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In a clinical study, volunteers are tested for a gene that has been found to increase the risk for a disease. The probability that a person carries the gene is 0.1. All volunteers are independent in terms of carrying the gene. (a) What is the probability that four or more people need to be tested to detect two with the gene? (b) What is the expected number of people to test to detect two with the gene?

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Answer:

a) 0.972

c) 20

Explanation:

Let X be the random variable of number of people that need to be tested to detect 2 with the gene.  X is a negative binomial random variable with te parameters: p=0.1  r= 2  The probability mass function of X is given by:

f(x) = (x-1/1) x 0.9 x-2 x o.1 2, x>= 2

calculate:  

p(x>=4) = 1-P (x=3) - P (x=2) =0.972

E(X) = 2/0.1 = 20

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