If your system is prepared in the state ##\hat{\rho}## then the probability for the ##x##-component of the spin being ##\sigma_x## is
$$P(\sigma_x)=\langle \sigma_x|\hat{\rho}|\sigma_x \rangle.$$
If your spin is prepared in a pure state, represented by the ket ##|\psi \rangle##, then the statistical operator is the projection operator ##\hat{\rho}=|\psi \rangle \langle \psi|##, and the general formula gives you the Born rule for pure states, as it should be: