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February 24th, 2013, 09:19 AM   #1
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Conditional Probability

A hospital receives 2/5 of its ’flu vaccine from company A and the remainder from company B. Each shipment contains a large number of vials of vaccine. From A, 3% of the vials are ineffective and from B, 2% are ineffective. A hospital tests n=25 randomly from selected vials from one shipment and finds that 2 are ineffective. What is the conditional probability that this shipment came from company A? [Ans: 0.54229]

{The answer I get is 0.5 exactly assuming that the testing of n=25 and 2 of out that is ineffective is just useless additional info. It's probably not useless, what do I do with the part where n=25 and 2 out that is ineffective?}
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February 25th, 2013, 09:35 AM   #2
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Re: Conditional Probability

This is a wrong example. The questioner doesn't understand what conditional probability means, maybe he should visit this hospital. Let's see what we need for a conditional probability. First of all we need the set of the elementary events. If we have a die the set of the elementary events obvious: {1,2,3,4,5,6}. Secondly, we need the probability of each elementary events. By a die we assume the probability is equable, i.e. P([we get 4]) = 1/6. We can have also a loaded die with probabilities 10%, 10%, 10%, 10%, 10%, 50% respective to the set of elementary events. If we have those then we can talk about probability of (non-elementary) events. An event = a subset of the set of elementary events. For example {2,4,6} is a subset, so it is an event. We can easily calculate the probability of this event. At a normal dice it's 50%, at a loaded die it's 70%. If we want to talk about conditional probability first of all we need two events, i.e. two subsets. And these two generate a third, the mutual events, i.e. the section of the two subsets. For example what is the probability of getting x>3 if we know x is odd. One subset is {4,5,6}, the other subset is {1,3,5}, the mutual subset is {5}, the answer is 10% in both cases.

Back to hospital let's ask ourselves what is the set of the elementary events? Well, it is a 2-member set: {"the shipment came from A", "the shipment came from B"}. What are the regarding probabilities? It has been said: {2/5,3/5}. Now focus on the final question, it is talking about a conditional probability. But what is the conditional subset?? Undefined. There is a condition, but that doesn't refer to this set of elementary events. It does refer to another set of events, which is irrelevant, it refers out from the frame of the question. It is like we asked what is the probability of odd number if we know there are more plastic die than bone. So the strict answer is 2/5. The example states there is a chance of 2/5 that the shipment comes from A and it asks what is the chance that the shipment comes from A if an irrelevant fact is true. 2/5.
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February 25th, 2013, 07:08 PM   #3
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Re: Conditional Probability

Originally Posted by csak
This is a wrong example.
True, so perhaps you could suggest how it should have been worded (given how it was intended to be answered).
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