Help with a Poisson distribution question

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trickslapper

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Sep 17, 2010
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I'll just post the exact question then give my thoughts and the answer:

In a certain population, the number of colds a person gets in a year has a P(3) distribution. A new anti-cold drug lowers lambda from 3 to .75 and is effective for 8/10 people. At the beginning of last year, the entire population was given the drug. At the end of last year, one person was selected at random from the population and was found to have had only one cold during the year. What is the probability that the drug was effective for this person?

The answer from the back of the book is: .905.

-Is the questions telling me that lambda is .75? In other words can i just ignore that first sentence?
-How does the 8/10 people come into play?
-And how do you this problem lol?

thanks!

Edit: i figured it out, end up using bayes formula... I really hate this class my professor doesn't speak good english so i'm always having to play catch up with this class
 
Walk your way through it. Picking a finite population may help you get a grip on it. Try N = 1000 people.

Folks for whom the remedy works = 1000 * 0.80 = 800
Folks for whom the remedy does not = 1000 * 0.20 = 200

When the remedy works, lambda = 0.75
When the remedy fails, lambda = 3.00

For the 800, P(1 cold) = 0.75*exp(-0.75) = 0.354, thus 800*0.354 = 283.2 of the original 1000
For the 200, P(1 cold) = 3.00*exp(-3.00) = 0.149, thus 200*0.149 = 29.8 of the original 1000

We are ready to answer the question. 283.2/(283.2+29.8) = 283.2/313 = 0.905

Now, think through it again without assuming a finite population. You may wish to set up a two-way contingency table. This will also help you get over the fact that we cut people up. You should have been initially concerned about the concept of 283.2 people. Normally, we don't think of counting people that way. This is just a theoretical discussion. They won't mind being subdivided a little! We could have picked 10,000 and avoided the problem. It's an arbitrary choice and is of no consequence.

Are we getting anywhere?
 
Yes, Bayes formula is important, here. That is what I had in mind when I said, "think through it again without assuming a finite population."

Good work!
 
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