The best description I was ever given of the probabilistic method: "It's not like we're searching for a needle in a haystack here. We're searching for hay in a

The best description I was ever given of the probabilistic method: "It's not like we're searching for a needle in a haystack here. We're searching for hay in a haystack... But every time we reach in, we find a needle" - Laszlo Babai This was said of the Ramsey number (Example 1) which is very hard to construct examples of (AFAIK there isn't an algorithm to do it), even though a randomly generated graph will be an example with probability tending toward 1.