The number of times that a person contracts a cold in a given year is a Poisson

Question

The number of times that a person contracts a cold in a given year is a Poisson random variable with parameter lambda = 5. Suppose that a new wonder drug (based on large quantities of vitamin C) has just been marketed that reduces the Poisson parameter to lambda = 3 for 75% of the population. For the other 25 percent of the population the drug has no appreciable effect on colds. If an individual tries the drug for a year and has 2 colds in that time, how likely is it that the drug is beneficial for him or her?

Explanation / Answer

let B be the event that the drug is benificial

P(B) =0.75

P(B^)=1-0.75=0.25

Let X represent the number of colds on a year

we have poission distribution formulea is P(x; ) = (e-) ( x) / x!

P(x=2 | B) = 3^2 / 2! (e^-3) = 9/2 (e^-3)

P(X=2 |B^) = 5^2 / 2! (e^-5) = 25/ 9 (e^-5)

now final answer is given by

P(B| X=2) =        P(x=2 | B) *P(B) / [ ( P(x=2 | B) *P(B) + P(X=2 |B^) *P(B^)]

               =     9/2 (e^-3) * 0.75 / [ 9/2 (e^-3) * 0.75+ 25/ 9 (e^-5) * 0.25]

             = 0.88864

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