Fish from two tributaries into one net

Question

A net sits where two tributaries join. Fish drift in from the first tributary as a Poisson process at $10$ per hour and from the second, independently, at $14$ per hour.

What is the probability that exactly $3$ fish enter the net in the next $5$ minutes?

Accepted Answer

Two independent Poisson inflows combine into one Poisson process with the summed rate: \lambda = 10 + 14 = 24 	ext{ per hour}. Five minutes is 	frac{5}{60} = 	frac{1}{12} of an hour, so the mean number of fish in the window is \mu = \lambda t = 24 	imes 	frac{1}{12} = 2. The chance of exactly 3 fish is P(N = 3) = e^{-2}\,\frac{2^3}{3!} = e^{-2}\,\frac{8}{6} = 	frac43\,e^{-2} \approx 0.180. About an 18\% chance. Whether the fish came from the left or right tributary never mattered once we merged the two flows into a single rate. Takeaway. Superposition turns "two sources into one net" in…

Fish from two tributaries into one net

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