Choose randomly 3 points on a circumference of a circle.
What is the probability that the center of the circle
lies within the triangle ABC?
Source: Simplified from a 3-dimensional problem on a Putnam exam.
The first two points chosend are equally likely be anywhere from zero to 180° apart. The center of the circle will be within the triangle if the third point is anywhere on the arc that's antipodal to the arc between the first two points. The antipodal arc is the same length as the original arc itself.
The probability then is the fraction of the circle covered by the arc. That's anywhere from zero to one half, in a uniform distribution, making the average probability equal to 1/4.
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Posted by Charlie
on 2024-04-09 08:39:57 |
solution
