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.