Points On A Circle

Probability Level 4

Consider the unit circle x2+y2=1 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) (1,0) ?


Technical details: We pick a point on the circumference uniformly at random in the following manner.
1. First select θU[0,1] \theta \sim U[0,1] , the uniform distribution on the unit interval
2. Next, we pick the point p=(cos(2πθ),sin(2πθ)) p = \left( \cos ( 2 \pi \theta) , \sin ( 2 \pi \theta) \right) .

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