Question

In a large population of people, some individuals are infected with a rare disease and some are not. Let POS represent the event that an individual is infected, and let NEG represent the event that an individual is not infected.  

A blood test can be administered to try to find out whether an individual is infected. Let represent the event that the test result is positive [i.e., the test indicates that the individual is infected] and let represent the event that the test result is negative [i.e., the test indicates that the individual is not infected].

The blood test is not perfectly accurate. It gives incorrect results with the following probabilities:

If a person really is infected, the test result is positive 99.9 percent of the time, and the test result is negative 0.1 percent of the time.

If a person really is not infected, the test result is negative 99 percent of the time, and the test result is positive 1 percent of the time.  

Suppose that 0.6 percent of all individuals in the population are infected with the disease (i.e., if one member of the population is chosen randomly, there is a probability of 0.006 that s/he will be infected),.

a) If one member of the population is selected randomly to take the blood test, what is the probability that the result will be positive?

b) Suppose one member of the population is selected randomly to take the blood test and that the test result is positive. Given this test result, what is the probability that the individual tested really is infected with the disease?

c) Does your answer to (b) raise any questions in your mind about whether testing randomly chosen individuals for rare diseases would produce useful information?

d) Suppose that one member of the population is selected randomly, and is given the blood test twice. Let represent the event that the result of the first test is positive, and let represent the event that the first test is negative. Similarly, let represent the event that the result of the second test is positive, and let represent the event that the second test is negative.

If an infected person is given this test twice:

-- the first test result is positive 99.9 percent of the time and negative 0.1 percent of the time

-- the second test result is positive 99.9 percent of the time and negative 0.1 percent of the time, regardless of the outcome of the first test

If a person who is not infected is given this test twice:

-- the first test result is negative 99 percent of the time and positive 1 percent of the time

-- the second test result is negative 99 percent of the time and positive 1 percent of the time, regardless of the outcome of the first test

[Something like this might be the case if errors in the results of the test were due to random mistakes made in the labs that do the testing, rather than an inherent characteristic of the individual being tested.]

Suppose a randomly selected member of the population has been given the test twice, and that both results were positive. Given these test results, what is the probability that the individual really is infected?

e) In the two-test scenario described in part (d) of this problem, are the events and independent? Your answer should be based on the definition of statistical independence, not just a loose verbal argument.

Answer 1