Question

Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distances between the points are less than the radius of the circle?

$\dfrac{1}{36}$

$\dfrac{1}{24}$

$\dfrac{1}{18}$

$\dfrac{1}{12}$

$\dfrac{1}{9}$

Solution:

A chord has length less than the radius exactly when the arc it subtends is less than $60^\circ,$ since a chord of a $60^\circ$ arc equals the radius.

All three pairwise chords are shorter than the radius precisely when the three points all lie within some arc of $60^\circ.$

The probability that $n$ random points all lie within some arc of angle $L$ is $n\left(\dfrac{L}{2\pi}\right)^{n-1}.$ With $n = 3$ and $L = \dfrac{\pi}{3}$ (that is, $\dfrac{L}{2\pi} = \dfrac{1}{6}$ ), the probability is $3\left(\frac{1}{6}\right)^2 = \frac{1}{12}.$

Thus, the correct answer is D.

2003 AMC 12B Problem 25

Solution:

Problem 25 in Other Years