A manufacturer buys 40% of a certain
part from one supplier and the rest from a second
supplier. It notes that 2% of the parts from the first
supplier are defective, and 3% are defective from
the second supplier. What is the probability that a
part is defective? use bayes theorem

We can use **Bayes' Theorem** along with the **Law of Total Probability** to find the probability that a randomly chosen part is defective.

### Let’s define the events:
- \( S_1 \): The part comes from the first supplier.
- \( S_2 \): The part comes from the second supplier.
- \( D \): The part is defective.

### Given information:
- \( P(S_1) = 0.40 \) (40% of parts from the first supplier),
- \( P(S_2) = 0.60 \) (60% of parts from the second supplier),
- \( P(D | S_1) = 0.02 \) (2% of parts from the first supplier are defective),
- \( P(D | S_2) = 0.03 \) (3% of parts from the second supplier are defective).

We want to calculate \( P(D) \), the overall probability that a part is defective.

Using the **Law of Total Probability**:

\[
P(D) = P(D | S_1)P(S_1) + P(D | S_2)P(S_2)
\]

### Substituting the given values:

\[
P(D) = (0.02)(0.40) + (0.03)(0.60)
\]

\[
P(D) = 0.008 + 0.018 = 0.026
\]

So, the probability that a randomly chosen part is defective is **0.026**, or **2.6%**.

Would you like more details on Bayes' Theorem or further clarification? Here are 5 related questions you might find interesting:

1. How is Bayes' Theorem applied to multiple suppliers?
2. What if more suppliers with different defect rates were added?
3. How can this concept be extended to conditional probabilities in a manufacturing system?
4. Can you explain how the Law of Total Probability applies in other real-world contexts?
5. What if we change the percentage of defective parts for either supplier? How would that affect \(P(D)\)?

**Tip**: Bayes' Theorem is useful for updating probabilities as new information becomes available.

A manufacturer buys 40% of a certain part from one supplier and the rest from a second supplier. It notes that 2% of the parts from the first supplier are defective, and 3% are defective from the second supplier. What is the probability that a part is defective? Use Bayes' theorem.

Learn how to calculate the probability of a defective part using Bayes' Theorem. This problem explores the application of probability concepts like the Law of Total Probability and Bayes' Theorem, suitable for students in Grades 11-12.

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