Question

Customers arrive at Lisa’s Hair Salon during a day in any particular business hour according to a Poisson distribution with a rate of ? = 3 per hour. Now the salon has two barbers who can each service a customer in exactly 30 minutes. Suppose a customer, Amy, arrives at 2:00pm and finds both barbers idle.

(a) What is the probability that we will observe customers waiting before 2:30pm?

(b) What is the probability that Amy observes the next customer (i.e. the customer after her) arrival before 2:15pm?

(c) What is the probability that Amy will find the shop empty when she leaves?

Answer 1

Solution

Back-up Theory

If a random variable X ~ Poisson(λ), i.e., X has Poisson Distribution with mean λ then

probability mass function (pmf) of X is given by P(X = x) = e ^{– λ}.λ^x/(x!) …………….……..(1)

where x = 0, 1, 2, ……. , ∞

Values of p(x) for various values of λ and x can be obtained by using Excel Function,

Statistical, POISSON……………….. ………………………………………………………. (1a)

If X = number of times an event occurs during period t,

Y = number of times the same event occurs during period kt, and X ~ Poisson(λ),

then Y ~ Poisson (kλ) …………………………………………………………………..….. (2)

Now to work out the solution,

Let X = number of customers who arrive at Lisa’s Hair Salon in an hour.

Then, we are given X ~ Poisson(λ), where λ = 3 …………………………………………… (3)

Part (a)

Let Y = number of customers who arrive at Lisa’s Hair Salon in ½ hour (i.e., 30 minutes).

Then, vide (2) and (3), Y ~ Poisson(1.5) [30 minutes = ½ hour …………………………… (4)

Given, Amy arrives at 2:00 pm and finds both barbers idle, we will observe customers waiting before 2:30 pm if number of customers arriving in 30 minutes (2:00 to 2:30) is 2 or more, noting that one barber would be busy servicing Amy for 30 minutes since service time is 30 minutes and the second barber can take care of one more customer arrival.

So, probability that we will observe customers waiting before 2:30 pm

= P(Y ≥ 2)

= 0.4422 [vide (1a)] Answer

Part (b)

Let Z = number of customers who arrive at Lisa’s Hair Salon in ¼ hour (i.e., 15 minutes).

Then, vide (2) and (3), Y ~ Poisson(0.75) [15 minutes = ¼ hour …………………………… (5)

So, probability that Amy observes the next customer (i.e. the customer after her) arrival before 2:15 pm

= P(Z = 1)

= 0.3543 [vide (1a)] Answer

Part (c)

Probability that Amy will find the shop empty when she leaves

= P(no arrival in 30 minutes)

= P(Y = 0)

= 0.2231 [vide (1a)] Answer

DONE