Average probability of bit error is

$P(e)=Q(\sqrt[\phantom{2}]{2{}_b})$

Remember; output SNR of MF$/$correlator: $\frac{2{}_b}{N_0}$

$P_b$ depend on SNR only, not the waveform $g(t).$

In terms of distance;

$d_{12}=2\sqrt[\phantom{2}]{E_b}=>P_b=Q(\sqrt[\phantom{2}]{\frac{d_{12}^2}{2N_0}})$

HW: What if $s_1$ and $s_2$ are not equiprobable, i$.$e$.P(s_1)=q,P(s_2)=(1-q)?$

$P(e)$ for MAP receiver?

$P(e)$ for ML receiver?