Probability

 
If 3 points of a triangle lie on the circumference of a Circle, then what is the Probability of the triangle being an acute angle triangle.
Gator Competitive Exam 2013




Answer:

If one of the sides of the triangle form the diameter, then the triangle is a right angle triangle. If all the coordinates of the triangle lie in one side of semi  circle , then the triangle is obtuse angle triangle.
So, only if 2 points of the triangle lie on same semicircle, and the third point lies in the other semi circle can we have an acute angle triangle.

So, Probability of the point lying on one side of semicircle = 1/2
Since the occurrence of one is independent of the other, so, probability that all three coordinates will lie in the same semicircle = 1/2 x 1/2 x 1/2 = 1/8
Since there are 2 semicircles so the  probability of all three coordinates lying in the same semi circle will be = 2 x 1/8 = 1/4

So, probability of any 2 points lying in a semicircle = 1 - 1/4 = 3/4

So, Probability of an acute angle triangle = 3/4= 0.75

The same may be demonstrated using random coordinates in Excel sheet.
semicircle, -0.5 < x < 0.5 and y > 0
sampling size = 1169, all three points in one side of semicircle = 293 (average value) So, Probability of all three points in same semicircle = 293/ 1169 = 0.25
So, 2 points lying in the same semicircle = 1 - 0.25 = 0.75