Get inspired by the success stories of our students in IIT JAM MS, ISI MStat, CMI MSc DS. Learn More

This is a very beautiful sample problem from ISI MStat PSB 2004 Problem 4 based on finding the probability using Uniform distribution . Let's give it a try !!

Two policemen are sent to watch a road that is long. Each of the two policemen is assigned a position on the road which is chosen according to a uniform distribution along the length of the road and independent of the other's position. Find the probability that the

policemen will be less than 1/4 kilometer apart when they reach their assigned posts.

Uniform Distribution

Basic geometry

Let X be the position of a policeman and Y be the position of another policeman on the road of 1km length .

As it is given that chosen according to a uniform distribution along the length of the road and independent of the other's position hence we can say that and and X,Y are independent .

Now we have to find the probability that the policemen will be less than 1/4 kilometer apart when they reach their assigned posts , which is

nothing but .

So , let's calculate the probability here some sort of geometry will help to calculate it easily !

In general we have 0<X<1 and 0<Y<1 and hence the total probability is the area of the square

And in favourable case we have . so, it's basically the area covered by ACBDEF = Area covered by square - area of the triangles BGD and AFH = - = .

Therefore

Calculate the same under the condition that road is of length (b-a) , b>a and both are positive real number .

This is a very beautiful sample problem from ISI MStat PSB 2004 Problem 4 based on finding the probability using Uniform distribution . Let's give it a try !!

Two policemen are sent to watch a road that is long. Each of the two policemen is assigned a position on the road which is chosen according to a uniform distribution along the length of the road and independent of the other's position. Find the probability that the

policemen will be less than 1/4 kilometer apart when they reach their assigned posts.

Uniform Distribution

Basic geometry

Let X be the position of a policeman and Y be the position of another policeman on the road of 1km length .

As it is given that chosen according to a uniform distribution along the length of the road and independent of the other's position hence we can say that and and X,Y are independent .

Now we have to find the probability that the policemen will be less than 1/4 kilometer apart when they reach their assigned posts , which is

nothing but .

So , let's calculate the probability here some sort of geometry will help to calculate it easily !

In general we have 0<X<1 and 0<Y<1 and hence the total probability is the area of the square

And in favourable case we have . so, it's basically the area covered by ACBDEF = Area covered by square - area of the triangles BGD and AFH = - = .

Therefore

Calculate the same under the condition that road is of length (b-a) , b>a and both are positive real number .

Cheenta is a knowledge partner of Aditya Birla Education Academy

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

JOIN TRIALAcademic Programs

Free Resources

Why Cheenta?

Online Live Classroom Programs

Online Self Paced Programs [*New]

Past Papers

More