Let $X_1 = \min(\max(\eta_1 +0.5 + a,0),1) $
Let $X_2 = \min(\max(\eta_2+0.5+b,0),1) $
where ${a,b} \in \mathbb{R}$ and $\eta_i\sim U[-d,d]$ for $i=1,2$ (where U represents the Uniform distribution) and $X_i$ must be between 0 and 1.
I need to find: $P(X_1+X_2\geq 1)$
My idea is to break it down into pieces. I.e. start with $P(X_1=1)\cup P(X_2=1)$ and then add to that $P(X_1+X_2\geq 1) | X_i\in[0,1]$
Finding the first term should be straight forward (I think, since they are independent distributions).
As for the second term, using an answer to a previous question of mine, I get it looks something like:
$\int_{-0.5-b}^{0.5-b}\left(\int_{-a-b-\eta_2}^{0.5-a}\frac{1}{4d^2} \, d\eta_1\right) \, d\eta_2 $
Before I attempt to solve this, can someone let me know if I'm on the right track