On a typical weekday morning, customers arrive at the post office independently and at a rate of 3 per 10 minute period.
(i) State, in context, a condition needed for the number of customers who arrived at the post office during a randomly chosen period of 30 minutes to be well modelled by a Poisson distribution.
(ii) Find the probability that no more than 4 customers arrive between 11.00 a.m. and 11.30 a.m.
(iii) The period from 11.00 a.m. to 11.30 a.m. on a Tuesday morning is divided into 6 periods of 5 minutes each. Find the probability that no customers arrive in at most one of these periods.
The post office opens for 3.5 hours each in the morning and afternoon and it is noted that on a typical weekday afternoon, customers arrive at the post office independently and at a rate of 1 per 10 minute period. Arrivals of customers take place independently at random times.
(iv) Show that the probability that the number of customers who arrived in the afternoon is within one standard deviation from the mean is 0.675, correct to 3 decimal places.
(v) Find the probability that more than 38 customers arrived in a morning given that a total of 40 customers arrived in a day.
(vi) Using a suitable approximation, estimate the probability that more than 100 customers arrive at the post office in a day.