1. ## Help with a Probability question please?

A hospital receives 2/3 of its flu vaccine shipments from Company A and the remainder from Company B. Each shipment contains a very large number of vaccine vials. For Company A’s shipments, 10% of the vials are ineffective, while for Company B’s shipments, 2% of the vials are ineffective.

(a) In a sample of 30 randomly selected vials from Company A, what is the expected number of ineffective vials? Motivate your answer fully.
(b) The hospital tests 30 randomly selected vials from a shipment and finds that exactly one vial is ineffective. What is the probability that this shipment came from Company B?

After checking all the conditions, this is what I've come up with for part (a):

p(a)=P{x=a} = (30Ca)(0.10)^a(0.90)^(30-a) if a = 0,1,2,...,30
0 otherwise

I'd really appreciate any help you guys could offer. I'm a little confused here.

2. Originally Posted by tuheetuhee
A hospital receives 2/3 of its flu vaccine shipments from Company A and the remainder from Company B. Each shipment contains a very large number of vaccine vials. For Company A’s shipments, 10% of the vials are ineffective, while for Company B’s shipments, 2% of the vials are ineffective.

(a) In a sample of 30 randomly selected vials from Company A, what is the expected number of ineffective vials? Motivate your answer fully.
(b) The hospital tests 30 randomly selected vials from a shipment and finds that exactly one vial is ineffective. What is the probability that this shipment came from Company B?

After checking all the conditions, this is what I've come up with for part (a):

p(a)=P{x=a} = (30Ca)(0.10)^a(0.90)^(30-a) if a = 0,1,2,...,30
0 otherwise

I'd really appreciate any help you guys could offer. I'm a little confused here.
(a) Although you're sampling without replacement, the very large number of vials in the population clearly suggests you can approximate things by a binomial distribution:

Let X be the random variable number of ineffective vials in a sample of 30 from Company A.

X~Binomial(p = 0.1, n = 30).

E(X) = ......

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(b) You require the concept of conditional probability:

$\Pr(\text{Company B} \, | \, \text{1 vial is ineffective}) = \frac{\Pr( \text{Company B and 1 vial is ineffective})}{\Pr(\text{1 vial is ineffective})}$

Let Y be the random variable number of ineffective vials in a sample of 30 from Company B.

Y~Binomial(p = 0.02, n = 30).

Then Pr(Company B and 1 vial is ineffective) = Pr(1 vial is ineffective | company B) x Pr(Company B) = Pr(Y = 1) x (1/2) = Pr(Y = 1)/2.

Pr(1 vial is ineffective) = Pr(Company B and 1 vial is ineffective) + Pr(Company A and 1 vial is ineffective) = Pr(Y = 1)/2 + Pr(X = 1)/2.

Therefore $\Pr(\text{Company B} \, | \, \text{1 vial is ineffective}) = \frac{\Pr(Y = 1)}{\Pr(Y = 1) + \Pr(X = 1)} = \, ......$