The probability of a Randomly selected adult in the US being

The probability of a Randomly selected adult in the US being infected 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

The probability of a single person being infected with HIV is 0.006.

Probability of one person having HIV is independent of another person having HIV.

24 people are tested, so the probability of at least 3 people having HIV is the complementary probability of 2 people having HIV+1 person having HIV+0 persons having HIV.

It could be written mathemetically as,

P(at least 3 people have HIV ) = 1 - [P(exactly 2 people have HIV)+ P(exactly one person has HIV)+P(no one has HIV)]

The probability of a person not infected with HIV should be . 1 - 0.006 = 0.994

P(exactly two person having HIV) = C(24,2)(0.006)²(0.994)²²

P(exactly one person having HIV) = C(24,1)(0.006)(0.994)²³

P(no one has HIV) = C(24,0)(0.994)²⁴

P(at least 3 people have HIV ) = 1 - [ C(24,2)(0.006)²(0.994)²² + C(24,1)(0.006)(0.994)²³ + C(24,0)(0.994)²⁴ ]

P(at least 3 people have HIV ) = 0.0004