Points On A Circle
Consider the unit circle \( x^2 + y^ 2 = 1 \).
Choose 3 points uniformly at random on the circumference, which divides the circle into 3 arcs.
What is the expected length of the arc that contains the point \( (1,0) \)?
Technical details: We pick a point on the circumference uniformly at random in the following manner.
1. First select \( \theta \sim U[0,1] \), the uniform distribution on the unit interval
2. Next, we pick the point \( p = \left( \cos (\theta \pi) , \sin (\theta \pi) \right) \).