Geometric probability

Geometric probability . In mathematics, specifically in theory of stochastic events, it refers to results of the probability (measurement of a stochastic event) of belonging (position) of a point of a closed figure of the line, plane or space, also belongs to a sub-figure (subset) of it. On the line we have the segments (closed intervals) and their subsegments; on the plane, particularly the simple and convex flat figures and their flat sub-figures; in space polyhedra, cones, cylinders, and spatial spheres and sub-figures, inscribed or built within their respective interiors.

Suppose that segment m is a subset of segment M. In segment M a point is drawn randomly, accepting that the probability that the point is located in segment m is proportional to the length of this segment and does not depend on its location. with respect to segment M we define the


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  • 1 Geometric probability on the line
  • 2 Geometric probability in the plane
  • 3 Geometric probability in space
  • 4 Social practice
  • 5 Sources

Geometric probability on the line

We are only going to relate segments and their segments, although we can work on a finite collection of disjoint segments two by two and a subcollection of them. The probability of belonging of a point of segment M is at the same time on its subsegment m, is obtained by the formula

P = length of m: length of M (g1)

  • which is obviously the probability that the point belongs to segment m

Example A segment M of length 48 cm contains a segment m of 32 cm. find the probability that a randomly indicated point in segment M is also in M. It is assumed that such probability does not depend on the position of m in M, but is proportional to the length of m.

Resolution Let event H = {a point of segment M marked randomly, it is also in segment m} using formula g1 the probability of event H results

P (H) = 32: 48 = 1/3

Geometric probability in the plane

The probability of belonging of a point of the plane figure F is also in the subfigure f that is part of F, it is a ratio of areas.

  • The probability is assumed to be proportional to the area of ​​f and does not depend on its position in figure F or the shape of f and F.
  • We define the probability that the point belongs to f is

P = S f  : S F ; here S  denotes the respective areas of the figures f and F. Example Inside a circle of radius r a point is marked at random. Find the probability that this point belongs to the interior of a regular hexagon inscribed in a circle. Such probability is assumed to be proportional to the area of ​​the triangle and does not depend on the arrangement of the triangle in the circle. Resolution

  • Area of ​​the circle is 22/7 r 2= S c
  • Area of ​​the inscribed hexagon 1.5r 2× (3) 5 = S h
  • Event H = {A randomly marked circle point also belongs to the inscribed hexagon}

P (H) = S h ÷ S c = 1.5r 2 × (3) 0.5 ÷ 22/7 r 2 = S c = 0.8266061 …

Geometric probability in space

We are interested in finding the probability of a randomly selected point in a spatial figure K being at the same time in its spatial subfigure k of it.

  • The probability is assumed to be proportional to the volume of k and does not depend on its position in figure K or the shape of k or y.
  • We define the probability that the point is also in k is

P = V k  : V K ; Here we have found the quotient of the respective volumes of the given sub-figure and figure. Example At the centers of each of 6 the faces of a cube of side L are the vertices of an octahedron. Find the probability that a point inside the cube, marked randomly, is also inside the octohedron. That probability is assumed to be proportional to the volume of the octohedron and does not depend on its particular position in the cube. Resolution

  • Cube volume L 3= V c
  • Volume of the octohedron is 1 / 6L 3= V or
  • Event D = {a point of the cube marked at random is also in the octohedron with respective vertices of the 6 faces of the cube}

P (D) = V or ÷ V c = 1 / 6L 3 ÷ L 3 = 1/6

Social practice

Road accident

  • Consider the coastal road from Lima to Tumbes, it has a length of 1271.6 km and the one from Trujillo to Chiclayo is 207.7 km. The four cities are assumed to be towns (points) on the same highway and are in succession Lima-Trujillo-Chiclayo and Tumbes. It is reported that a truck collision has occurred on the coastal highway, we want to find the probability that it occurred at a point on the Trujillo-Chiclayo section.


  • The road can be assumed as a simple curve, without auto-intersections and the image of an interval, and the geometric probability of the line can be applied.
  • Let the event A = {point of the highway from Lima to Tumbes be also between Trujillo and Chiclayo}
  • The probability of the event is P (A) = long (Trujillo-Chiclayo) ÷ long (Lima- Tumbes) = 207.7 / 1271.6 = 0.163259

Plane crash We are reported the plane crash in a place in Spain, if we want to know the probability that the place of the plane crash is also in Catalonia, we obtain it by finding the quotient of the surfaces of Catalonia and Spain, which are 32,108 km 2 and 505,990 km 2 respectively; applying the geometric probability in the plane.

  • The event C = {Spain’s place of the crash of an airplane that is also in Catalonia}
  • The probability of C is P (C) = Surface of Spain ÷ Surface of Catalonia = 32,108 ÷ 505,990 = 0. 0634558

Native solar If you know the country where a person was born and you want to know the probability that they were also born in an administrative region of the aforementioned country, the flat geometric probability can be applied, considering the inhabitant as a point and the country and its region administrative, such as flat figure and subfigure


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