Suppose we have the following problem:
The dot is at $\mathbf{a}=(0,2)$ and the lines have directions $(1,1)$ and $(-1,1)$, both passing through the origin.
Suppose that now the dot suffers from additive white Gaussian noise, i.e.
$$\mathbf{a_w}=(0+W_1,2+W_2)$$
where both $W_i$ have zero mean and variance $\sigma^2$. They are independent.
I want to calculate the probability $p_e$ that $\mathbf{a_w}$ escapes the zone marked by the red lines. I've thought of doing the following:
$$p_e=\mathbb{P}(\mathbf{a_w}\mathrm{\ escapes\ in\ the\ horizontal\ axis})+\mathbb{P}(\mathbf{a_w}\mathrm{\ escapes\ in\ the\ vertical\ axis})-\mathbb{P}(\mathbf{a_w}\mathrm{\ escapes\ in\ both})$$
due to the probability of the union property. This would lead to:
$$\begin{align} p_e&=\mathbb{P}(|W_1|>2)+\mathbb{P}(W_2<-2)-\mathbb{P}(|W_1|>2)\mathbb{P}(W_2<-2)\\ &=2Q\left(\frac{2}{\sigma}\right)+Q\left(\frac{2}{\sigma}\right)-2Q\left(\frac{2}{\sigma}\right)Q\left(\frac{2}{\sigma}\right)\\ &=3Q\left(\frac{2}{\sigma}\right)-2Q^2\left(\frac{2}{\sigma}\right) \end{align}$$
Is this approach correct? Is there any other less "homemade" way to do this (i.e. more formal, in some sense)?
You're in luck, since the product of Gaussians in Cartesian coordinates with the same variance is rotationally invariant (since the exponent is a multiple of $r^2$, the squared distance from the origin). So you can rotate your coordinate system by $\frac\pi4$ to make the red lines axis-parallel and then do exactly the sort of inclusion–exclusion calculation that you tried in the unrotated system (except that now both factors in each product will be half-infinite).