Question

1. A person takes a test to detect the occurence of a disease. The test’s characteristics are such that a person testing positive has actually a 70% chance actually having contracted the disease. Meaning that a person not having contracted the disease has a 30% chance of testing positive to it.

   The test’s results are used to allow a person to exit a quarantine – because having the disease makes them immune to it and they are no longer in risk of being infected. But as we saw, 30% of those presumed to have had the disease, have not actually gone through it. If they allowed into the general population, they risk being infected.

   What would you do to ensure that a smaller number of those “false positives”, i.e., people who test positive without having contracted the disease, are allowed to exit the quarantine?

Answer 1

One way is to repeat the test twice or thrice to reduce the number of “false positives”.

We will treat the person testing positive if it has positive results in all the tests.

For two tests,  probability of false positives = P(Test positive | Does not have the disease) * P(Test positive | Does not have the disease) = 0.3 * 0.3 = 0.09

Thus, if two tests are conducted, the probability of false positives is reduced to 0.09

For three tests,  probability of false positives = P(Test positive | Does not have the disease) * P(Test positive | Does not have the disease) * P(Test positive | Does not have the disease) = 0.3 * 0.3 * 0.3 = 0.027

Thus, if three tests are conducted, the probability of false positives is reduced to 0.027