The probability of a randomly selected adult in one country

The probability of a randomly selected adult in one country being infected with a certain virus is 0.003. In tests for the virus, blood samples from 10 people are combined. What is the probability that the combined sample tests positive for the virus? Is it unlikely for such a combined sample to test positive? Note that the combined sample tests positive if at least one person has the virus.

Solution

The probability of a randomly selected adult in one country being infected with a certain virus is 0.003. In tests for the virus, blood samples from 10 people are combined. What is the probability that the combined sample tests positive for the virus? Is it unlikely for such a combined sample to test positive? Note that the combined sample tests positive if at least one person has the virus.

n=10

p=0.003

we have to calculate probability of at least one person has the virus.

First we calculate probability of no person has the virus

P( x=0)

P(X=x) = (_nC_x) p^x (1-p)^n-x   

P(X=0) = (₁₀C₀) 0.003⁰(1-0.003)^10-0   

P(X = 0) = 0.9704

probability of at least one person has the virus = 1-0.9704 = 0.0296

probability that the combined sample tests positive for the virus = 0.0296

It is unlikely for such a combined sample to test positive because P=0.0296 which is < 0.05