The term geometrical probability has been applied to an amazing assortment of problems that deal with distributions of objects in space, the space usually being one-, two-, or three-dimensional. The objects are usually points or straight lines, but could be more complicated figures. Many of the key results of geometrical probability were developed in the late 1800s, for use in applications far removed from urban systems. Through 1970 or so, ideas of geometrical probability had been useful in astronomy, virology, biology, forestry, atomic physics, search theory, crystallography, and sampling theory.

In urban applications, geometrical probability concepts help us to analyze interrelationships among objects distributed probabilistically throughout an urban environment. They may be people requiring some kind of on-scene service (e.g., pickup by a taxicab, on-scene medical assistance, a visit by a social worker); resources of an urban service system (e.g., buses, police cars, mail carriers); vehicles in an urban traffic pattern; places of residence, perhaps categorized by demographic variables (e.g., age, income, education, race); economic goods; and so on. Many of the probabilistic problems that we might confront in an urban spatial setting can be analyzed using the concepts of derived distributions developed in Section 3. 1. However, there are several important results in geometrical probability that we will find useful to consider separately. Some of the introductory concepts are considered in this section in the context of examples; Sections 3.4-3.6 and 3.8 develop the more general results.