Question

A test to determine whether a certain antibody is present is 99.2​% effective. This means that the test will accurately come back negative if the antibody is not present​ (in the test​ subject). The probability of a test coming back positive when the antibody is not present​ (a false​ positive) is 0.008. 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?

Answer 1

a) 95.29% probability that the test comes back negative for all six ​people

b) 4.71% probability that the test comes back positive for at least one of the six ​people

Explanation:

For each person who do not have the antibody and are given the test, there are only two possible outcomes. Either the test comes back negative, or it comes back positive. The probability of the test coming back negative for a person is independent of any other person. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

99.2​% effective

So
`p = 0.992`

six randomly selected people

This means that
`n = 6`

​(a) What is the probability that the test comes back negative for all six ​people?

This is P(X = 6).

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 6) = C_(6,6).(0.992)^(6).(0.008)^(0) = 0.9529`

95.29% 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?

Either it comes negative for all six people, or it comes positive for at least one of them. The sum of the probabilities of these events is 100%. So

p + 95.29 = 100

p = 4.71

4.71% probability that the test comes back positive for at least one of the six ​people