# The Borel-Cantelli Theorem

Let $\{A_j\}_{j=1}^\infty$ be an infinite sequence of events.

If the $A_j$ are mutually independent events and $\sum_{j=1}^\infty P(A_j)$ diverges, then $P(\{ A_n {i.o.} \} ) = 1$. Where $\{ A_n{i.o} \}$ reads “$A_n$ infinitely often”.

So, what does it mean? Well, we can think of it like this:

Suppose you have your brand new email account, and you are thinking about a really safe password, one that is “unbreakable”. Well, the above theorem says that no matter how super long and complicated you make your password, eventually it can be “broken”, so it is rather a question of asking “How long will it take for someone to break my password?” instead.

Another way to think about it:

You have a coin, and you are going to start flipping that coin until you get Tails, Tails, Head, in that order, consecutively, three times. You will eventually get it, it’s just a matter of when, not if.

There are many other examples that we can come up with, but I suggest that you come up with some of them. See you next time!