Question

Customers arrive at Lily’s Barber Shop during a day in any particular business hour according to a Poisson distribution with a rate of ? = 2 per hour. Now the shop has two barbers who can each service a customer in exactly 15 minutes. Suppose a customer, Lisa, arrives at 3:00pm and finds both barbers idle. Find

(i) The probability that we will observe customers waiting before 3:15pm.

(ii) The probability that Lisa will find the shop empty when she leaves.

Answer 1

a) Given, lisa arrives at 3:00pm and finds both barbers idle, we will observe customers waiting before 3:15pm.

if number of customers arriving in 15 minutes (3:00 to 3:15) is 2 or more, noting that one barber would be busy servicing Lisa for 15 minutes since service time is 15 minutes and the second barber can take care of one more customer arrival.

probability that we will observe customers waiting before 3:15pm.

The provided mean is λ=2.

We need to compute Pr(X≥2). Therefore, the following is obtained:

Pr(X≥2) = 1−Pr(X<2)

=1−Pr(X≤1)

= 1 - 0.406

= 0.594

b) Probability that Lisa will find the shop empty when she leaves.