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Geometric Probability

The study of the probabilities involved in geometric problems, e.g., the distributions of length, area, volume, etc. for geometric objects under stated conditions.

The following table summarized known results for picking geometric objects from points in or on the boundary of other geometric objects, where Delta(3) is the Robbins constant.

type of selectionquantitymeandistribution known?point probability known?
line line pickinglength1/3yes-
isosceles triangle line pickinglength1/(30)[2+4sqrt(2)+(4+sqrt(2))sinh^(-1)1]-
equilateral triangle line pickinglength1/(20)(4+3sqrt(3))-
square line pickinglength1/(15)(2+sqrt(2)+5sinh^(-1)1)yes-
circle line pickinglength2-(16)/(pi^2)yes-
disk line pickinglength(128)/(45pi)yes-
tetrahedron line pickinglength?-
cube line pickinglengthDelta(3)yes-
triangle triangle pickingarea1/(12)
square triangle pickingarea(11)/(144)yesyes
pentagon triangle pickingarea1/(180)(9+2sqrt(5))
hexagon triangle pickingarea(289)/(3888)
circle triangle pickingarea3/(2pi)
disk triangle pickingarea(35)/(48pi^2)
tetrahedron triangle pickingarea?-
cube triangle pickingarea?-
ball triangle pickingarea9/(77)pi-
sphere triangle pickingarea?-
tetrahedron tetrahedron pickingvolume(13)/(720)-(pi^2)/(15015)
cube tetrahedron pickingvolume(3977)/(216000)-(pi^2)/(2160)
ball tetrahedron pickingvolume(12)/(715)pi
sphere tetrahedron pickingvolume4/(105)pi
octahedron tetrahedron pickingvolume?

See also

Bertrand's Problem, Buffon-Laplace Needle Problem, Buffon's Needle Problem, Circle Inscribing, Computational Geometry, Integral Geometry, Point Picking, Stochastic Geometry, Sylvester's Four-Point Problem

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References

Ambartzumian, R. V. (Ed.). Stochastic and Integral Geometry. Dordrecht, Netherlands: Reidel, 1987.Isaac, R. The Pleasures of Probability. New York: Springer-Verlag, 1995.Kendall, M. G. and Moran, P. A. P. Geometrical Probability. New York: Hafner, 1963.Kendall, W. S.; Barndorff-Nielson, O.; and van Lieshout, M. C. Current Trends in Stochastic Geometry: Likelihood and Computation. Boca Raton, FL: CRC Press, 1998.Klain, D. A. and Rota, G.-C. Introduction to Geometric Probability. New York: Cambridge University Press, 1997.Santaló, L. A. Introduction to Integral Geometry. Paris: Hermann, 1953.Santaló, L. A. Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley, 1976.Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, 1978.Stoyan, D.; Kendall, W. S.; and Mecke, J. Stochastic Geometry and Its Applications, 2nd ed. New York: Wiley, 1987.Weisstein, E. W. "Books about Geometric Probability." http://www.ericweisstein.com/encyclopedias/books/GeometricProbability.html.

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Geometric Probability

Cite this as:

Weisstein, Eric W. "Geometric Probability." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GeometricProbability.html

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