---
tag: 0924BCD257
kind: concept
name: Geometric probability
msc: [60D05]
uses: [0ZYBZ34RT2]
sources:
  - url: https://en.wikipedia.org/wiki/Geometric_probability
    locator: lead section
same_as:
  wikidata: Q327164
  wikipedia: Geometric probability
---

## Summary

Geometric probability studies random geometric objects, such as random points, lines or shapes in the plane or in space.
In the simplest case, a point is chosen uniformly in a region $\Omega$ of finite positive area.
Then the probability of a subregion $A$ is the area of $A$ divided by the area of $\Omega$.
Classic problems are [[Buffon's needle problem|06X5PF6YFP]] and the [[Bertrand paradox|0T0JWJ3M47]].

## Remarks

The same rule works with length on a line and with volume in space.
Mathematicians studied such problems from the 17th century on. Their modern form is part of integral geometry and stochastic geometry.

## Explanations

### L1 human

In geometric probability you measure chances with lengths and areas instead of with counts.
Throw a dart at random at a square board.
The chance to hit the inscribed circle is the area of the circle over the area of the square, that is $\pi/4$.