We can solve this problem using Bayes' Theorem, which helps us find the probability of an event given some prior information.

In this case, we want to find the probability that the defective part came from supplier $S1$ given that it is defective. This can be expressed as:

$P(S1 \mid D)$

Where:

• $S1$ is the event that the part came from supplier S1.
• $D$ is the event that the part is defective.

Bayes' Theorem formula is:

$P(S1 \mid D) = \frac{P(D \mid S1) P(S1)}{P(D)}$

Where:

• $P(D \mid S1)$ is the probability that a part is defective given that it came from S1.
• $P(S1)$ is the prior probability that a part came from S1.
• $P(D)$ is the total probability that a part is defective, regardless of the supplier.

Step 1: Assign values to the known probabilities.

• $P(S1) = 0.50$ (since 50% of the parts come from S1)

• $P(S2) = 0.20$ (since 20% of the parts come from S2)

• $P(S3) = 0.30$ (since 30% of the parts come from S3)

• $P(D \mid S1) = 0.05$ (5% of the parts from S1 are defective)

• $P(D \mid S2) = 0.03$ (3% of the parts from S2 are defective)

• $P(D \mid S3) = 0.06$ (6% of the parts from S3 are defective)

Step 2: Find $P(D)$ (the total probability that a part is defective).

Using the law of total probability, we calculate $P(D)$ as follows:

$P(D) = P(D \mid S1)P(S1) + P(D \mid S2)P(S2) + P(D \mid S3)P(S3)$

Substituting the known values:

$P(D) = (0.05)(0.50) + (0.03)(0.20) + (0.06)(0.30)$

$P(D) = 0.025 + 0.006 + 0.018 = 0.049$

Step 3: Use Bayes' Theorem to find $P(S1 \mid D)$.

Now, apply the values to Bayes' Theorem:

$P(S1 \mid D) = \frac{(0.05)(0.50)}{0.049}$

$P(S1 \mid D) = \frac{0.025}{0.049} \approx 0.5102$

Conclusion:

The probability that the defective part was supplied by S1 is approximately 0.5102, or 51.02%.

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Would you like any further details or have any questions? Here are 5 related questions to consider:

1. How would the result change if the defect rates for the suppliers were different?
2. Can you apply Bayes' Theorem in other contexts outside defect rates, like medical tests?
3. What if the supplier S3 had a 10% defect rate? How would that impact the outcome?
4. How do we interpret the overall defect rate (P(D)) in a business context?
5. Can Bayes'