It’s flu season and for the past two days you’ve had a headache and sore throat. You learn that 90% of people who actually have the flu also have those symptoms, which makes you worry. Does that mean the chances of your having the flu is 90%? In other words, if there’s a 90% chance of having a headache and sore throat given that you have the flu, does that mean there’s a 90% chance having the flu given that you have a headache and sore throat?
We can use symbols to express this question as follows: Pr(Flu | Symptoms) = Pr(Symptoms | Flu) = 90%?
The answer is no. Why?
If you think about it you’ll realize that there are other things besides the flu that can give you a combination of a headache and sore throat, such as a cold or an allergy, so that having those symptoms is certainly not the same thing as having the flu. Similarly, while fire produces smoke, the old saying that “where there’s smoke there’s fire” is wrong because it’s quite possible to produce smoke without fire.
Fortunately, there’s a nice way to account for this.