Question

Please answer a-b

A clinical study shows the results of several people who have been analyzed to see if they have the gene that increases the risk of getting cancer. The study reveals that the probability that a person has the cancerous gene is 0.1. Experts estimate that when they find that five people who have the cancer gene is that a medicine called ABX should be applied.

a) What is the probability that 7 or more people have to be analyzed to determine that the ABX medicine should be applied?

b) On average how many people should be tested before applying the ABX medication

Answer 1

Let X be the number of people analysed before application of ABX medicine      
This is a Negative Binomial Distribution problem      
p = 0.1                    probability that a person has cancerous gene     
r = 5                        number of people with cancerous gene found (successes)     
       
X follows a Negative Binomial distribution with pdf      
X = 5, 6, 7, 8 …  
a) To find P(7 or more people will have to be analysed to determine if ABX has to be applied)      
that is to find P(X ≥ 7)      
We use Excel function NEGBINOM.DIST to find the probability      
P(X ≥ 7) = 1- P(X < 7)      
                   = 1 - P(X ≤ 6)      
                  = 1 - NEGBINOM.DIST(1, 5, 0.1, TRUE)      
                  = 0.999945      
P(7 or more people will have to be analysed to determine if ABX has to be applied) = 0.999945      
       
b) X follows a Negative Binomial distribution      
Average number of people to be analysed before applying ABX       
= Expectation of X      
= E(X)      

= 45      
Average number of people to be analysed before applying ABX = 45