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Math Help - Poisson distribution, seems easy enough but I can't figure out what I'm doing wrong

  1. #1
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    Question Poisson distribution, seems easy enough but I can't figure out what I'm doing wrong

    Ok heres the question verbatim
    An individual suffers from aparticular disease which leads to him having attacks from time to time. The number of attacks per month can be assumed to follow a poisson distribution with mean 2. He is included in a trial for a new treatment. In the trial there is a 50% probability that he will recieve a placebo which will not affect the rate of attacks and a 50% probability that he will recieve the new treatment which should reduce the mean number of attacks per month to one. In the first month of the trial, he has no attacks. Calculate the probability that he has received the new treatment.


    So first of all I found the probability that X=0 given that he took the placebo (p)

    Pr(X=0|p) = Poisson(0,2) = e^-2

    and the probability that X=0 given that he was given the new treatment (n)

    Pr(X=0|n) = Poisson(0,1) = e^-1

    Then using the Law of total probability and Pr(p)=Pr(n)=0.5

    Pr(X=0) = Pr(X=0|n)Pr(n) + Pr(X=0|p)Pr(p) = (e^-1)/2 + (e^-2)/2

    Now using Bayes' Theorem

    Pr(n|X=0) = Pr(n)/Pr(X=0) = (0.5)/(0.251607362) = 1.98722325

    Which obviously isn't possible, and I can't think where I've gone wrong. Please help quickly because this is driving me insane ; ;.
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    P(n | X=0) = P(n & X=0) / P(X=0)
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