Question

a test to determine whether a certain antibody is present is 99.4% effective. this means that the test will accurately come back negative if the antibody is not present (a false positive) is 0.006. suppose the test is given to six randomly selected people who do not have the antibody.
(a) what is the probability that the test comes back negative for all six people?
(b) what is the probability that the test comes back positive for at least one of the six people?
(a) p(all 6 tests are negative) = .9762 (round to four decimal places as needed.)

Explanation:

Step1: Determine the probability of a single test being negative for a person without the antibody

The probability of a test coming back negative when the antibody is not present is \(1 - 0.006=0.994\)

Step2: Use the multiplication rule for independent events

Since the tests are independent, for \(n = 6\) people who do not have the antibody, the probability that all \(6\) tests are negative is \(P(\text{all 6 tests are negative})=(0.994)^{6}\)

Calculate \((0.994)^{6}=0.994\times0.994\times0.994\times0.994\times0.994\times0.994\)

\(0.994^{2}=0.988036\), \(0.988036\times0.994 = 0.982107784\), \(0.982107784\times0.994=0.976215137\)

Answer:

\(0.9762\)