Question

A hospital obtains 40% of its flu vaccine from company A, 50% from Company B and 10% from Company C. From past experience it is known that 3% of the vials from A are ineffective, 2% from B are ineffective, and 5% from C are ineffective. The hospital tests five vials from each shipment, if at least one of the 5 is effective, find the conditional probability of that shipment having come from Company C.

Answer 1

To find the conditional probability of a shipment coming from Company C given that at least one of the vials is effective, we can use Bayes' theorem.

Step-by-step explanation:

To find the conditional probability of a shipment coming from Company C given that at least one of the vials is effective, we can use Bayes' theorem. Let's denote the events as follows:

• A: The shipment comes from Company A
• B: The shipment comes from Company B
• C: The shipment comes from Company C
• E: At least one of the 5 vials is effective

We are given the following information:

• P(A) = 0.4
• P(B) = 0.5
• P(C) = 0.1
• P(E|A) = 1 - 0.03 = 0.97
• P(E|B) = 1 - 0.02 = 0.98
• P(E|C) = 1 - 0.05 = 0.95

Using Bayes' theorem, the conditional probability of the shipment coming from Company C given that at least one of the vials is effective is:

P(C|E) = (P(C) * P(E|C)) / (P(A) * P(E|A) + P(B) * P(E|B) + P(C) * P(E|C))

Substituting the given values, we get:

P(C|E) = (0.1 * 0.95) / (0.4 * 0.97 + 0.5 * 0.98 + 0.1 * 0.95) = 0.019

Therefore, the conditional probability of the shipment coming from Company C given that at least one of the vials is effective is 0.019 or 1.9%.