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

> 
> 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?