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  4. 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

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Q. 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

KVPYKVPY 2009

Solution:

Two point on a circumference of circle of radius $r$
Distance between two points is atleast $r$
$\therefore $ Angle subtends to chord of length
atleast $r$ is greater than or equal to $60^{\circ}$
image
$\therefore $ Required pronabaility $=1-\frac{60}{180}$
$=1-\frac{1}{3}=\frac{2}{3}$