The calculation of such probabilities has been studied extensively by communications engineers under the name _$M$-ary orthogonal signaling_
where the model is that one of $M$ equal energy orthogonal signals 
being transmitted and the
receiver attempting to decide which one was transmitted by examining
the outputs of $M$ filters _matched_ to the signals. Conditioned
on the identity of the transmitted signal, the sample outputs of
the matched filters are (conditionally) independent unit-variance
normal random variables. The sample output
of the filter matched to the signal transmitted is a
$N(\mu,1)$ random variable while the outputs of all the other filters
are $N(0,1)$ random variables. 

The _conditional_ probability of a correct decision (which in the
present context is the event $E = \{X_0 > \max_n X_n\}$) conditioned
on $X_0 = \alpha$ is 
$$P(E \mid X_0 = \alpha) = \prod_{i=1}^n P\{X_i < \alpha\} 
= \left[\Phi(\alpha)\right]^n$$
where $\Phi(\cdot)$ is the cumulative probability distribution 
of a standard normal random variable, and hence the unconditional
probability is 
$$P(E) = \int_{-\infty}^{\infty}P(E \mid X_0 = \alpha)
\phi(\alpha-\mu)\,\mathrm d\alpha
= \int_{-\infty}^{\infty}\left[\Phi(\alpha)\right]^n
\phi(\alpha-\mu)\,\mathrm d\alpha$$

Work in progress