Question

an at - home covid - 19 test to detect the presence of the sars - cov - 2 virus has a sensitivity of 0.7997. this means that the test will come back positive if the virus is present 79.97% of the time. the probability of a test coming back negative when the virus is present (a false negative) is 0.2003. suppose that the at - home covid - 19 test is given to four randomly selected people who have the sars - cov - 2 virus.
(a) what is the probability that the at - home covid - 19 test comes back positive for all four people?
(b) what is the probability that the at - home covid - 19 test comes back negative for at least one of the four people?
(a) ( p(\text{all }4\text{ tests are positive}) = 0.4090 ) (round to four decimal places as needed.)
(b) ( p(\text{at least one negative})=square ) (round to four decimal places as needed.)

Explanation:

Step1: Use the complement rule

The complement of the event “at least one negative” is “all 4 tests are positive”. Let \(P(A)\) be the probability of “at least one negative” and \(P(B)\) be the probability of “all 4 tests are positive”. Then \(P(A)=1 - P(B)\).

Step2: Substitute the known value

We know from part (a) that \(P(B) = 0.4090\). So \(P(A)=1 - 0.4090\).

Answer:

\(0.5910\)