Fooled by randomness

I just finished reading Nassim Taleb's excellent book Fooled by Randomness which combines insights from finance, behavioral economics, probability and statistics. I will write more about this later, but here’s an interesting example from the book on how we lack intuition for basic probability calculations:

You are a doctor. A certain disease tends to afflict one in every thousand individuals in the population. There is a test for the disease, but in 5% of the cases it yields a false positive. A person comes in one day and tests positive. What are the odds he has the disease? Stop for a moment and try to answer the question.

The naïve answer (and the one that most physicians gave when posed this problem) is that he has the disease with 95% certainty. The correct answer is about 2% (notice the wide discrepancy here – the naïve estimate misses the mark by a HUGE amount). Why? Because the test has a markedly higher error rate than the base rate of occurrence of the disease in the population. Consider that in a population of 1000, 1 person on average will have the disease, but the test will generate on average 50 false positives. Therefore the probability that a given individual who tests positive has the disease is 1/51, which is about 2%.

Why does almost everyone venture an answer of 95% as the odds of the subject having the disease? My theory is that people confuse the probability of event A conditional on event B (p(A|B)) with the probability of event B conditional on event A (p(B|A)): the probability of testing positive given that one does not have the disease is 5% whereas the probability of not having the disease given that one tests positive is 98% and the confusion of these probabilities leads to the wrong answer.

This is certainly interesting – no one would expect a physician to intuit the subtlety of the problem, much less pull out a scratchpad and compute the right answer. Now change the framing of the problem. You are a juror. You know that one in every thousand people has committed a criminal act in their lifetime. A crime has been committed and a fingerprint obtained from the crime scene. The fingerprint was correlated against a random collection of fingerprints drawn from people in the city and a positive match was obtained with one individual. The fingerprint matching program has a false positive rate of 5%. What are the odds the individual committed the crime?

This kind of problem can crop up in a number of different real-world decision-making scenarios and our lack of intuition for probability can lead to drastically bad calls. Physicians presumably have some basic grounding in probability and statistics, but it is probably safe to assume most jurors are mathematically illiterate. Makes you wonder how many false positives are sitting out prison sentences.

The problem caught me out the first time it was presented to me, as it did a mathematician friend of mine. I observed a tendency on my part to jump to an answer (the wrong one, in this case). It was only when I discovered how drastically wrong I was, that I sat down and did the calculation. If the problem had been presented to me on a math exam, it would probably have engaged the analytic part of my brain and I would have patiently worked it out and arrived at the right answer. When it jumped out at me out of a context where I wasn’t expecting a tricky calculation, I jumped to an easy, ill-considered and completely wrong answer. This is probably what is referred to as engaging System 1 versus System 2.