## Points in a Circle

Draw one circle of radius 1 and a concentric circle inside with a radius of
1/2. We want to pick a random point in the large circle and determine the
odds the point is also in the inner circle. There are various ways to pick
the point.
- From the center of the circle, pick a random angle and a random distance
from 0 to 1.
- From the center of the circle, pick two points, each with a random angle
and distance from 0 to 1. Connect the two points and use the midpoint
of that line.
- Create a set of parallel lines exactly 2 units apart. Toss the
circle onto the lines. One will intersect the large circle. Use
the midpoint of that chord.
- Pick one point on the perimeter and pick a second point on the perimeter
by random angle from the center. Connect the two points and use the
midpoint of that chord.
- Pick one point on the perimeter and pick a second point on the perimeter
by random distance around the perimeter. Connect the two points and
use the midpoint of that chord.
- Place an ant at the center. The ant takes moves 1/100 units with each
step. Each step is in a random direction, reversing direction if it
hits the side of the outer circle. Pick the ant's position after 200
steps. (I don't know how to analyze this one.)

Source: Original. From college. Any other approaches lead to different results?

Only a few solutions were received. Still writing them up and waiting for
more. (I'm a little behind in my POTW management - sorry.)

Mail to Ken