Geometric probabilities

Probabilities of events related to the relative location of geometric figures placed at random on a plane or in a space. The simplest example may be stated as follows: A point is "thrown" at random into a planar domain $A$. What is the probability of this point "falling" into a domain $B$ lying inside $A$? If it is assumed that the probability sought depends only on the "shape" of the domain and not on its "location" inside $A$, one must conclude that it is uniquely determined and equal to the ratio of the area of $B$ to the area of $A$.

The assumption that the probabilities sought are invariant with respect to a group of transformations in a Euclidean space including translations, rotations and reflections is typical of most problems in geometric probability. The answer is usually obtained in the form of the ratio of an invariant measure of the set of "favourable cases" to an invariant measure of the set of "all possible cases" (cf. Integral geometry); the analogy with the classical definition of probability is evident. One may note that in the Bertrand paradox, which is connected with geometric probabilities, one answer only satisfies the condition of invariance.

The first example of computing geometric probabilities was the Buffon problem, which laid the foundations of the idea of randomness in geometry. The 200 year old history of the development of this idea consists of periods of enthusiasm and intensive studies alternating with periods during which the subject was underestimated and interest in it waned. The growing interest in the subject in the second half of the 20th century resulted in a large increase in the number of models studied (e.g. random sets, fields of straight lines, so-called thread fields, etc.), as a result of which the theory of geometric probabilities has become a new branch of probability theory — stochastic geometry.

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