Suppose we have $N+1$ i.i.d. random variables from a normal distribution with mean $0$ and variance $\sigma^{2}$. That is:
$$X_{i}\sim \mathcal{N}(0,\sigma^{2}),\quad i=1,\dots , N+1$$
Suppose $b>0$ is some known, positive number. We want to find the following probability: $$Pr\left(X_{N+1}+b \geq \max\left(X_{1},X_{2},\dots , X_{N}\right)\right)$$.
Notice that we do not know the realizations, so we basically need to find: $$Pr\left(\max\left(X_{1},X_{2},\dots , X_{N}\right)-X_{N+1}\leq b\right)$$.
How can we compute that? Would it be correct to derive the distribution of $\max\left(X_{1},X_{2},\dots , X_{N}\right)-X_{N+1}$, given the distribution of the maximum of $N$ normal random variables and given the distribution of $X_{N+1}$. Is this the only approach?