TeX source:

\Pr\{0\} = \sum_i p_i\Pr\{0|i\} = \sum_i p_i\langle0|\psi_i\rangle\langle\psi_i|0\rangle = \langle0| \cdot \sum_i p_i|\psi_i\rangle\langle\psi_i| \cdot |0\rangle = \langle0|\rho|0\rangle.