We can solve this problem using Bayes' Theorem, which is ideal for determining conditional probabilities. Here's the breakdown:

Given:

• 70% of the shipments come from supplier A (reliable), i.e., $P(A) = 0.7$.
• 30% of the shipments come from supplier B, i.e., $P(B) = 0.3$.
• Supplier A’s shipments typically contain 10% defectives, i.e., $P(D|A) = 0.1$.
• Supplier B’s shipments typically contain 20% defectives, i.e., $P(D|B) = 0.2$.
• Out of 20 parts selected, 1 part is found to be defective.

We need to calculate the probability that the shipment came from supplier A, given that 1 defective part was found. Mathematically, this is $P(A|D)$, where $D$ indicates that 1 defective part is found out of the 20.

Step 1: Applying Bayes' Theorem

$P(A|D) = \frac{P(D|A) \cdot P(A)}{P(D)}$

Where:

• $P(D|A)$: Probability of finding 1 defective part in 20 from supplier A.
• $P(D|B)$: Probability of finding 1 defective part in 20 from supplier B.
• $P(D)$: Total probability of finding 1 defective part in 20 from either supplier (A or B).

Step 2: Calculating $P(D|A)$ and $P(D|B)$

The number of defectives in a sample of 20 follows a binomial distribution. We are interested in the probability of exactly 1 defective part being found, i.e., $k = 1$ for a binomial distribution.

For supplier A: $P(D|A) = \binom{20}{1} (0.1)^1 (0.9)^{19} = 20 \times 0.1 \times (0.9)^{19}$

For supplier B: $P(D|B) = \binom{20}{1} (0.2)^1 (0.8)^{19} = 20 \times 0.2 \times (0.8)^{19}$

Step 3: Total probability $P(D)$

The total probability $P(D)$ is a weighted sum of the probabilities for supplier A and supplier B: $P(D) = P(D|A) \cdot P(A) + P(D|B) \cdot P(B)$

Step 4: Substituting into Bayes' Theorem

Finally, we substitute the values back into Bayes' Theorem: $P(A|D) = \frac{P(D|A) \cdot P(A)}{P(D)}$

Let's compute the numerical values.The probability that the shipment came from the more reliable supplier (Supplier A) given that one defective part was found is approximately 91.62%.

Would you like more details or have any questions?

Here are some related questions to deepen your understanding:

1. What would happen to the probability if two defective parts were found instead of one?
2. How does the size of the sample affect the probability outcome?
3. What is Bayes' Theorem, and how is it applied in real-world problems like this?
4. How would the result change if the defective rates for suppliers A and B were closer?
5. How can we generalize this approach to shipments with more than two suppliers?

Tip: Bayes' Theorem is particularly powerful for updating probabilities when new evidence is available. It's commonly used in medical diagnosis, spam filters, and reliability testing.