= Solution
The joint <Born rule> uses the <tensor product> of the two outcome bras. Its amplitude is
$$
(\langle0|\otimes\langle\theta_0|)|\psi\rangle_{AB}=\frac45\langle\theta_0|0\rangle=\frac45\cos\theta.
$$
The other source component vanishes on contraction with $\langle0|$ on A. Therefore
$$
\boxed{\Pr(A=0,B=\theta_0)=\frac{16}{25}\cos^2\theta.}
$$
This is a joint <probability>: $\Pr(A=0)=16/25$, while $\Pr(B=\theta_0\mid A=0)=\cos^2\theta$. Dropping the first factor would give only the <conditional probability>.
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