Today's Math Lesson - Conditional Probability

The instructor in the AI online class I am taking from Stanford was discussing conditional probability today, which is an old topic for me. One contest every year for the North Suburban Math League was only probability: simple probability for the Freshman team, geometric probability for the Sophomores, and more advanced probability for the Juniors and the Seniors. And in the lesson, the instructor does a problem that I first saw done by John Allen Paulos in 1988 in a book called Innumeracy: Mathematical Illiteracy and its Consequences.

It is still available on Amazon - although if you buy one it probably won't be signed by Dr. Paulos like mine is. What can I say - I'm a math geek. The example Dr. Paulos gives is intended to give people a more numerate understanding of drug testing - something to counteract the typical "Well, he tested positive and the test is 90% accurate, so he's probably guilty."

So, let's assume the test is 90 % accurate and let's guess that our population of drug users is 5% of the total population. That is, for any random group of 100 people, 95 are drug free and 5 are not. Let's suppose that our workplace has 1000 people. 95% of 1000 is 950, so 950 is our expected drug free population and the remaining 50 are drug users. So we give everyone the drug test.

Of the 50 drug users, 45 test positive. (Remember the test is 90% accurate - 90% of 50 is 45.) The other 5 drug users test negative.

Of the 950 people who are not drug users, 855 test negative (Again, 90% accurate and 90% of 950 is 855). That leaves 950 - 855 = 95 people in this group that test positive, even though they are drug free.

So, let's suppose you are contacted by your employer and told that you tested positive. What does that mean. Well, 140 people tested positive - the two big numbers above added together. And only 45 of those 140 people are drug users. So the probability that you are a drug user if you tested positive is 45 out of 140 or about 32%. Less than a third of the positive tests are accurate. But the company sells itself as being 90% accurate, which it is. You just have to understand that  90% accurate doesn't mean what it sounds like. Or to modify an earlier statement, he tested positive and the test is 90% accurate, so he's probably not guilty.

So what should the company do? Well, at the very least it should retest all the positive results. Should it fire everyone that gets two positive tests? The probability that you get two positive tests if you are drug-free is pretty small, but there are still going to be people fired who are not drug users but are labeled as such.

Well, one solution would be to focus on people whose performance has slipped or who are making a lot of mistakes at work. But that should really mean that they should start by seeing counselors to find out other reasons for behavior changes. Stress at home, maybe difficulties with a spouse or adolescent children, the beginning of a physical or mental illness that may need treating; in general, that would require an interest in the worker and his welfare. And that would require managers who are skilled at working with people. Nah, the test is simpler. Besides, it's 90% accurate. Let's just go with that.

And that's what Paulos meant by the consequences of mathematical illiteracy.