This is a problem that can be solved using Bayes' Theorem, which helps us find the conditional probability of an event given that another event has occurred.

Let’s define the events:

• $A_1$: Product came from machine 1
• $A_2$: Product came from machine 2
• $A_3$: Product came from machine 3
• $B$: The product is non-defective

We are asked to find the probability that a non-defective product came from machine 2, i.e., $P(A_2 \mid B)$.

Step 1: Known information

• $P(A_1) = 0.35$ (35% of products are from machine 1)
• $P(A_2) = 0.40$ (40% of products are from machine 2)
• $P(A_3) = 0.25$ (25% of products are from machine 3)
• Defective rates:
  • $P(\text{Defective} \mid A_1) = 0.02$
  • $P(\text{Defective} \mid A_2) = 0.01$
  • $P(\text{Defective} \mid A_3) = 0.12$

Thus, the probabilities of producing a non-defective product are:

• $P(B \mid A_1) = 1 - 0.02 = 0.98$
• $P(B \mid A_2) = 1 - 0.01 = 0.99$
• $P(B \mid A_3) = 1 - 0.12 = 0.88$

Step 2: Use Bayes' Theorem

Bayes' Theorem formula is: $P(A_2 \mid B) = \frac{P(B \mid A_2) P(A_2)}{P(B)}$

To find $P(A_2 \mid B)$, we first need $P(B)$, the total probability of a non-defective product.

$P(B) = P(B \mid A_1) P(A_1) + P(B \mid A_2) P(A_2) + P(B \mid A_3) P(A_3)$ Substituting the known values: $P(B) = (0.98)(0.35) + (0.99)(0.40) + (0.88)(0.25)$

Now, let’s calculate that:

$P(B) = 0.343 + 0.396 + 0.22 = 0.959$

Step 3: Find $P(A_2 \mid B)$

Now that we have $P(B) = 0.959$, we can substitute it back into Bayes' Theorem: $P(A_2 \mid B) = \frac{(0.99)(0.40)}{0.959} = \frac{0.396}{0.959}$

Let's calculate the final result:

$P(A_2 \mid B) \approx 0.4129$

Final Answer:

The probability that a non-defective product came from the second machine is approximately 0.4129.

Would you like more details or have any questions?

---

Here are 5 related questions to expand your understanding:

1. How would the answer change if the defective rates of the machines were different?
2. What is the probability that a defective product came from machine 3?
3. How would you calculate the probability that a product is defective overall?
4. What assumptions are made in using Bayes' Theorem in this problem?
5. Can you extend this problem to more than three machines?

Tip: When solving probability problems, especially those involving conditional probability, organizing the given data clearly and methodically applying Bayes' Theorem can greatly simplify the solution process.