QB OF PROBABILITY

VINAYAKA MISSIONS UNIVERSITY

VINAYAKA MISSION’S KIRUPANANDA VARIYAR ENGINEERING COLLEGE

DEPARTMENT OF MATHEMATICS

QUESTION BANK

PROBABILITY AND QUEUEING THEORY

II B. E., ELECTRONICS  AND COMMUNICATION ENGINEERING

BATCH (2008-2012)

ACADEMIC YEAR 2009 - 2010

UNIT – I

PROBABILITY AND RANDOM VARIABLES

PART A

1.   A is known to hit the target in 2 out of 5 shots where as B is known as the target in 3 out of 4 shots. Find the probability of the target being hit when they both try?

2.    If the probability that a communication system has high selectivity is 0.54 and the probability that it will have high fidelity is 0.81 and the probability that it will have both is 0.18. Find the probability that a system with high fidelity will have high selectivity.

3.   If A and B are events with.

4.   State the properties of probability density function.

5.    A and B are events such that.

6.   What is the chance that a leap year selected at random will contain 53 Sundays?

7.   If  , prove that.

8.   A bag contains 3 red and 4 white balls. Two draws are made without replacement. What is the probability that both the balls are red?

9.   A bag contains 8 white and 10 black balls. Two balls are drawn in succession. What is the probability that first is white and second is black?

10.  Four cards are drawn without replacement. What is the probability that they are all Aces?

11.  State Baye’s theorem.

12.  Define Cumulative distribution function.

13.  Let X be a random variable with E(X) = 1 and E[X(X-1)] = 4, Find V(X) and V (2-3X).

14.  Define probability mass function.

15.  Suppose that the random variable X assumes three values 0, 1 and 2 with probabilities 1/3, 1/6 and ½ respectively. Obtain the distribution function of X.

16.  If X is a random variable with pdf f(x) and “a” and “b” are constants then            

                    .

17.  Let events A and B be independent with P(A) = 0.5 and P(B) = 0.5, Find the chance that neither     

    A nor B occurs.

18.  If, a, h are constants.

19.  X and Y are independent random variables with variance 2 and 3. Find the variance of 3X+4Y.

20.  Given that the pdf of a Random Variable X is f(x) = Kx, 0<x<1 .Find k and P (X>0.5).

21.  Define moment generating function of a discrete and continuous random variable X.

22.  Define moments of a random variable X about any point and about origin.

23.  Prove that the Probability of an impossible event is Zero.

24.  Find the m.g.f. of a random variable X having the p.d.f.

25.  Define mean and variance in terms of expectations.

PART B

1.      From a group of 3 Indians, 4 Pakistanis and 5 Americans, a sub-committee of four people is selected by lots. Find the probability that the sub-committee will consist of 

(i)                 2 Indians and 2 Pakistanis

(ii)               1 Indian, 1 Pakistani and 2 Americans

(iii)             4 Americans                                                                                        (12 marks)

2.      If A and B are independent events, prove that 

 (i)  and B are independent

(ii) A and are independent

(iii)  and  are independent                                                                             (12 marks)

3.       (i)  An urn contains 5 balls. Two balls are drawn and are found to be white. What is the   

      probability of all the balls being white?                                                                      (6 marks)

 (ii) State and prove Baye’s theorem.                                                                               (6 marks)

4.      A random variable X has the following probability function.

Values of X	0	1	2	3	4	5	6	7	8
P(x)	a	3a	5a	7a	9a	11a	13a	15a	17a

(i)                 Determine the value of ‘a’

(ii)               Find

(iii)             Find the distribution function of X.                                                                    (12 marks)

5.      (a) For the following density function find

                (i) the value of ‘a’

                      (ii) mean and variance.                                                                                      (6 marks)

      (b) A man draws 3 balls from an urn containing 5 white and 7 black balls. He gets Rs.10 for    

            each white ball and Rs.5 for each black ball. Find his expectation.                           (6 marks)

6.      (i) The probability distribution function of a random variable X is

                       

             Find the cumulative distribution function of X.                                                        (6 marks)

      (ii)  A continuous random variable X has the probability density function

              Find the mean and variance of X.                                                                            (6 marks)

7.      Find the moment generating function of the random variable with the probability law

          Find the mean and variance.                                   (12 marks)

8.      (i) If  are independent random variables then

                                                                            (6 marks)

(ii) If a random variable X takes the value 1,2,3,4 such that          

       2P(X=1) =3P(X=2) =P(X=3) =5P(X=4). Find the probability distribution of X                                                                                                                                                            (6 marks)

9.      The contents of Urns I, II, and III are as follows. 2 white, 3 black and 4 red balls, 3 white, 2 black and 3 red balls and 4 white, 1 black and 3 red balls. An urn is chosen at random and two balls are drawn. They happen to be white and red. What is the probability that they come from urns I, II or III?                                                                                                      (12 marks)

10.  (a)    A random variable X has the probability density function

                       

                Find                           (6 marks)

    (b)     A continuous random variable X has probability density function

                   . Find ‘a’ and ‘b’ such that

                                                                                              (6 marks)

UNIT II

STANDARD DISTRIBUTIONS

PART A

1.      The mean and variance of a binomial distribution are 4 and 4/3. Find P (X≥1).

2.      10 coins are thrown simultaneously. Find the probability of getting atleast 7 heads.     

3.      A perfect cubic die is thrown a large number of times in sets of 8. The presence of a 5 or 6 is treated as a success. In what proportion of the sets can we expect 3 successes?

4.      If X is a Poisson variate such that, find P(X=0) and P(X=3).  

5.      If X and Y are independent Poisson variates with means respectively, find the probability that X=Y.

6.      Six coins are tossed 6400 times. Using the Poisson distribution, what is the approximate probability of getting six heads 10 times?

7.      Using Poisson distribution, find the probability that the ace of spades will be drawn from a pack of well shuffled cards atleast once in 104 consecutive trials.

8.      Define geometric distribution and write its mean and variance.

9.      If the probability that a target is destroyed on any one shot is 0.5, what is the probability that it would be destroyed on 6^th attempt?

10.  If the probability is 0.05 that a certain kind measuring device will show excessive drift, what is the probability that the sixth of these measuring devices tested will be the first to show excessive drift?

11.  Prove that for uniform distribution.

12.  Show that for a uniform distribution  , the m.g.f about origin is

13.  If X is uniformly distributed with mean 1 and variance 4/3, find P [ X < 0].

14.  Subway trains on a certain line run every half hour between mid-night and six in the morning. What is the probability that a man entering the station at a random time during this period will have to wait at least twenty minutes?

15.  State and prove memoryless property of the exponential distribution.

16.  The time (in hours) required to repair a machine is exponentially distributed with parameter =1/2. What is the probability that a repair takes atleast 10 hours given that its duration exceeds 9 hours?

17.  Define Gamma distribution

18.  Define Weibull distribution.

19.  The life time of a component measured in hours is Weibull distribution with parameter.  Find the mean life time of the component.

20.  Define transformation of one-dimensional random variable.

21.  The pdf of a random variable X is f(x) =2x, 0<x<1, find the pdf of Y=3x+1.

22.  If ‘X’ is exponentially distributed with parameter 1, find the probability distribution function of.

23.  State Chebychev’s inequality.

24.  If ‘X’ is exponentially distributed with parameter λ, find the probability distribution function of.

25.  The amount of time that a camera will run with out having to be reset is a random variable having exponential distribution with q = 50 days. Find the probability that such a camera will have to reset in less than 20 days

PART B

1.   (i) 6 dice are thrown 729 times. How many times do you expect atleast three dice to show a five or a six?                                                                                                                   (6 marks)

    (ii) A random variable X has a uniform distribution over (-3, 3), compute

                                                                           (6 marks)

2.   (i) If X is a Poisson variate such that P(X=2) =9P(X=4) +90P(X=6).  Find the variance.

                                                                                                                                                (6 marks)

    (ii) If the life X (in years) of a certain type of car has a Weibull distribution with the parameter, find the value of the parameter, given that probability that the life of the car exceeds 5 years is. For these values of, find the mean and variance of X.                                                                                                                                   (6 marks)

3.   (i) Find the mean and variance of geometric distribution.                                                 (6 marks)

    (ii) With mean 120 days. Find the probability that such a watch will

(a)         have to be set in less than 24 days and

(b)         not have to be reset in atleast  180 days                                                            (6 marks)

4.   (i)    If X is a uniformly distributed r.v. in, find the pdf of Y = tanX.              (6 marks)

     (ii)   A car hire firm has two cars which it hires out daily. The number of demand for a car on

            each day is distributed as poisson variate with mean 1.5. Obtain the proportion of days on

             which a) there was no demand b) demand is refused

                                                                        (6 marks)

5.       (i) Find the mean and variance of uniform distribution.                                                  (6 marks)

      (ii) A die is cast until 6 appears. What is the probability that it must be cast more than five times?                                                                                                                               (6 marks)

6.      (i) Obtain the moment generating function of Gamma distribution and hence find its mean and Variance.                                                                                                                  (6 marks)

      (ii) A pair of dice is rolled 900 times and X denotes the number of times a total of 9 occurs. Find P(80≤X≤120) using Chebychev’s inequality                                                 (6 marks)

7.      (i) Find the moment generating function of exponential distribution and hence find its mean and variance.                                                                                                                 (6 marks)

      (ii) The random variable X has the pdf f(x) = e^-x , x≥0. Find the density function of

a)        Y = 2X + 3

b)        Y = X²                                                                                 (6 marks)

8.      (i) Define Weibull distribution and find its mean and variance.                                     (6 marks)

      

(ii)        Fit a Poisson distribution to the following data and calculate the theoretical frequencies.                                                                                                                                       (6 marks)

Deaths	0	1	2	3	4
Frequency	122	60	15	2	1

                                                           

                                               

9.      (i) A random variable X has probability density function. Use Chebychev’s inequality to show that  and show also that the actual probability is given by.                                                                                                                    (6 marks)

      (ii) If  is the probability density function of a random variable X, find the probability density function of Y = 2X – 3.                                                                      (6 marks)

10.  (i) Two unbiased dice are thrown. If X is the sum of the number showing up, prove that     , using Chebychev’s inequality.                                                            (6 marks)

  (ii) If the random variable X is uniformly distributed over (-1,1). Find the density function of .                                                                                                           (6 marks)

UNIT III

TWO DIMENSIONAL RANDOM VARIABLES

PART A

1. If X and Y are random variables having the joint density function .
2. Find the acute angle between the two lines of regression.
3. Find k if the joint probability density function of a bivariate random variable (X, Y) is given by

                       

4. Two random variable X and Y have joint probability density function

                       

Find E(XY).

5. State the Liapounoff’s form of Central of theorem.
6. State the Lindberg-Levy’s form of Central limit theorem.
7. Explain scatter diagram
8. Write the properties of joint distribution function.
9. Define marginal density function for the continuous random variables X and Y.
10. Define joint pdf for continuous random variable X and Y.
11. Differentiate between correlation and regression
12. Define covariance.
13. Find the marginal density functions of X if.
14. Define correlation and given an example.
15. Prove that.
16. The two regression equations of the variables x and y are x=19.93-0.87y and y=11.64-0.50x Find mean of x
17. If ,i=0, 1,2,…50 are independent random variables each having a Poisson distribution with parameter , evaluate .
18. Define two dimensional discrete and continuous random variables.
19. Explain when we say the two random variable X and Y are independent.
20. Define conditional probability distribution of the two random variable X and Y.
21. The following regression equations were obtained from a correlation table;

                  y = 0.516x + 33.73;

                        x = 0.512y + 33.52,   Find the value of the correlation coefficient

22. State the properties of regression coefficient
23. Define joint pdf of two discrete random variables X and Y.
24. If X and Y are independent, prove that COV(X,Y) = 0
25. If the joint pdf of (x,y) is given by f(x,y)=k, 0≤X<Y≤2,Find k

PART B

1. (i) From the following joint distribution of X and Y, find the marginal distributions.                                                                                                                                                         

X Y	0	1	2
0	3/28	9/28	3/28
1	3/14	3/14	0
2	1/28	0	0

(6 marks)                    

(ii) The joint density function of X and Y is                                                                                                                                                               Verify whether X and Y are independent                                                        (6 marks) 

2. If X and Y are two random variables with joint pdf                                                                                                                            

Find     i)     

                        ii)

                                              ii)                                                                 (12 marks) 

3. a) Given the following joint density function

                                             

            i) Find the marginal density functions of X and Y.

ii) Find the conditional density function of Y given X=x.                 (6 marks)

            b) Calculate Karl Pearson’s coefficient of correlation                                                          

Price                :           10        11        13        15        18

Demand          ::          60        52        48        40        30                                (6 marks)

                     

4. Two random variables X and Y have the joint pdf                                                                                                                             Find the COV(X,Y)                                                                                        (12 marks)
5. In a partially destroyed laboratory record of an analysis of correlation data, the following results are only legible

Variance of X = 9

Regression equations are

            8X – 10Y + 66 = 0

              40X – 18Y - 214 = 0

                        Find     i)          the mean values of X and Y

ii)                  the correlation coefficient between X and y

iii)                the standard deviation of Y                                        (12 marks)

6. Let X, Y and X be uncorrelated random variables with zero means and standard deviations 5, 12 and 9 respectively. If U = X+Y and V=X+Y, find the correlation coefficient between U and V.                                                                                                             (12 marks)
7. (i) Calculate the correlation coefficient for the following heights (in inches) of fathers X and    

     their sons Y

X	65	66	67	67	68	69	70	72
Y	67	68	65	68	72	72	69	71

(6 marks)

            (ii) Let X and Y have j.d.f . Find the marginal density functions.

                  Find the conditional density function of Y given X=x.                                       (6 marks)

8. If the joint pdf of (X, Y) is given by.         (12 marks)
9. (i) The life time of a certain brand of a tube light may be considered as a random variable with mean 1200 h and standard deviation 250 h. Find the probability, using central limit theorem, that the average life time of 60 lights exceeds 1250h.                                 (6 marks)

(ii) If X and Y each follow an exponential distribution with parameter 1 and are

     independent, find the pdf of  U = X – Y.                                                             (6 marks)

10. (i) The joint distribution of X and Y is given by

                       

                        Find the marginal distributions.                                                                      (6 marks)

            (ii) Given            

                  (a) Find C (ii) Find f(x)  (iii) Find f(y/x).                                                              (6 marks)

                                                              UNIT IV

RANDOM PROCESSES, MARKOV CHAIN

PART A

1.Define irreducible Markov chain?

2.What is continuous random sequence? Give an example.

3.What is stochastic matrix? When is it said to be regular?

4.State any four properties of Poisson process.

5.What is meant by steady-state distribution of Markov Chain?

6.Define preventive maintenance downtime.

7.A system consists of 3 identical unit s connected in parallel. The unit reliability factor is 0.9. If the unit failures are independent of one another and the successful operation of the system depends on the satisfactory performance of any one unit, determined the system reliability.

8.What is a Markov chain?

9.What will be the superposition of n independent Poisson processes with respective average rates ?

10.Define instantaneous availability A(t) of a component. How does it differ from reliability of component?

11.Define strict sense and wide sense stationary process.

Show that the sum of two independent Poisson process is a Poisson process.

12.The transition probability matrix of a Makov chain with three state 0,1 and 2 is    

With initial distribution .Find .

13.Define one-step transition probability.

14.Give comparison between random variables and random process.

15.Define Poisson process.

16. A radar emits particles at the rate of 5 per minute according to Poisson distribution. Each particles emitted has probability 0.6. Find the probability that 20 particles are emitted in a 4 minute period.

17.Consider the Markov chain with TPM given by

           

Show that it is ergodic.

18.Let  be a Stochastic matrix. Check whether it is regular.

19.Define Birth and Death process.

20.Define renewal process.

21.State Chapman-Kolmogorov theorem.

22.Define reliability function of time t.

23.Define Hazard rate function.

24.Give three properties of reliability function of t.

25.Define limiting distribution.

PART B

1. a) Explain the classification of Random process. Give an example to each class.

(6 marks)

   b) Explain the properties of auto correlation.                                              (6 marks)

2.(i)Prove that the random process X(t) = A cos(ωt + θ) is a wide sense stationary where A and ω are constants and θ is uniformly distributed random variable over( 0,2п).

(6 marks)

    (ii) Derive the expectation of N(t).                                                             (6 marks)

3.a) Prove that the difference of two independent Poisson process is not a Poisson process

    (6 marks)

b)     Prove that i)                                                                                                                                ii)                                                 (6 marks)

4.The probability distribution of the process {X(t)} is given by

     

Show that it is not stationary.                                                                           (12 marks)

5.Two random process X(t) and Y(t) are defined by X(t)= Acost + B sint and Y(t)= B cost – Asint. show  that  X(t) and Y(t) are jointly wide-sense stationary if A and B are uncorrelated random variables with zero means and the same variances and  is constant .             (12 marks)

6.(i)Three boys A,B and C are throwing a ball to each other. ‘A’ always throws the ball to ‘B’ and ‘B’ always throws the ball to ‘C’ but ‘C’ is just as likely to throw the ball to ‘B’ as to ‘A’. Show that the process is Markovian. Find the transition matrix and classify the states.                      (6 marks)

(ii) Write a note on Binomial process.                                                                                  (6 marks)

7.(i)A man either drives a car or catches a train to go to office each day. He never goes 2 days in a two by train but if he drives one day, then the next day he is just as likely to drive again he is just as likely to drive again he is to travel by train. Now suppose that on the first day of  the week, the man tossed a fair die and drove to work if and only if a ‘6’ appeared. Find (i) the probability that he takes train on the third day (ii) the probability that he drives to work in the long run.      (8 marks)

(ii) Find the hazard rate function corresponding to the Weibull distribution given by

                                                                                                (4 marks)

8.What is the reliability of the system in the following figure? P(A)=P(B)=P(C)=0.8 (parallel redundancy) P(D)=0.95,P(E)=0.85. How would the reliability improve further if sub system ‘E’ is also made parallel redundant? Show that configuration of this system.                                    (12 marks)

9.State the postulates of Poisson process. Discuss any two properties of Poisson process.

(12 marks).

10.(a) If customers arrive at a counter in accordance with a Poisson process with a mean rate of 3 per minute, find the probability that the interval between 2 consecutive arrivals is

      (i) more than 1 minute

       (ii) between 1 minute and 2 minutes

       (iii) 4 minutes or less                                                                                          (6 marks)

    (b) Derive birth and death process                                                                       (6 marks)        

 

UNIT V

QUEUEING THEORY

PART A

1.Write down Pollaczek-Kinchine formula.

2.What is the probability that a customer has to wait more than 15 minutes to get his service completed in (M/M/1): queue system if .

3.For (M/M/1) :model, write down the Little’s formula.

4.For  model, write down the formula for (a) average number of customers in the queue(b) average waiting time in the system.

5.What is the probability that an arrival to an infinite capacity 3 server Poisson queueing system with  enters the service without waiting?

6.Obtain the steady state probabilities of an queueing model.

7.Write Kendal’s notation.

8.Derive average number of customers in the system in M/M/1.

9.A car mechanic finds that the time spent on his jobs has an exponential distribution with mean 30 minutes. If he repairs the car as how it comes in, arrival rate is Poisson with an average rate of 10 per 8 hour day. What is the repairmen’s expected idle time each day?

10.In a super market, the average arrival rate of a customer is 10 in every 30 minutes following a Poisson process. The average time taken by the Manager to list and calculate the purchase is 2.5 minutes, which is exponentially distributed. What is the probability that the queue length exceeds 6?

11.A T.V. repairmen finds that the time spent on his job has an exponential distribution with mean 30 minutes. If the repaired set arrive on an average of 10 per 8-hour day with Poisson. What is the repairmen’s idle time each day?

12.In a single server queueing system with Poisson input and exponential service times, if the mean arrival rate is 3 calling units per hour, the expected service time is 0.25 hour and the maximum possible number of calling units in the system is 2. Find .

13.Derive average number of customers in the queue of a M/M/C model.

14.Derive waiting time  customers in the system of a M/M/C model.

15.Write the probability an arrival has to get the service without waiting in M/M/C model.

16.Derive the probability that someone will be waiting in M/M/C model.

17.Derive the probability that the number of customers in the system exceeds k In M/M/1 model.

18.Write six characteristics of queueing process.

19.Write steady-state value of in M/M/1 model.

20. Derive average number of customer in the queue in M/M/1 model.

21.Derive the probability that an arrival has to  wait in M/M/C model.

22.A barbershop has space to accommodate only 10 customers. He can serve only one person at a time. If a customer comes to his shop and finds it full he goes to the next shop. Customers randomly arrive at an average rate per hour and the barber service time is exponential with an average of 5 minutes per customers.

23.A person repairing radios finds that the time spent on the radio sets has been exponential distribution with mean 20 minutes. If the radios are repaired in the order in which they come in and their arrival is approximately Poisson with an average rate of 15 for 8 hour day, what is the repairman’s expected idle time each day?

24.Trains arrive at the yard every 15 minutes and the service time is 33 minutes. If the line capacity of the yard is limited to 4 trains, find the probability that the yard is empty.

25.A branch of Punjab National Bank has only one typist. Since the typing work varies in length (number of pages to be typed), the typing rate is randomly distributed approximating a Poisson distribution with mean service rate of 8 letters per hour. The letters arrive at a rate of 5 per hour during the entire 8-hour work day. If the typewriter is valued at Rs 1.50 per hour, determine (i) Equipment utilization (ii) average system time.

PART B

1.A bank has two tellers working on savings account. The first teller handles withdrawals only. The first teller handles with drawls only. The second teller handles deposits only. It has been found that the service time distributes for both deposits and withdrawals are exponential with mean service time of 3 minutes per customer. Depositors are found to arrive in Fashion throughout the day with mean arrival rate of 16 per hour. Withdrawers also arrive in a fashion with mean arrival rate of 14 per hour. What would be the effect on the average waiting time for the customers if each teller could handle both withdrawals and deposits.                                                                                                          (12 marks)

2.In a heavy machine shop, the overhead crane is 75% utilized. Time study observations gave the average slinging time as 10.5 minutes with a standard deviation of 8.8 minutes. What is the average calling rate for the services of the crane and what is the average delay in getting service? If the average service time is cut to 8.0 minutes, with a standard deviation of 6.0 minutes, how much reduction will occur, on average, in delay of getting served?                                                                                                          (6 marks)

3.A repairman is to be hired to repair machines which breakdown at an average rate of 3 per hour. The breakdown follow Poisson distribution. Non-productive time of machine is considered to cost Rs.16/hour. Two repairman have been interviewed. One is slow but cheap while the other is fast and expensive. The slow repairman charages Rs.8 per hour. The fast repairman demands Rs.10 per hour and services at the average rate of 6 per hour. Which repairman should be hired?                                                                      (6 marks)

4.A two person barber shop has 5 chairs to accommodate waiting customers. Potential customers who arrive when all 5 chairs are full, leave without entering the barber shop. Customers arrive at the average rate of 4 per hour and spend an average of 12 minutes in the barber’s chair. Compute  and average number of customers in the queue.

(12 marks)

5.Derive the formula for

     (i) average number of customers in the queue                                    (12 marks)

     (ii) average waiting time of a customer in the queue for  (M/M/1):model

6.On average 96 patients per 24 hour day require the service of an emergency clinic. Also on average a patient  requires 10 minutes of attention. Assume that the facility can handle only one emergency at a time. Suppose that it costs the clinic Rs.100 per patient treated to obtain an average servicing time of 10 minutes and that each minute of decrease in this average time would cost Rs.10 per patient treated. How much would have to be budgeted by the clinic to decrease the average size of the queue from  patients to  patient?

                                                                                                                        (6 marks)

7.There are three typists in an office. Each typist can type an average of 6 letters per hour. If letters arrive for being types at the rate of 15 letters per hour. What fraction of time all the typists will be busy? It is the average number of letters waiting to be typed? (6 marks)

8.A petrol pump station has 2 pumps. The service times follow the exponential distribution with mean of 4 minutes and cars arrive for service is a Poisson process at the rate of 10 cars per hour. Find the probability that a customer has to wait to service. What is the probability that the pumps remains idle?                                           (6 marks)

9.(a)In the railway marshaling yard, goods trains arrive at a rate of 30 trains per day. Assume that the inter-arrival time follows exponential distribution and the service time distribution is also exponential with an average 36 minutes. Calculate the following:

   (i) the mean queue size

    (ii) the probability that the queue size exceeds 10

 (b)if the input of trains increase to an average of 33 per day, what will be the change in

    the above qualities?                                                                         (12 marks)

10.At a Beauty Parlour shop, with one beautician, ladies arrive according to Poisson distribution with mean arrival rate of 5 per hour and hair design was exponentially distributed with an average design taking 10 minutes. As it is a very good parlour, customers do have patience to wait, find

(i) Average number of ladies in the shop and the average number of waiting to do the hair design.

(ii) % of time as arrival can walk inside the parlour without having to wait.

(iii)% of ladies who has to wait prior to getting into the chair for hair design.(12 marks)