Assume that customer arrivals at a barber shop are random and independent of one another, and the number of customer arrivals at a barber shop and the time until the next customer arrives is independent.
(a) In city A, on average, 3 customers arrive at a barber shop every hour. Using an appropriate probability distribution,
(i) find the probability that at least 5 customers arrive at a barber shop every hour.
(ii) A sample of 25 barber shops in city A was obtained. Find the probability that at least 3 barber shops were visited by at least 5 customers.
(iii) A customer has just arrived in a barber shop. Find the probability that the time, until the next customer arrives will be at most 2 hours (from now).
(b) Let X be the number of female customers who arrive at a barber shop before the first male customer arrives. The probability of a female customer arriving at the barber shop is 0.75.
(i) Find P(X ≥ 8).
(ii) In a random sample of size 36, find the normal approximation for P(2.5 ≤ ᄁ ≤ 3.5), where ᄁ is the sample mean.