You have a stick of 1 meter long Let A and B be random point

You have a stick of 1 meter long. Let A and B be \"random points\" placed on the stick.

Let the distances of A and B from the left end of the stick be denoted by X and Y respectively. The stick is divided into 3 pieces (by breaking the stick at points A and B).

What is the probability that the resulting three pieces can be used to form a \"Triangle\"?

Hint: A triangle is possible if the length of any piece is less than the sum of the lengths of the other two pieces. Clearly identify the \"base\" of the joint pdf and the region of interest.

Solution

Let draw a diagram

---------------y-----------

------x-----

-------------------------------------------- rope 1 meter long

A B

length of pieces =

oA = x , 0B = y-x , oC = 1-y

Now Accroding to the traingle properties.A triangle is possible if the length of any piece is less than the sum of the lengths of the other two pieces.

x+ y -x > 1-y ==> y > 1-y ==> y > 1/2

x - y + x > 1-y ==> 2x > 1 ==> x > 1/2

y - x > 0 , 0c = 1 - y < 1-1/2 < 1/2