What's the probability ?

Question

Given a square of side's lenght equals to "s", we choose randomly a point inside the square. So, what's the probability that the distance from the point to the center of the square be greater than s/2 ?
Please explain the solution.

Answer 1

The points exactly s/2 from the centre lie on a circle which touches the four sides. Inside the circle the distance of the point from the centre is less than s/2, and outside, it is greater. The area of the circle is (pi/4) times that of the square, so that is the probability of it being smaller. By subtraction, the probability of it being greater is 1 - pi/4 or approximately 0.2146.

Answer 2

Draw the square and then draw a circle of radius s/2 centered inside the square. Calculate the area of the circle and subtract it from the area of the square. The probability that the point will be > s/2 units from the center of the square (and circle) will be this area divided by the area of the square.


Doug