Counting only the trucks at a toll

Question

Vehicles pass a toll booth as a Poisson process at rate $30$ per hour. Independently, each vehicle is a truck with probability $0.2$ (otherwise a car).

What is the probability that exactly $4$ trucks pass in the next $30$ minutes?

Accepted Answer

When you keep only some events of a Poisson process by flipping an independent coin for each one, the survivors still form a Poisson process. This is called thinning, and the thinned rate is the original rate times the keep-probability. So the truck stream is Poisson with rate \lambda{	ext{truck}} = 30 	imes 0.2 = 6 	ext{ per hour}. Now just work with the truck process. Thirty minutes is half an hour, so the mean number of trucks is \mu = \lambda{	ext{truck}}\, t = 6 	imes 	frac12 = 3. The chance of exactly 4 trucks is P(N = 4) = e^{-3}\,\frac{3^4}{4!} = e^{-3}\,\frac{81}{24} = 3.375\,e…

Counting only the trucks at a toll

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