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 $3/2.$

maximum Oct 6, 2023

#1**+1 **

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 \(\alpha \))

Alan Oct 6, 2023