$X,Y,Z$ are i.i.d $U(0,1)$. What is $P(X+Y+Z > 1)$? 
Assume that $X,Y$ and $Z \sim U(0,1)$. Assume further $X,Y$ and $Z$ are mutually independent.
  What is $P(X+Y+Z>1)$?


I've written down the density function of $X,Y$ and $Z$. I've drawn a picture. I understand the range of a new variable $U=X+Y+Z$ should be between $0$ and $3$. But I'm stuck on how to find the distribution of $U$.

$$P(X+Y+Z\leq 1)=\mu\{(x,y,z)\in[0,1]^3:0\leq x+y+z\leq 1\}=\frac{1}{6}$$
since the middle term is the volume of a simple pyramid.
