Editing Combinatorics (section)

===Probabilistic combinatorics===
[[Image:Self avoiding walk.svg|thumb|right|150px|[[Self-avoiding walk]] in a [[Lattice graph|square grid graph]].]]

{{Main|Probabilistic method}}

In probabilistic combinatorics, the questions are of the following type: what is the probability of a certain property for a random discrete object, such as a [[random graph]]? For instance, what is the average number of triangles in a random graph? Probabilistic methods are also used to determine the existence of combinatorial objects with certain prescribed properties (for which explicit examples might be difficult to find) by observing that the probability of randomly selecting an object with those properties is greater than 0. This approach (often referred to as ''the'' [[probabilistic method]]) proved highly effective in applications to extremal combinatorics and graph theory. A closely related area is the study of finite [[Markov chains]], especially on combinatorial objects. Here again probabilistic tools are used to estimate the [[Markov chain mixing time|mixing time]].{{clarify|date=November 2022}}

Often associated with [[Paul Erdős]], who did the pioneering work on the subject, probabilistic combinatorics was traditionally viewed as a set of tools to study problems in other parts of combinatorics. The area recently grew to become an independent field of combinatorics.