Due to this question, I'm wondering about a list of some interesting examples of when the probability that something was going to occur was $0$ and occurs anyways.
I suppose a really basic example could be that the probability that a random number picked between $1$ and $n$ is prime tends to be $0$ as $n\to\infty$, but there are still an infinite amount of primes.
However, I'm interested in less trivial cases (preferably a list) that might very well blow my mind.
Notice: This is not the same as something impossible to occur nor is it the same as something unlikely to occur. Please see Zero probability and impossibility for some explanation.