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Suppose that the number of times that Vampire Bill gets attacked by Hep V in a given year is a Poisson random variable with parameter λ = 5. Now, he discovered a new drug (based on large quantities of vitamin C) that has just been marketed to protect from Hep V. It claims to reduce the Poisson parameter to λ = 3 for 75% of the population. For the rest of the population, the drug has no appreciable effect on Hep V. If Bill tries the drug for a year and only had 2 Hep V attacks in that time, how likely is it that the drug was beneficial for him?

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Manetho

Answer:

Explained

Explanation:

let probability of getting attached by Hep(V) without benefit of new drug =P(A)

probability of getting attached by Hep(V) with new drug =P(B)

hence probability that Hep V attack are 2 given he is without benefit of drug =P(C|A) =[tex]\small e^{-\lambda }\lambda ^{x}/x! where \small \lambda=5[/tex] ,x=2

=0.084

and probability that Hep V attack are 2 given he is with new drug =P(C|B) [tex]=\small e^{-\lambda }\lambda ^{x}/x! where \small \lambda=3[/tex] ,x=2

=0.224

hence probability of Hep V attack are 2 =P(C) =P(A)*P(C|A) +P(B)*P(C|B) =0.25*0.084+0.224*0.75=0.189

from above probability that drug is beneficial given he had 2 Hep V attacks =P(B)*P(C|B)/P(C) =0.224*0.75/0.189 =0.8889

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