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I have the following question: Buses arrive at a city according to a Poisson process with a rate of 5 per hour. What is the probability that the fifth bus of the day arrives after midday given they start arriving at 9 a.m.

What I think is the correct way to go about this question is that I calculate: $ P(N(0,3)\le4) $ The probability that less than or equal to four buses occur between 9am to midday. Because we have to reach the 4th bus in that time before the 5th bus can arrive. Provided I am correct with that notion the formula would look like: $ \sum_i_=_0^4 {15^i e^{-15}\over i!} $.

Thank you

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up vote 1 down vote accepted

a short answer: yes your answer is correct. the number of arrival before 12 is distributed as Poisson (15), so you need this to be strictly smaller than 5.

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