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Half a Circle, Half a Sphere (Posted on 2009-10-28) Difficulty: 3 of 5
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?

See The Solution Submitted by Brian Smith    
Rating: 3.0000 (3 votes)

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Some Thoughts choosing random points | Comment 1 of 4

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
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