If U1, U2, U3 are iid Uniform[0,1], then what is the probability of U1+U2>U3? This is what I have so far.The distribution function of U1+U2=Z is 
$$
f_Z(z) = z_{I[0,1]}+(2-z)_{I(1,2]}
$$
I also have $$P[U_1+U_2>U_3] = E[I(U_1+U_2>U_3)]=E[E[I(U_1+U_2>U_3)|U_3]]=E[P[U_1+U_2>U_3|U_3]]$$
But I'm not sure how to calculate this expectation, or the probability inside the brackets. Help would be much appreciated!
 A: There is a simple intuitive interpretation of your problem: Fate selects a point $(x,y,z)$ according to a uniform distribution in the cube $C:=[0,1]^3$. What is the probability $P$ that $z< x+y$? As in this view the probability measure coincides with ordinary euclidean volume the probability $P$ is equal to the volume of the set $$A:=\{(x,y,z)\in C\>|\>z<x+y\}\ .$$
Drawing a figure we see that $A$ looks somewhat complicated, but the set
$$A':=C\setminus A=\{(x,y,z)\in C\>|\>z\geq x+y\}$$
has a simpler description: It is (in three ways!) a pyramid of height $1$ over a base triangle of area ${1\over2}$. It follows that the the probability $P$  is given by
$$P=1-{\rm vol}(A')=1-{1\over3}\cdot{1\over2}={5\over6}\ .$$
A: You can compute the distribution of $\frac{U_1+U_2}{U_3}$
which is possible whatever their respective distributions are. Suppose
now $X = \frac{U_1+U_2}{U_3}$ and $F_X$ be the c.d.f of $X$, we have simply
$$P(U_1+U_2>U_3) = P(\frac{U_1+U_2}{U_3}>1) = 1-F_X(1)$$
If you are unfamiliar with the computation of the distribution of transformation of random variables, here's the result:
Let $\zeta = \frac{\xi}{\eta}$, $f_\zeta, f_\xi, f_\eta$ being their respective p.d.f's. Then
$$f_\zeta(t)=\int_{-\infty}^{\infty}| y |\ f(yt,t)dy$$
where $f(x,y)$ is the joint p.d.f. of $\xi$ and $\eta$, which equals to $f_\xi(x)f_\eta(y)$ if they are independent. 
If $\zeta = \xi + \eta$, we have
$$f_\zeta(t) = \int_{-\infty}^{t}f(t-x,x)dx$$
Detailed proof is easily found on a probability textbook. 
Concretely, for this problem, you need to compute $U_1 + U_2$ first, finding the p.d.f of it, then proceed to compute the p.d.f and c.d.f of $\frac{U_1 + U_2}{U_3}$.
