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Two points on a circle of radius $1$ are chosen at random.  Find the probability that the distance between the two points is at most $3/2.$

 Oct 6, 2023
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Look at the diagram:

Find angle \(\alpha \) using the cosine rule    (i.e. \(cos(a) = (b^2 + c^2 - a^2)/(2bc)\) )

 

Take the ratio of the arc of the circle subtended by \(2\alpha \)  to that of the circumference of the whole circle to get the probability.

i.e. probability = \(2\alpha / (2\pi) \)

 

(Note that the lower point could be on the opposite side of the circle hence \(2\alpha \) not just \(\alpha \))

 Oct 6, 2023
edited by Alan  Oct 6, 2023
edited by Alan  Oct 6, 2023

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