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?

The answer is $\int _0^{1/2} \int_0^{x+1/2} \, dy \, dx+\int _{1/2}^{1} \int_0^{1} \, dy \, dx=7/8$. You get this by using the fact that joint density is given by $f_{X,Y} (x,y) =1$ for $0\leq x\leq 1,0\leq y\leq 1$ and $0$ for other values of $x$ and $y$. You just have to figure out the range for $x$ and $y$ from the inequality $y-x \leq 1/2$
