Suppose that you get a strange rash on your arm and you are quite concerned that you might have some rare disease – one that occurs randomly in the general population in only 1 in every 10,000 people. You decide to get tested, and the test comes back positive. To your dismay, the doctor tells you that the test is very accurate, being correct 99% of the time, regardless of whether the result is positive or negative.
What are your chances that you actually have the disease? Is it approximately:
Thankfully, and perhaps counter-intuitively, the answer is #4 – you have less than 1 percent chance that you have the disease.
You can figure this out using something called Bayes’ theorem, which helps us determine the probability of event A given event B, written P(A|B), in terms of the probability of B given A, written P(B|A), and the probabilities of A and B:
P(A|B)=P(A)P(B|A) / P(B)
In our medical test case, “event A” is the event you have this disease, and “event B” is the event that you test positive. Therefore, we would say that P(B|A)=.99, P(A)=.0001, and P(B) can be found by conditioning on whether event A does or does not occur:
P(B)=P(B|A)P(A)+P(B|not A)P(not A)
or .99*.0001+.01*.9999, which is less than 1 percent.
The explanation behind this counter-intuitive result is that the disease is so rare that there are a very large number of false positives. If you think about what will happen with a population of a million people, then 100 people will have the disease (1 in 10,000), and 99 of those 100 will be correctly diagnosed (99% accuracy). 999,900 people will NOT have the disease, but out of those 9999 (1%) will get a false positive. So, the chance that you actually have the disease is about 99/(99+9999), which gives you 0.0098, or about 1%.
This of course wouldn’t be the case if the disease wasn’t randomly distributed throughout the population – for example, in case of inherited traits or environmental factors.