Probability error for antipodal systems and orthogonal systems
For equally likely
binary signals $s_0(t)$ and $s_1(t)$ of energies $E_0$ and $E_1$ respectively
in an AWGN channel with two-sided power spectral density $\frac{N_0}{2}$, the
error probability is given by
$$P_e = Q\left(\sqrt{\frac{E_0 + E_1 - 2 \langle s_0, s_1 \rangle}{2N_0}}\right)
\tag{1}$$
where
$\displaystyle \langle s_0, s_1 \rangle = \int_{-\infty}^\infty s_0(t)s_1(t)\,\mathrm dt$ is the inner product of the two signals.

For antipodal signals, $E_0 = E_1 = E$ and $\langle s_0, s_1 \rangle = -E$ giving
everyone's favorite answer $Q\left(\sqrt{2E/N_0}\right)$ while for equal
energy orthogonal signals, the inner product is $0$ giving everybody's second
favorite answer $Q\left(\sqrt{E/N_0}\right)$. For some details of the proof
of $(1)$,
see this answer that I wrote
some time ago.  For even more detail,
see 
this Lecture Note of mine.