= Solution
Bosonic <creation operators> commute, so the cross terms cancel:
$$
a_1^\dagger a_2^\dagger|0\rangle
=\frac12[(b_1^\dagger)^2-(b_2^\dagger)^2]|0\rangle
=\frac{|2,0\rangle-|0,2\rangle}{\sqrt2}.
$$
Thus
$$
\boxed{P(2,0)=P(0,2)=\frac12,\qquad P(1,1)=0.}
$$
The square root of two in $(b^\dagger)^2|0\rangle=\sqrt2|2\rangle$ is essential. Destructive <interference> removes the one-particle-per-output amplitude; the particles do not independently choose ports.
For <photons> this is the ideal <Hong-Ou-Mandel effect>. Its laboratory test combines one <photon> from each input at a balanced <beam splitter> and scans relative arrival delay while measuring output coincidences. Spatial, <polarization> and spectral indistinguishability must be controlled; distinguishable <photons> restore a nonzero coincidence baseline. The original experiment used down-converted <photons>, a translated <beam splitter> and coincidence counting, with an independently estimated accidental-coincidence background.<Measurement of Subpicosecond Time Intervals between Two Photons by Interference by C. K. Hong and others>[Physical Review Letters 59 (1987), pp. 2044–2046, figure 1 and equations (2), (9): setup, balanced splitting cancellation, delay scan and accidental subtraction]{ref}
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