A study considered risk factors for HIV infection among IV d

A study considered risk factors for HIV infection among IV drug users. It found that 40% of users who had <= 100 injections (light users) per month and 55% of users who had >100 injections (heavy users) per month were HIV positive.

1. Suppose we have a group of 10 light users and 10 heavy users. What is the probabiity that exactly 3 of the 20 users are HIV positive?

2. What is the probability that at least 4 of the 20 users are HIV positive?

3. Is the distribution of the number of HIV positive among the 20 users binomial? Why or why not?

Solution

P(a light user is HIV positive ) = 0.4

P(a heavy user is HIV positive) = 0.55

P(3 out of 20 users are positive) = P(3 light users are HIV positive) +P(2light users and 1 heavy user is HIV positive)+ P( 1 light user and two heavy user are HIV positive) +P(3 heavy users are HIV positive)

P(3 out of 20 users are positive) = C(10,3)(0.4)³*(0.6)⁷(0.45)¹⁰ +C(10,2)(0.4)²*(0.6)⁸C(10,1)(0.45)⁹(0.55)+ C(10,2)(0.4)¹(0.6)⁹C(10,1)(0.45)⁸(0.55)²+C(10,3)(0.6)¹⁰(0.55)³(0.45)⁷

P(3 out of 20 users are positive) = 0.002

2.

P(4 out of 20 users are HIV positive) = P(4 light users are HIV positive)+P(3light users and 1 heavy user are HIV positive)+P(2 light user and 2 heavy user are HIV positive)+P(1 light user and 3 heavy user are HIV positive)+P(4 heavy users are HIV positive)

P(4 out of 20 users are HIV positive) = Bin(10,4,0.4)Bin(10,0,0.55) + Bin(10,3,0.4)Bin(10,1,0.55) + Bin(10,2,0.4)Bin(10,2,0.55) + Bin(10,1,0.4)Bin(10,3,0.55)+Bin(10,0,0.4)Bin(10,4,0.55)

P(4 out of 20 users are HIV positive) = 0.008

3. The distribution is not binomial , but it is a combination of binomial.