Given n points drawn randomly on the circumference of a circle, what is the probability they will all be within any common semicircle?

A search for probability points semicircle yields a page (http://www.math.niu.edu/~rusin/known-math/98/hemisphere) asking about such problems in various dimensions, but including a statement about the two-dimensional case (actually 1-dimensional considered along the circumference):

"The case n = 0 yields the known probability that all k points lie on a semicircle: k/2^(k-1)."

In fact the formula k/2^(k-1) fits with both the numerical integration solution as well as the simulation.

Only my estimate that the value .03513 for 9 points, which I took as 1/32, really should have been 9/256.

The .01951 for 10 points is close to the 10/512 theory predicts.

Posted by Charlie on 2003-05-23 03:51:22