3 ducks have landed in a circular pond. What is the probability there is a diameter such that the semicircle on one side contains all of them?
Repeat for 4 ducks.
Note: consider the ducks to be randomly chosen points in a circle.
(In reply to
Randomize by XY coordinates instead of polar by Larry)
There is a method of picking points inside a circle that is more efficient than the "dartboard" approach of picking a point in a square and seeing if it hits the circle. Over at Mathworld they have a short article on Disk Point Picking, which is exactly what we are doing to choose a random spot for a duck.
From the article:
Let R1 be a random number uniformly picked on the interval [0,1) and R2 be a random number uniformly picked on the interval [0,2*pi)
The Cartesian coordinate (x.y) for a random point inside a circle of unit radius can be given as x=sqrt(R1)*cos(R2) and y=sqrt(R1)*sin(R2).
From this parameterization for Cartesian points (x,y), it is pretty easy to find the polar coordinate version for (rad,theta) is rad=sqrt(R1) and theta=R2. In this form it becomes much more apparent that all we need is the random values for theta, which equivalently represent random points on the circumference.
If you really wanted to you could play with "bad" randomly generated radii, like the rookie mistake of using rad=R1 as mentioned in the article. Or get really creative. Any function which passes through (0,0) and (1,1) and is increasing between those two points is a contender. Weird radii distributions could be had from rad=2^x-1, or rad=sin(pi*x/2). So long as the values for theta are uniformly distributed we expect the same results for whatever radii are generated.
re: Randomize by XY coordinates instead of polar
