Problem: Suppose that you are in an airplane that has been hijacked by a group of terrorists, and that exactly 1% of the people on the airplane are terrorists. Luckily, there is a police officer on the plane who is 99% accurate at detecting terrorists: If you are a terrorist, then he will classify you as a terrorist 99% of the time, and if you are not a terrorist, then he will classify you as innocent 99% of the time. He classifies the first person as a terrorist. What is the probability that the first person is actually a terrorist

Solution: Now, at first you may feel that the odds of the first person being a terrorist is extremely high. After all, the police officer has a 99% accuracy. However, this is not the case at all, as we shall see below that the probability of the first person actually being a terrorist is just 50% !

Now, let’s first consider the probability of the first person being innocent and him being classified as a terrorist. The probability of the first person being innocent is 99%. However, the probability that the first person is classified as a terrorist given that he is innocent is just 1%, as the police officer has 99% accuracy. So the probability of him being innocent and classified as a terrorist is 0.99 * 0.01 = 0.0099 = 0.99%.

Now, let’s consider the probability of the first person being a terrorist and him being classified as a terrorist. The probability of the person being a terrorist is 1%. The probability that he is classified as a terrorist given that he is a terrorist is 99%, owing to the police officer’s 99% accuracy. So the probability of him being a terrorist and him being classified as a terrorist is 0.01 * 0.99 = 0.0099 = 0.99%.

So both of these cases have the same probability! When we say that the police officer classifies the first person as a terrorist, the universal set of all possible outcomes is restricted to the two cases considered above, and since both of these cases are equally likely, both of them must have the same probability. Since the probabilities must add up to 100% and both of the probabilities are equal, both of the probabilities must be 50% !!

So the probability that the person was actually a terrorist is just 50%, as stated earlier. This at first seems like a pretty counterintuitive result as despite the police officer’s superhuman accuracy, there is just a 50% chance that the person is actually a terrorist. However, as our argument illustrates, this is the truth. This problem comes from an extremely interesting topic in mathematics called as conditional probability, which I would urge you to explore further if you are a fellow math enthusiast.

Don’t forget to check my blog next week for another fascinating problem!!