Question

 A chord connecting two points randomly chosen on a circunference of a disk of radius 1 cuts it into two segments. Select the best approximation to the mathematical expectation of the area of the smaller segment.

1. O.2

2. 0.6

3. 0.4

4 0. 8

5. 1

Answer 1

Let the circle of centre O and radius 1 unit . A and B are two points on the circumference which cuts the circle into two segments. The portion under AB is a minor segment.

Area of smaller segment = Area of sector AOB - Area of Triangle AOB

Area of sector AOB for angle $\theta$ $= \theta \times \dfrac{\pi r^2}{360}$

Area of Triangle AOB $= \dfrac{1}{2} \times base \times height.$

$= \dfrac{1}{2} \times OA \times BM$

$= \dfrac{1}{2} \times r \times rsin\theta = \dfrac{1}{2}r^2sin\theta$

Area of smaller segment $= \dfrac{\theta \pi r^2}{360}- \dfrac{1}{2}r^2sin \theta$