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.

1212=6,6+6=1212\cdot\frac{1}{2}=6,\quad 6+6=12
Detector efficiency scan rowClicked and missed bars close the count ledger.1/2clicked1/2misseddetector

Two-thirds efficiency clicks eight of twelve

Efficiency 2/3 sends 8 photons to clicked and 4 to missed.

1223=8,8+4=1212\cdot\frac{2}{3}=8,\quad 8+4=12
Detector efficiency scan rowThe total arrivals stay fixed while efficiency changes.2/3clicked1/3misseddetector

Three-quarters efficiency clicks nine of twelve

Efficiency 3/4 gives 9 clicked photons and 3 missed photons.

1234=9,9+3=1212\cdot\frac{3}{4}=9,\quad 9+3=12
Detector efficiency scan rowEfficiency is a gate on counts, not a detector label.3/4clicked1/4misseddetector