Solution to the Drug Test

thescienceofdeductionandmore:

To remind you, here’s the problem:

Suppose that a test for using a particular drug is 99% sensitive and 99% specific. That is, the test will produce 99% true positive results for drug users and 99% true negative results for non-drug users. Suppose that 0.5% of people are users of the drug. What is the probability that a randomly selected individual with a positive test is a drug user?

 Solution:

One not familiar with Baye’s Theorem might think the probability is 99%, but they would be wrong.  The probability the person is a drug user is around 1/3.  The reason the probability is so low is because of all of the “false positives”.

If you don’t want to use the formula in Baye’s theorem to calculate the exact probability, here’s a quick way to approximate and see why the probability is so low.

Let’s assume there is a population of 200 random people being given the test. Since .5% of people are drug users, let’s assume 1 of the 200 people is a drug user.  Now let’s see how accurate this test is in identifying the 1 drug user.

The drug user will take the test, and since the test is 99% accurate, she will almost certainly test positive.  So let’s assume she tests positive.  The 1% chance she will test negative doesn’t materially change our analysis.

Now let’s test the other 199 people, none of whom is a drug user. If the test were 100% accurate, none of them will test positive.  But it is only 99% accurate.  That means 1% of them will receive wrong test results and will test positive.  1% of 199 is 1.99, which is around 2.

So if we test 200 people, a total of around 3 will test positive. 1 of them will be a true positive, a drug user.  2 of them will be a false positive, a non-drug user.  So if a person tests positive, the probability they are actually a drug user is around 1/3. 

September 01, 2018 at 11:22PM