A test to determine whether a certain antibody is present is 99.6 % effective. This means that the test will accurately come back negative if the antibody is not present (in the test subject) 99.6 % of the time. The probability of a test coming back positive when the antibody is not present (a false positive) is 0.004. Suppose the test is given to four randomly selected people who do not have the antibody. (a) What is the probability that the test comes back negative for all four people? (b) What is the probability that the test comes back positive for at least one of the four people?

Part (a) The probability that the test comes back negative for all four people is: 0.98401 (approximately)

Part (b) The probability that the test comes back positive for at least one of the four people is: 0.01599 (approximately)

Step-by-step explanation:

Consider the provided information.

The test will accurately come back negative if the antibody is not present (in the test subject) 99.6 % of the time.

99.6% can be written as 0.996 which is the probability of test is negative.

The probability of a test coming back positive when the antibody is not present (a false positive) is 0.004.

Part (a)

The probability that the test comes back negative for all four people is:

P (all 4 tests are negative) = (0.996) (0.996) (0.996) (0.996)

P (all 4 tests are negative) = 0.98401 (approximately)

Part (b)

The probability that the test comes back positive for at least one of the four people:

At least one means it may be 1 or 2 or 3 or all four gets a positive test. In other word it can be say that all 4 tests are negative.

As we know the probability of all tests are negative is 0.98401. So subtract this number from 1.

P (at least one of 4 people) = 1-0.98401

P (at least one of 4 people) = 0.01599 (approximately)