Question

a test to determine whether a certain antibody is present is 99.9% effective. this means that the test will accurately come back negative if the antibody is not present (in the test - subject) 99.9% of the time. the probability of a test coming back positive when the antibody is not present (a false positive) is 0.001. suppose the test is given to seven randomly selected people who do not have the antibody. (a) what is the probability that the test comes back negative for all seven people? (b) what is the probability that the test comes back positive for at least one of the seven people? (a) p(all 7 tests are negative)= (round to four decimal places as needed )

Explanation:

Step1: Identify probability of single - negative test

The probability that a test comes back negative for a person without the antibody is $p = 0.999$.

Step2: Use multiplication rule for independent events

Since the tests are independent, for $n = 7$ people, the probability that all 7 tests are negative is $P(\text{all 7 tests are negative})=p^n$. Substituting $p = 0.999$ and $n = 7$ into the formula, we get $P=(0.999)^7$.

Step3: Calculate the value

$(0.999)^7=0.9930$.

Step4: Calculate probability of at least one positive

The probability that at least one test is positive is the complement of the event that all tests are negative. Let $A$ be the event that at least one test is positive and $B$ be the event that all tests are negative. Then $P(A)=1 - P(B)$. We know $P(B)=(0.999)^7 = 0.9930$, so $P(A)=1 - 0.9930=0.0070$.

Answer:

(a) $0.9930$
(b) $0.0070$