Three points on the circumference of a circle are chosen at random. What is the probability the points are all on the same semicircle?

Four points on the surface of a sphere are chosen at random. What is the probability the points are all on the same hemisphere?

Presumably you want to make any point as likely as any other point to be chosen as a given random point.

Choosing a random point on a circle is easy: a uniform distribution of angle from 0 to 360 degrees.

With a sphere, lets get a frame of reference by using the earth considered as a perfect sphere. The longitude can be chosen as for a circle, and the latitude as the arcsin(rand#) where rand# is chosen on a uniform distribution between -1 and 1. This is the basis for making equal-area cylindrical maps of the world centered on the equator.

Posted by Charlie on 2009-10-28 11:13:08