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 and the Bertrand paradox.

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 $\pi/4$.