Question

Three barbers work at a barbershop. Based on estimations, the barbershop is idle 1 time out of 15; 2/15 of the time there is one customer; 3 times out of 15 there are two customers; and 4/15 of the time, there are three customers. Each customer yields a net revenue of 10 dollars.

Let X be a random variable defined as the number of customers

a) Determine the probability distribution of X

b) Determine the cumulative distribution function of X

c) Calculate the probability that: i) All three barbers are working. ii) At least one of the barbers is working

Answer 1

Total number of barbers = 3

Probability of barber shop being Idle( or 0 customers) , P(0)= 1/15

Probability of barber shop with 1 customers , P(1)= 2/15

Probability of barber shop with 2 customers, P(2) = 3/15

Probability of barber shop with 3 customers, P(3) = 4/15

A. Probability Distribution of X is given by,

P(X=x) = (x + 1) /15

B. Cumulative distribution function of x is given by,

CDF = P(X<=x) = P(X<=3) =   = 1/15 + 2/15 + 3/15 + 4/15 = 10/15

C. 1. All three barbers will be working, when there are atleast 3 customers in the barber shop

The probability that there are more than 3 customers in the shop is given by,

P(X>=3) = 1 - P(X<3) = 1 - (P(0)+P(1)+P(2) ) = 1- ( 1/15 + 2/15 + 3/15) = 1 - (6/15) = 9/15

Therefore, the probability that all the three barbars are working is given by , P(X>=3) = 9/15.

2. So, when there is only one customer, one barber has to work.

Therefore, the proabability that atleast one of the barbers is working = 1 - probabiilty of no barber working = 1 - P(0)

= 1 - 1/15 = 14/15