Probability Prob: A hospital receives 1/5 of its flu vaccine shipments from Company X

A hospital receives 1/5 of its flu vaccine shipments from Company X and the remainder of its shipments from other companies. Each shipment contains a very large number of vaccine vials. For Company Xs shipments, 10% of the vials are ineffective. For every other company, 2% of the vials are ineffective. The hospital test 30 randomly chosen vials from a shipment and finds that one vial is ineffective. What is the probability that this shipment came from Company X?


I was hoping that someone can check my work

WOLOG we can assume that the other companies is just company Y

Keeping with the 1:4 ratio we can assume that 20 vials came from X and 80 came from Y

Then 2 ineffective vials are from X and 1.6 came from Y

Then P(an ineffective vial came from X) = 2/3.6 = 10/18 = 5/9

Then P(that this shipment came from Company X) = ₃₀C₁(5/9)¹(4/9)²⁹

Thank you.

Jomo said:

I am having trouble posting the question. I will keep trying. Thanks.

A hospital receives 1/5 of its flu vaccine shipments from Company X and the remainder of its shipments from other companies. Each shipment contains a very large number of vaccine vials. For Company Xs shipments, 10% of the vials are ineffective. For every other company, 2% of the vials are ineffective

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It would help if you would state what question you are trying to answer!
I presume that it is "what is the probability a vial came from company X given that it is ineffective?"

I was hoping that someone can check my work

WOLOG we can assume that the other companies is just company Y

Keeping with the 1:4 ratio we can assume that 20 vials came from X and 80 came from Y

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Where did you get "20" and "80"? Were you given that there are 100 vials or are you assuming 100 vials for simplicity?
You should say that.

Then 2 ineffective vials are from X and 1.6 came from Y

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Personally, I would have used 1000 vials, 200 from company X and 800 from other companies just because I don't like decimal fractions! With 1000 vials, there are 20 ineffective vials from company X and 16 from other companies.

Then P(an ineffective vial came from X) = 2/3.6 = 10/18 = 5/9

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Equivalently, out of my 36 ineffective vials, 20 were from company X so the probability a vial came from company X given that it was ineffective is 20/36= 5/9 just as you got.

Then P(that this shipment came from Company X) = ₃₀C₁(5/9)¹(4/9)²⁹

Thank you.

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Then I do not know what question you are trying to answer! Is it "Given that one vial out of 30 was ineffective, what was the probability the shipment came from company X?"

HallsofIvy said:

It would help if you would state what question you are trying to answer!
I presume that it is "what is the probability a vial came from company X given that it is ineffective?"


Where did you get "20" and "80"? Were you given that there are 100 vials or are you assuming 100 vials for simplicity?
You should say that.


Personally, I would have used 1000 vials, 200 from company X and 800 from other companies just because I don't like decimal fractions! With 1000 vials, there are 20 ineffective vials from company X and 16 from other companies.


Equivalently, out of my 36 ineffective vials, 20 were from company X so the probability a vial came from company X given that it was ineffective is 20/36= 5/9 just as you got.


Then I do not know what question you are trying to answer! Is it "Given that one vial out of 30 was ineffective, what was the probability the shipment came from company X?"

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As I wrote at the beginning I am having trouble posting the problem. It is normal text that will not post. I even just tried posting the problem here in hopes that you would be kind enough to post it for me but it would not go through.Your last line is correct.

HallsofIvy said:

It would help if you would state what question you are trying to answer!
I presume that it is "what is the probability a vial came from company X given that it is ineffective?

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Yes, that is the question.

Where did you get "20" and "80"? Were you given that there are 100 vials or are you assuming 100 vials for simplicity?
You should say that.

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I feel I did say that we can assume 20 and 80 by writing that we can assume that 20 vials came from X and 80 came from Y

Personally, I would have used 1000 vials, 200 from company X and 800 from other companies just because I don't like decimal fractions! With 1000 vials, there are 20 ineffective vials from company X and 16 from other companies.

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I respect that position

Then I do not know what question you are trying to answer! Is it "Given that one vial out of 30 was ineffective, what was the probability the shipment came from company X?"

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Yes. Please understand that I just needed (and requested via the 1st line of my post) some more time to complete the post. In the end I found that at least in this post I could not write the word s e l e c t e d. Strange!!

Can someone please verify my work? After all this problem I had with getting the question posted I just want to bump this post to the top. Thanks.

Jomo said:

Can someone please verify my work? After all this problem I had with getting the question posted I just want to bump this post to the top. Thanks.

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I agree with Prof. Halls, I do not understand the numbers involved. But his advise, "Given that one vial out of 30 was ineffective, what was the probability the shipment came from company X?" I will setup for you. \(\displaystyle V\) means the vial is bad.
\(\displaystyle \begin{align*}\mathcal{P}(X|V) &=\dfrac{\mathcal{P}(X\cap V)}{\mathcal{P}(V)}\\ &=\dfrac{\mathcal{P}(V|X)\mathcal{P}(X)}{\mathcal{P}(V)}\\ &=\dfrac{\mathcal{P}(V|X)\mathcal{P}(X)}{\mathcal{P}(V\cap X)+\mathcal{P}(V\cap Y)}\\ &=\dfrac{\mathcal{P}(V|X)\mathcal{P}(X)}{\mathcal{P}(V|X)\mathcal{P}(X)+\mathcal{P}(V|Y)\mathcal{P}(Y)} \end{align*}\)

pka said:

I agree with Prof. Halls, I do not understand the numbers involved. But his advise, "Given that one vial out of 30 was ineffective, what was the probability the shipment came from company X?" I will setup for you. \(\displaystyle V\) means the vial is bad.
\(\displaystyle \begin{align*}\mathcal{P}(X|V) &=\dfrac{\mathcal{P}(X\cap V)}{\mathcal{P}(V)}\\ &=\dfrac{\mathcal{P}(V|X)\mathcal{P}(X)}{\mathcal{P}(V)}\\ &=\dfrac{\mathcal{P}(V|X)\mathcal{P}(X)}{\mathcal{P}(V\cap X)+\mathcal{P}(V\cap Y)}\\ &=\dfrac{\mathcal{P}(V|X)\mathcal{P}(X)}{\mathcal{P}(V|X)\mathcal{P}(X)+\mathcal{P}(V|Y)\mathcal{P}(Y)} \end{align*}\)

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pka, Thanks! I see it clearly now! You are always so elegant in your responses.