## 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.
1. From the center of the circle, pick a random angle and a random distance from 0 to 1.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.)
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