With coherent BPSK, the two possible transmitted signals correspond to opposite points with equal energy $E_b$.

After matched filtering or correlation, the receiver obtains a scalar decision variable. In AWGN it can be modeled as a Gaussian random variable centered at

$$+\sqrt{E_b}$$

or

$$-\sqrt{E_b},$$

depending on the transmitted bit. With equally likely bits, the optimal threshold is zero.

Define the Gaussian tail function

$$Q(x)=\frac{1}{\sqrt{2\pi}}\int_x^\infty e^{-u^2/2},du.$$

The coherent BPSK bit error probability is

$$P_b=Q!\left(\sqrt{\frac{2E_b}{N_0}}\right).$$

Increasing $E_b/N_0$ separates the two Gaussian decision distributions relative to their noise spread, so errors become rapidly less likely.

This formula is specific to the stated assumptions: coherent BPSK, AWGN and equally likely independent bits. Different modulations, fading channels or receiver uncertainties produce different error probabilities.