what is the probabilty that sum of two random numbers between A and B is less than third number C

Question

What is the probabilty that sum of two random numbers uniformly distributed in $[A,B]$ is less than a fixed $C$?

I have tried answering this question using graph method to find the area under the curve, but I couldn't find out the answer despite long hours of efforts. Can someone help?

Answer 1

You can indeed do this graphically. Consider A=0 and B=1 for simplicity. Your situation can be represented by this :

So the probability you want is the area of the colored area:

$$P( X+Y > C) = 1-\frac{C^2}{2}$$

Edit : with the correct graph it's better