One way of doing this is by considering a randomly selected thing from a finitesized universe. If the probability that the random thing satifies certain properties is greater than zero, then this proves the existence of a thing that satisfies the properties. It doesn't matter if the probability is astronomically small; any probability strictly greater than zero will do. (Also, showing that the probability is zero can be used to prove the nonexistence of such an object).
Another way to use the probabilistic method is by calculating the expected value of some random variable. If it can be shown that the random variable can take on a value less than the expected value, this proves that the random variable can also take on some value greater than the expected value.
[This is a still a stub article because of its lack of examples.]
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