Photons a detector actually catches

Question

Photons strike a detector as a Poisson process at rate $100$ per second. Because the detector is imperfect, each photon is independently registered with probability $0.05$ (its quantum efficiency) and otherwise missed.

What is the probability that exactly $2$ photons are registered in a $0.5$ -second interval?

Accepted Answer

Registering each photon independently with a fixed probability is exactly thinning. The registered photons form their own Poisson process with rate \lambda{	ext{reg}} = 100 	imes 0.05 = 5 	ext{ per second}. Over half a second the mean number registered is \mu = \lambda{	ext{reg}}\, t = 5 	imes 0.5 = 2.5. The chance of exactly 2 registrations is P(N = 2) = e^{-2.5}\,\frac{2.5^2}{2!} = 3.125\,e^{-2.5} \approx 0.257. So about a 26\% chance. The huge stream of missed photons is irrelevant to the count of the ones that made it through. Takeaway. Detector efficiency is just thinning: multiply …

Photons a detector actually catches

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