods?
4. What are the properties of definite and indefinite integrals?
5. Can you explain the concept of eigenvalues and eigenvectors?

**Tip:** Always double-check your calculations, especially when dealing with probabilities and multiple steps in solving problems.

Explore a probability problem involving defective bolts produced by machines A, B, and C in a factory setting. Calculate probabilities using Bayes' Theorem to determine the likelihood that a defective bolt came from each machine. Ideal for advanced high school and college students studying probability theory.

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