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 tests are positive})=square ) (round to four decimal places as needed.)

Explanation:

Step1: Use the multiplication rule for independent events

Since the tests are independent, if the probability of a single - test being positive (when the virus is present) is \(p = 0.7997\), and we have \(n = 4\) independent tests. The multiplication rule for independent events \(A_1,A_2,\cdots,A_n\) is \(P(A_1\cap A_2\cap\cdots\cap A_n)=P(A_1)\times P(A_2)\times\cdots\times P(A_n)\)

Step2: Calculate the probability

For \(n = 4\) and \(P(\text{positive test})=0.7997\) for each test, we have \(P(\text{all 4 tests are positive})=(0.7997)^4\)

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Answer:

\(0.4090\)