The probability of a randomly selected audlt in the US being

The probability of a randomly selected audlt in the U.S. being ingected with HIV is 0.006. In tests for HIV, blood samples from 24 people are combined. What is the probability that at least 3 people have HIV?

Solution

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where     
k = number of successes in trials        
n = is the number of independent trials        
p = probability of success on each trial

P( X < 3)   = P(X=2) + P(X=1) + P(X=0)
= ( 24   2 ) * 0.06^2 * ( 1- 0.06 ) ^22 + ( 24   1 ) * 0.06^1 * ( 1- 0.06 ) ^23 +
( 24   0 ) * 0.06^0 * ( 1- 0.06 ) ^24 +     
= 0.8282

P( X > = 3 ) = 1 - P( X < 3) = 0.1718