The number of colds per year of an individual can be described by the Poisson di
ID: 2922171 • Letter: T
Question
The number of colds per year of an individual can be described by the
Poisson distribution. Suppose that a new drug reduces the average number of colds per
year from 3 to 1 and works for 75 percent of the population. A woman decides to take the
drug during two years, starting on January 1st 2008. Find
1) the probability that she will have exactly one cold during the two years.
2) the probability that the drug is working for her given that she has at most 3 colds
during the two years.
3) the probability that she will have exactly two colds in year 2010 given that she had
at most 3 colds from January 1st 2008 until December 31st 2009.
Explanation / Answer
The new drug reduces the average number of colds per year from 3 to 1 for 75 % percent of population.
so now we can see here that there are two poisson distribution is working on.
FIrst for 75% population, where average number colds per year is 1. 1=1
Second for 25% population , where average number of colds per year is 3. 2=3
Q.1 the probability that she will have exactly one cold during the two years.
so if drug affects positively, average number of cold in two year = 21=2
if drug is ineffecitve, average number of cold in two year = 22= 6
so , Pr(X = 1) = Pr(Drug effective) * Pr(X = 1) + Pr( Drug ineffective) * Pr(X = 1)
= 0.75 * POISSON (X =1; 2) + 0.25 * POISSON(X = 1; 6)
= 0.75 * 0.271 + 0.25 * 0.0149 = 0.207
(2) the probability that the drug is working for her given that she has at most 3 colds during the two years.
Pr(drug is working l She has at most 3 colds in two years)
Pr(She has at most 3 colds in two years) = Pr(Drug working) * Pr( Colds <=3) + Pr(Drug not working) * Pr(Colds <=3)
= 0.75 * POISSON (Colds <=3 ; 2) + 0.25 * Poisson (Colds <=3 ;6)
= 0.75 * 0.857 + 0.25 * 0.1512
= 0.68
Pr(drug is working l She has at most 3 colds in two years) =
Pr(Drug working) * Pr( Colds <=3) / [Pr(drug is working l She has at most 3 colds in two years)]
0.75 * 0.857 / (0.75 * 0.857 + 0.25 * 0.1512 ) = 0.945
(3) the probability that she will have exactly two colds in year 2010 given that she had at most 3 colds from January 1st 2008 until December 31st 2009.
Answer : Pr(Colds = 2 l at most 3 colds in 2 years in the effect of drug)
as these two events are independent to each other probability of having exactly two colds in year 2010 is independent of colds occurred in last 2 years in the effect of drug.
Pr(Colds = 2 l at most 3 colds in 2 years in the effect of drug) = Pr(Colds = 2 in 2010)
= POISSON (Colds = 2; 6) = 0.0446