Question

Two points are randomly chosen on the circumference of a circle of radius $r$. The probability that the distance between the two points is at least $r$ is equal to

• A. $\frac{\pi }{2}$
• B. $\sin r$
• C. $\frac{2}{\pi }$
• D. $\frac{2}{3}$

Answer 1

The correct option is D $\frac{2}{3}$

Choose a random point on the circumference. The second point can be placed anywhere on the circumference except for within $60$ degrees to the left or right of the initial point(i.e., the angle swept by the line joining the centre and the initial point, to the left or right, must be greater than or equal to $60$ degrees).

The 1 marked in the figure is the initial point chosen at random(apologies for the untidy figure)

And, since $360$ degrees is the sample space, the probability is $\frac{(360-120)}{360}=\frac{2}{3}$