Determining the average probability of error
Note that
$$
\mathbb{P}(n_m+i_m&lt;-1\cap i_m = \frac{-1}{2})
$$
is equal to
$$
\mathbb{P}(n_m &lt; -0.5)\mathbb{P}(i_m = -0.5) = 0.25\mathbb{P}(n_m &lt; -0.5)
$$
since $n_m$ and $i_m$ are independent.
Another way to obtain the solution is to add the the probability of these three disjoint events:

$n_m &lt; -1.5$,
$i_m = 0$ and $-1.5 &lt; n_m &lt; -1$,
$i_m = -0.5$ and $-1 &lt; n_m &lt; -0.5$

In this case, we can add the three probabilities because the events are disjoint. To keep them disjoint, it is important to keep the noise ranges disjoint too; otherwise, you would be adding them multiple times.