Problems of the following type, and their solution techniques, were first studied in the 18th century, and the general topic became known as geometric probability.
* (
Buffon's needle
In mathematics, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon:
:Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. ...
) What is the chance that a needle dropped randomly onto a floor marked with equally spaced parallel lines will cross one of the lines?
* What is the mean length of a random chord of a unit circle? (cf.
Bertrand's paradox).
* What is the chance that three random points in the plane form an acute (rather than obtuse) triangle?
* What is the mean area of the polygonal regions formed when randomly oriented lines are spread over the plane?
For mathematical development see the concise monograph by Solomon.
[
]
Since the late 20th century, the topic has split into two topics with different emphases.
Integral geometry In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformati ...
sprang from the principle that the mathematically natural probability models are those that are invariant under certain transformation groups. This topic emphasises systematic development of formulas for calculating expected values associated with the geometric
objects derived from random points, and can in part be viewed as a sophisticated branch of
multivariate calculus.
Stochastic geometry
In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of spatial point processes, hence notions of Palm conditioning, which exten ...
emphasises the random geometrical objects themselves. For instance: different models for random lines or for random tessellations of the plane; random sets formed by making points of a
spatial Poisson process
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
be (say) centers of discs.
See also
*
Wendel's theorem
References
*Daniel A. Klain, Gian-Carlo Rota - Introduction to Geometric Probability.
*Maurice G. Kendall, Patrick A. P. Moran - Geometrical Probability.
Eugene Seneta, Karen Hunger Parshall, François Jongmans - Nineteenth-Century Developments in Geometric Probability: J. J. Sylvester, M. W. Crofton, J.-É. Barbier, and J. Bertrand
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Fields of geometry
Probability theory