1/3 is a nice number for this case

Discrete Mathematics Level 3

Circles \(L\) and \(l\) share the same center. Circle \(L\) has radius \(3r\) and circle \(l\) has radius \(r.\) Points \(A, B, C\) are chosen on the circumference of circle \(L\) uniformly at random.

What is the probability that the centroid of \(\triangle ABC\) lies in the interior of circle \(l\)?

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