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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(2): Solution | Comment 9 of 21 |
(In reply to re: Solution by Bryan)

The derivation of the .75 probability depends on the probability distribution at the previous level (2 points) being uniform from 0 to pi. But the probabilities of the fourth point added are not going to be the result of a uniform distribution at level 3, but rather a distribution that favors the higher portion of the possible range, as spans can increase but never decrease. My numerical integration also shows that while 75% of cases had the first 3 points within a 180-degree span, 18.7% were within a 90-degree span and 56.2 % were a span between 90 degrees and 180 degrees. So multiplication by .75 from each stage to the next is not warranted.
  Posted by Charlie on 2003-05-19 16:15:15

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