On my homework, there is a problem:
A hospital receives two-fifths of its flu vaccine from Company A and the remainder from Company B. Each shipment contains a large number of vials of vaccine. From Company A, 3% of the vials are ineffective; from Company B, 2% are ineffective. A hospital test n = 15 randomly selected vials from one shipment and finds that 2 are ineffective. What is the conditional probability that this shipment came from Company A?
The given answer is $.568$. I keep getting $.5$.
I think I'm confused what to do with the $2/15$ inefficient. I didn't use that at all when I calculated, and I can't figure out when to use it.
My calculation: $$P(I) = P(A)P(I|A) + P(B)P(I|B) = (.4)(.03) + (.6)(.02) = .024$$ $$P(A|I) = [P(A)P(I|A)]/P(I) = [(.4)(.03)]/.024 =.5.$$