Answer: The probability that the test comes back negative for all seven people is approximately 98.6%.
The probability that the test comes back positive for at least one of the seven people is approximately 1.4%.
Explanation:
(a) To find the probability that the test comes back negative for all seven people, we can multiply the probabilities of each person testing negative.
Given that the test is 99.8% effective, the probability of a person testing negative is 0.998. Since the test is given to seven people who do not have the antibody, we can calculate the probability as:
P(negative for all) = (0.998)^7
Calculating this probability, we get:
P(negative for all) ≈ 0.986
Therefore, the probability that the test comes back negative for all seven people is approximately 0.986.
(b) To find the probability that the test comes back positive for at least one of the seven people, we can calculate the complement of the probability that all seven people test negative.
The complement of the probability that all seven people test negative is the probability that at least one person tests positive. So, we have:
P(at least one positive) = 1 - P(negative for all)
Substituting the previously calculated value of P(negative for all), we get:
P(at least one positive) = 1 - 0.986
Calculating this probability, we find:
P(at least one positive) ≈ 0.014
Therefore, the probability that the test comes back positive for at least one of the seven people is approximately 0.014.
I hope this helps :)