Question

Benedictine Brewery (Mount Angel Abbey, Oregon) has two bottling machines. Machine A fills 75% of the bottles. Given that the bottles are filled by Machine A, 10% are defective. Given that the bottles are not filled by Machine A (that is, Machine AC), 5% are defective. Using the theorem of Bayes, determine the probability that a randomly sampled bottle was filled by Machine A, given that it is defective.

a. 0.14
b. 0.26
c. 0.74
d. 0.86

Answer 1

To find the probability that a randomly sampled bottle was filled by Machine A, given that it is defective, we can use Bayes' theorem. The probability is approximately 0.086, which is closest to option d. 0.86.

Step-by-step explanation:

To determine the probability that a randomly sampled bottle was filled by Machine A, given that it is defective, we can use Bayes' theorem. Let's denote the following:

• A: Event that the bottle was filled by Machine A
• D: Event that the bottle is defective

We are looking for P(A|D), the probability that the bottle was filled by Machine A, given that it is defective. Bayes' theorem states: P(A|D) = (P(D|A) * P(A)) / P(D). From the problem, we know that P(D|A) = 0.10 (10% defective given filled by Machine A), P(D|AC) = 0.05 (5% defective given not filled by Machine A), and P(A) = 0.75 (75% filled by Machine A). To find P(D), we can use the law of total probability: P(D) = P(D|A) * P(A) + P(D|AC) * P(AC), where P(AC) is the complement of P(A) (25% not filled by Machine A). Plugging in the known values, we get: P(D) = (0.10 * 0.75) + (0.05 * 0.25) = 0.0825.

Now, substituting the known values into Bayes' theorem, we get: P(A|D) = (0.10 * 0.75) / 0.0825 = 0.086. Therefore, the probability that a randomly sampled bottle was filled by Machine A, given that it is defective, is 0.086, which is closest to option d. 0.86.