Q27: Three points are taken at random on the circumference of... - Statistical Methods & Experimental Designs - ICAR AIEEA PG Statistical Sciences 09 July 2023 (Shift 1) Paper | Agrigyan

Mathematical derivation of geometric probability:

• Incorrect: $1/3$ is not the value.
• Incorrect: $1/4$ is the complementary probability that the triangle formed contains the center of the circle.
• Correct: For $n$ independent points placed uniformly at random on the circumference of a circle, the probability that all $n$ points lie within some semicircle is given by the classic formula:
  $$P = \frac{n}{2^{n-1}}$$
  For $n = 3$ points:
  $$P = \frac{3}{2^{3-1}} = \frac{3}{2^2} = \frac{3}{4}$$
  Alternatively, three points do not lie in a semicircle if and only if the center of the circle lies inside the triangle formed by the three points, which has probability $1/4$. Thus, $P = 1 - 1/4 = 3/4$.
• Incorrect: $2/3$ is incorrect.