= Solution
Applying both final splitters gives
$$
\boxed{|\Psi_{II}\rangle=-\frac12|\gamma\rangle+\frac14\left(-3|cc\rangle+i|cd\rangle+i|dc\rangle-|dd\rangle\right).}
$$
The charged-particle coincidence probabilities are $9/16,1/16,1/16,1/16$ in the displayed order, and the annihilation <probability> remains $1/4$. In particular
$$
\boxed{P_{II}(D^+,D^-)=\frac1{16}.}
$$
Conditioning on surviving pairs would change this number to $(1/16)/(3/4)=1/12$; it would not change the amplitudes of the unconditioned all-trials state above.
\Image[/past-exam-of-the-quantum-information-course-of-the-university-of-cambridge/2013/paper-1-hardy-probabilities.png]
{title=Hardy interferometer: unconditional coincidence and annihilation probabilities}
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