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Cafe shop opens at 8:00. Customers arrive at rate of 35 customers per hour. Find the probability that between 80 and 100 customers arrive between 10:00 and 13:00.

Just looking if I am correctly thinking.

I have modeled the question like this: $$P(N_{5} - N_{2} <= 100 | N_{5} - N_{2} > 79) = P(N_{3} <=100 | N_{3} > 79)$$

Now, I would apply $ P(A|B) = \frac {P(A∩B)}{P(B)} $, so $ P(B) = P(N_{3} > 79)$ and $P(A∩B) = P(N_{3} <= 100)P(N_{3}>79)$

Would it be the correct way to solve the problem?

cineel
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vekeras
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    I don't really know why you're conditioning at all. It is just $P(N_5-N_2 \in [80,100])$ (possibly adjusting endpoints depending on what "between" means to you) and $N_5-N_2$ is Poisson(105) distributed. – Ian Dec 05 '21 at 15:43
  • How could I proceed with $P(N_{5} - N_{2} ∈ [80,100])$? Is it simply $P(N_{5} - N_{2} ∈ [80,100]) = P(N_{3} >= 80)P(N_{3} <=100)$? – vekeras Dec 05 '21 at 15:56
  • There's no conditioning to do, you just sum up values of the PMF. – Ian Dec 05 '21 at 15:56
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    How are $N_5$ and $N_2$ defined? – cineel Dec 05 '21 at 15:58

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