A test can be very reliable and yet a positive result may not mean what it seems. The mistake is to look only at the test's accuracy and forget how many healthy people there are before testing begins.
The Test That Seems Impossible
Riddle statement
A disease affects 1% of the population.
There is a test that is correct 99% of the time:
if you are sick, it gives a positive result 99% of the time;
if you are healthy, it gives a negative result 99% of the time.
A person takes the test and gets a positive result.
Approximately, what is the probability that this person is actually sick?
Show solution
Solution
The intuitive answer is usually 99%, but the real probability is much lower.
Imagine 10,000 people.
Since the disease affects 1%, approximately:
people will be sick.
The test correctly detects 99% of them, so it will give a positive result to:
sick people.
But there will also be:
healthy people.
The test gives a false positive for 1% of healthy people, so among them there will be approximately:
false positives.
Therefore, among all people who test positive, there are:
99 true positives; 99 false positives.
In total:
positive results.
Of those, the truly sick people are:
So the probability of being sick after a positive result is:
Answer: approximately 50%.
The test is very good, but the disease is rare. That is why the false positives among healthy people can match the true positives among sick people.