Several exact efficiency gates show detected and missed photons as one count ledger. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Half efficiency clicks six of twelve
The detector receives 12 photons. Efficiency 1/2 gives 6 clicked and 6 missed.
12⋅21=6,6+6=12
Two-thirds efficiency clicks eight of twelve
Efficiency 2/3 sends 8 photons to clicked and 4 to missed.
12⋅32=8,8+4=12
Three-quarters efficiency clicks nine of twelve
Efficiency 3/4 gives 9 clicked photons and 3 missed photons.