Editing Probabilistic method (section)

==Introduction==
If every object in a collection of objects fails to have a certain property, then the probability that a random object chosen from the collection has that property is zero. Thus, by [[contraposition]], if the probability that a random object chosen from the collection has that property is nonzero, then some object in the collection must possess the property.

Similarly, showing that the probability is (strictly) less than 1 can be used to prove the existence of an object that does ''not'' satisfy the prescribed properties.

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.

Alternatively, the probabilistic method can also be used to guarantee the existence of a desired element in a sample space with a value that is greater than or equal to the calculated expected value, since the non-existence of such element would imply every element in the sample space is less than the expected value, a contradiction.

Common tools used in the probabilistic method include [[Markov's inequality]], the [[Chernoff bound]], and the [[Lovász local lemma]].