Question

MEDICAL TESTS. Medical tests are used to indicate whether a person has a particular disease. The sensitivity of a test is the probability that a person having the disease will test positive (indicating the person has the disease, i.e., it’s the probability the test will pick up the fact that a person has the disease). The specificity of a test is the probability that a person not having the disease will test negative. A test for a certain disease has been in use for many years. Based on the "track record" of this test, its sensitivity is 0.934 and its specificity is 0.968. Finally, in the population, 1 in 500 people has the disease. What is the probability that a person who tests positive actually has the disease?

Answer 1

The probability that a person who tests positive actually has the disease is 5.52%.

Explanation:

One out of 500 people have the disease. That means 0.2% of the population (0.002).

Of this proportion, according to the sensitivity of the test, 93,4% (0.934) of them will test positive (true positives).

Also, the rest of the population, that represents 99.8% (0.998), don't have the disease, but, according to the specificity of the test, (1-0.968)=0.032 will test positive (false positives).

We have:

True positives tests = 0.002 * 0.934 = 0.001868, and

False positives tests = 0.998 * 0.032 = 0.031936

We can calculate the probability that a person who tests positive actually have the the disease as the division between the true positives and the total amount of positives:

P = True positives / (True positives + False positives)

P = 0.001868 / (0.001868+0.031936) = 0.055259733 = 5.52%