Question

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.

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

Answer 1

Here we have:

P(light user is HIV positive ) = 0.4

P(heavy user is HIV positive) = 0.55

Now

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

= 0.008

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