In probability theory, the Borel–Kolmogorov paradox (sometimes known as Borel's paradox) is a paradox relating to conditional probability with respect to an event of probability zero (also known as a null set). It is named after Émile Borel and Andrey Kolmogorov.

A great circle puzzle

Suppose that a random variable has a uniform distribution on a unit sphere. What is its conditional distribution on a great circle? Because of the symmetry of the sphere, one might expect that the distribution is uniform and independent of the choice of coordinates. However, two analyses give contradictory results. First, note that choosing a point uniformly on the sphere is equivalent to choosing the longitude <math>\lambda</math> uniformly from <math>[-\pi,\pi]</math> and choosing the latitude <math>\varphi</math> from <math display="inline">[-\frac{\pi}{2},\frac{\pi}{2}]</math> with density <math display="inline">\frac{1}{2} \cos \varphi</math>. Then we can look at two different great circles:

  1. If the coordinates are chosen so that the great circle is an equator (latitude <math>\varphi = 0</math>), the conditional density for a longitude <math>\lambda</math> defined on the interval <math>[-\pi,\pi]</math> is <math display="block"> f(\lambda\mid\varphi=0) = \frac{1}{2\pi}.</math>
  2. If the great circle is a line of longitude with <math>\lambda = 0</math>, the conditional density for <math>\varphi</math> on the interval <math display="inline">[-\frac{\pi}{2},\frac{\pi}{2}]</math> is <math display="block">f(\varphi\mid\lambda=0) = \frac{1}{2} \cos \varphi.</math>

One distribution is uniform on the circle, the other is not. Yet both seem to be referring to the same great circle in different coordinate systems.