Kolmogorov's zero–one law

Imagine you are playing a game with a coin. You might think that if you flip the coin a lot of times, you would eventually get an even number of heads and tails. But Kolmogorov's zero–one law teaches us that this might not be true at all!

In fact, Kolmogorov's zero–one law says that if you flip a coin an infinite number of times, there's only two possible outcomes: you either get all heads, or all tails. There is no in-between.

Think of it like building a sandcastle. You can make the sandcastle bigger by adding more and more sand. But eventually, you'll get to a point where the castle is as big as it can possibly get. Similarly, Kolmogorov's zero–one law says that once you flip a coin an infinite number of times, the result is either going to be all heads or all tails. You can flip the coin as many times as you want, but you'll never get an even number of heads and tails.

This law applies to many other situations too, not just flipping a coin. It tells us that if we have a very large number of events, and we look at all of them together, we can only either get one particular outcome or the other, with no in-between.