What can I say about $P\bigl[Y-X \le\frac{1}{2}\bigr]$ if $X$ and $Y$ are independent $U[0,1] $ variables? Let $X$ and $Y$ be two random independent variables with uniform distribution on $[0,1]$. 
What can I say about $P\bigl[Y-X \le\frac{1}{2}\bigr]$?
I tried doing the following:
$$P\Bigl[ Y \le X + \frac{1}{2}\Bigr]$$
Let $X + \frac{1}{2} = Z$. $Z$ is uniform on $\bigl[\frac{1}{2},\frac{3}{2}\bigr]$
Then I evaluated $\int_{\!\frac{1}{2}}^1t\, dt$ but the result is wrong. 
Can you give me some suggestions?

Hint. Draw a picture of the unit square and shade the area that matters. Then you can find the answer without integrals (even without pencil and paper after you see the picture).
