Answer to Question #187555 in Statistics and Probability for Sahil

Question #187555

 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


1
Expert's answer
2021-05-07T11:18:02-0400

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"


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