Compute expected arc length on a circle
Problem Description
Consider a unit circle (radius 1, centered at the origin).
Three points are chosen independently and uniformly at random on the circumference of this circle. These three points partition the circle into three distinct circular arcs.
Let L be the length of the arc (measured along the circumference) that contains the fixed point (1,0) (which corresponds to an angle of 0 radians or 0 degrees).
Your task is to compute the expected value of L, denoted as E[L].
AI interview prep
Core dilemma
Transforming a geometric probability problem into a tractable analytical or simulation-based solution, while correctly handling the conditional nature of the 'arc containing a fixed point' and the circular symmetry.
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Interview prep
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Compute expected arc length on a circle
Problem Description
Consider a unit circle (radius 1, centered at the origin).
Three points are chosen independently and uniformly at random on the circumference of this circle. These three points partition the circle into three distinct circular arcs.
Let L be the length of the arc (measured along the circumference) that contains the fixed point (1,0) (which corresponds to an angle of 0 radians or 0 degrees).
Your task is to compute the expected value of L, denoted as E[L].