Question

Parts and materials for skis made by Company C are supplied by two suppliers. Supplier​ A's materials make up 27​% of what is​ used, with Supplier B providing the rest. Past records indicate that 22​% of Supplier​ A's materials are defective and 9​% of Supplier​ B's materials are defective. Since it is impossible to tell which supplier the materials came from once they are in​ inventory, the manager wants to know which supplier more likely supplied the defective materials the supervisor has brought to his attention. Provide the manager this information.

Answer 1

Answer:

Supplier B more likely supplied the defective materials.

Explanation:

This exercise is solved in four steps:

1. Statistical events are defined:

A = (provider A)

B = (provider B)

D = defective materials

From the problem statement, 27% of the materials used by Company C are provided by supplier A. Therefore:

P (A) = 0.27.

The remaining 73% is provided by supplier B. Therefore:

P (B) = 0.73.

2. Conditional probabilities are established. In other words, what is the probability that the materials are defective? Remember that the "defect" is the condition that most interests the manager.

According to the example, 22% of materials from supplier A are defective. We can formalize this as follows:

P (D / A) = 0.22

On the other hand, 9% of supplier B materials are defective:

P (D / B) = 0.09

3. It will be determined what is the probability that each supplier has provided defective products by applying Bayes´ theorem.

3.1 The probability of this event will be found for supplier A:

The Bayes´ Theorem for this case is:

$P (A / D) = \frac{P(A) P(D/A)}{P(A) P(D/A) + P(B) P(D/B)}$

We replace with the data obtained in the previous points (1 and 2):

$P (A/D) = \frac{(0.27)(0.22)}{(0.27)(0.22)+(0.73)(0.09)}$

$P (A/D)= \frac{0.0594}{0.0594+0.0657}$

$P (A/D) = \frac{0.0594}{0.1251}$

P (A / D) = 0.474

That means that approximately 47.4% of defective materials come from supplier A.

3.2 The probability of this event for provider B will be found.

The Bayes´ Theorem for this case is:

$P (B/D) = \frac{P(B) P(D/B)}{P(A) P(D/A) + P(B) P(D/B)}$

We replace with the data obtained in the previous points (1 and 2):

$P (B/D) = \frac{(0.73)(0.09)}{(0.27)(0.22)+(0.73)(0.09)}$

$P (B/D)= \frac{0.0657}{0.0594+0.0657}$

$P (B/D) = \frac{0.0657}{0.1251}$

P (B / D) = 0.525

That means that approximately 52.5% of the defective materials come from supplier B.

4. Compare the conditional probabilities.

If we compare P (A / D) and P (B / D), we can see that the largest is P (B / D) (47.4 < 52.5). Therefore, supplier B is more likely to have supplied defective materials.