Probabilities?

Question

Customers's arrival rate is 1/2 per min. Find the probability that the 250th customer enters on the last hour of an 8-hr working day.

Answer 1

We have here a Poisson process. Since we're taking it over a long time period, we can use the Normal approximation.

P(250th customer in last hour)
= P(customers after 8 hours >= 250) - P(customers after 7 hours >= 250)
The number of customers after 8 hours is distributed as Poi(240), and the number after 7 hours is distributed as Poi(210). So we can approximate these probabilities as
P(Z >= (249.5-240)/√240) - P(Z >= (249.5-210)/√210)
= P(Z >= 0.613222) - P(Z >= 2.725759)
= 0.269865 - 0.003208
= 0.267 (3 d.p.) or 26.7%.