Probability of Points Lying in a Disk

[marcusl]

Imagine points in a plane with a uniform density. To use a simple model, consider just a finite region such as the unit circle and randomly place M points into that unit circle. By uniform density, I mean that any infinitesimal area anywhere in the unit circle has the same probability of containing a point. How would I find the probability as function of M of finding one or more points within a circle of radius r≤1?

BTW, this is not a homework problem.

 

[FactChecker]

In any subset, S, of finite area, A, the uniform density is 1/A. For a circle, Cr, of radius r <=1, the probability that a uniform random point in S is also in Cr is the ratio of the areas P = pi*r^2 / A. If the points are all independently placed, the odds that none are in Cr is (1-P)^M. The probability that at least one is in Cr is (1-(1-P)^M) For the set S = unit circle, P = r^2, so the answer is (1-(1-r^2)^M)

 

[marcusl]

Thanks. It seems so obvious once you explained it!