How far is a random point on Earth from the North Pole?

Question

A point is chosen uniformly at random on the surface of a sphere of radius 1. 1. What is the expected straight-line distance from the point to the North Pole, measuring the chord through the interior of the sphere, not along the surface? 2. Is the answer the same as the expected distance between two independent random points on the sphere?

Accepted Answer

The expected chord length is 4/3, and yes, it is the same as for two random points. The chord in terms of height. Put the North Pole at (0, 0, 1). A point on the unit sphere at height z is at distance the square root of (2 minus 2z) from the pole (by the cosine rule, or from (1 minus z) squared + (1 minus z squared) = 2 minus 2z). At the pole itself z = 1 and the distance is 0; at the South Pole z = minus 1 and the distance is 2. The distribution of height. By Archimedes' theorem, the area of a sphere between two horizontal planes depends only on the distance between the planes. So for a unif…

How far is a random point on Earth from the North Pole?

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