1 - (1-(1/2)^(n-2))^n

where n is the # of points on the sphere.

The question will become a lot easier if you restate it as the following:

What is the probability in finding at least one point such that all the other points on the sphere are on one side of the great circle going through this point.

When n=2, the probability= 1, when n=infinity, it becomes 0.

In his Scientific American column which was titled "Curious Maps", Martin Gardner ponders the fact that most of the land mass of the Earth is in one hemisphere and refers to a paper which models continents by small circular caps. He gives the above result.

See "The Probability of Covering a Sphere With N Circular Caps" by E. N. Gilbert in Biometrika 52, 1965, p323.

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