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?

• A. \( \frac{1}{36} \)
• B. \( \frac{1}{24} \)
• C. \( \frac{1}{18} \)
• D. \( \frac{1}{12} \)

Hint:

In combinatorics, consider the geometric constraints when selecting points randomly on a circle. Use area and geometric probability to calculate the required probabilities.

Answer 1

The Correct Option is D

The probability of choosing points randomly such that all pairwise distances are less than the radius of the circle can be found using combinatorial geometry.
The area that satisfies the condition is calculated, and the total area is normalized to find the probability.
The final result is \( \frac{1}{12} \).