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Don't Be a Square (Posted on 2003-05-19) Difficulty: 4 of 5
Given n points drawn randomly on the circumference of a circle, what is the probability they will all be within any common semicircle?

See The Solution Submitted by DJ    
Rating: 4.4667 (15 votes)

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re: Numerical Answers | Comment 10 of 21 |
(In reply to Numerical Answers by Charlie)

The algorithm I originally posted produces a distorted portion of the probability distribution above 180 degrees, but that below 180 degrees is unaffected so the probablities shown should still be accurate within the limitations of numerical integration.

The problem is exemplified by a span of 2 points spaced 170 degrees apart. The algorithm assumes an incremental probability that this would expand, with the third point, to, say, 250 by that third point being external to the original pair and 80 degrees from one of them. However, the distance between the original two is still 170, leaving an occupied span of 360 - 170 = 190 degrees.

This possibility of the new point flipping the occupied span to the opposite side again affects only the distribution within the part above 180 degrees, and not the probability of 180 or less versus 180 or more.
  Posted by Charlie on 2003-05-19 17:59:22

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