Let $(X_1,X_2,X_3)$ three independent random variables with uniform distribution $[0,1]$.
- Let $A=(X_1+X_2<1)$. Find $\mathbb{P}(A)$.
$\rightarrow \mathbb{P}(A)=\int_{0}^{1}[\int_{1-x_2}^{1}dx_1]dx_2=\frac{1}{2}$.
- Let $B=(X_1+X_2+X_3<1)$. Find $\mathbb{P}(B)$.
$\rightarrow \mathbb{P}(B)=\int_{0}^{1}dx_3[\int_{0}^{1-x_3}dx_2[\int_{0}^{1-x_3-x_2}dx_1]]=\frac{1}{6}$
- Find $\mathbb{P}(A\cap B^c) $.
I think that first two points are correct, but I'm stuck at point 3). Anyone can help me? Thanks in advance.