# Small probabilities add, big ones don’t

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2024-05-10 14:30:07

A video has been making the rounds in which a well-known professor [1] says that if something has a 20% probability of happening in one attempt, then it has a 40% chance of happening in two attempts, a 60% chance in happening in three attempts, etc.

This is wrong, but it’s a common mistake. And one reason it’s common is that a variation on the mistake is approximately correct, which we will explain shortly.

It’s obvious the reasoning in the opening paragraph is wrong when you extend it to five, or especially six, attempts. Are you certain to succeed after five attempts? What does it even mean that you have a 120% chance of success after six attempts?!

But let’s reduce the probabilities in the opening paragraph. If there’s a 2% chance of success on your first attempt, is there a 4% chance of success in two attempts and a 6% chance of success in three attempts? Yes, approximately.

In words, the probability of A or B happening equals the probability of A happening, plus the probability of B happening, minus the probability of A and B both happening. The last term is is a correction term. Without it, you’re counting some possibilities twice.