Inspired by <a href="http://mathoverflow.net/questions/2857/non-trivial-colouring-of-the-edges-of-an-infinite-complete-graph">this question</a>, I was curious about a comment in <a href="http://en.wikipedia.org/wiki/Kolmogorov%27s_zero-one_law">this article</a>:

> In many situations, it can be easy to
> apply Kolmogorov's zero-one law to
> show that some event has probability 0
> or 1, but surprisingly hard to
> determine which of these two extreme
> values is the correct one.

Could someone provide an example?