Past exam of the computer science course of the University of Oxford / 2019 / Quantum information / 2 / b / i / Solution

The joint Born rule uses the tensor product of the two outcome bras. Its amplitude isThe other source component vanishes on contraction with $⟨0|$ on A. ThereforeThis is a joint probability: $\mathrm{Pr}(A=0)=16/25$, while $\mathrm{Pr}(B={\theta }_0\mid A=0)={\cos }^2\theta$. Dropping the first factor would give only the conditional probability.