Probabilistic methods prove existence results in a nonconstructive fashion, by showing the chance of randomly selecting a solution is greater than zero.
Probabilistic methods of proof, championed by Paul Erdős, establish existence results by a nonconstructive argument, showing that an object satisfying a requirement may be randomly selected with positive probability. Although probability is used in such arguments, the results established are certain.
The basic Probabilistic Method can be described as follows: In order to prove the existence of a combinatorial structure with certain properties, we construct an appropriate probability space and show that a randomly chosen element in this space has the desired properties with positive probability. This method was initiated by Paul Erdős, who contributed so much to its development over a fifty year period, that it seems appropriate to call it "The Erdős Method."
