CASE STUDY - Probability in PERT

This is in continuation of PERT in my previous hub. As stated therein, PERT is probabilistic in nature. It is used in those projects which are being developed for the first time. Hence, the developer has no previous experience and seeks guidance from the knowledgeable persons. On the other hand, CPM is deterministic in nature as all estimates are based on own experience or track-records.

But it must be remembered that many internal and external events may frustrate the predictions. This happens frequently and is taken as norm rather than exception especially in unstable economic and political environments.

In order to make realistic estimates, PERT obtains three estimates of different scenarios as shown below:

  • Optimistic Time Estimates if everything turns favorable.
  • Pessimistic Time Estimates if all goes bad.
  • Mostly likely time which would be experienced in normal conditions.

Time Estimation Formula

Since Optimistic and Pessimistic conditions would be far less than normal conditions, a weight of one each is assigned to Optimistic and Pessimistic Times. In case of mostly likely time, a weight of 4 is assigned. This is a standard practice. The result is divided by total weight of 6, to find out weighted average which would serve as Time Estimate or Te as in the formula shown on the right-hand side:


Time Expected (Te)

A Construction Project

Let us start with construction program of a yatch. Being our first venture, we would prepare a PERT and obtain necessary estimates from designers, yatch builders and other knowledgeable persons like carpenters, welders and electricians in their respective fields.

The three estimates, optimistic, most likely and pessimistic, are given in table titled Basic Question. In the next table Time Expected (Te) has been calculated based on the formula given previously.

As stated before, PERT uses a "Weighted Average" of three time-estimates to calculate Time Expected (Te) for a particular task. These estimates are not wild guesses but have come from reliable sources. When it comes to masonry work, who can better estimate time required for making a brick wall than an experienced mason.

Various researchers have criticized use of “weighted averages” in time estimates. They argue that in this way, time would often be underestimated. But whenever one tries to predict future, one is confronted with many problems. To be realistic, one should make meticulous efforts and double check every figure.

Once we know the Te for each task, the rest is like CPM i.e. (i) the boxes representing variou activities would be placed keeping in view the predecesssor and successor activities, (ii) clear cut linkages shown between the activities, (iii) forward passes made to workout project duration, (iv) all possible paths identified and (iv) the longest path, being the Critical Path highlighted with red-line. This has been shown in the net work given below:

PERT Network - Activity On Node (AON)

Variance Calculation

PERT FORMULAE: Variance & Standard Deviation


Since PERT recognizes uncertainty in estimates of durations, it gives rough estimates about final completion. Now what-if analysis can be conducted like what is the probability of completion if project is delayed by certain period of time. Please note that probability of being completed by critical time is 50%. If more days are added the probability would increased and can be quantified by using normal curve method. For this we need a z-table and a standard deviation. PERT has special formula for calculating Standard Deviation. First, it would identify the activities on the critical path. Second, it would calculate variance for each activity on the path. Finally, square-root would be worked of sum total of variances of activities on the critical path. Necessary working is shown in the right hand side table.

Normal Curve with properties

Probability of completion


Normal distribution is natural distribution. I teach Project Management to a class of 40 students. Since all the students are reading from the same books, are being taught by the same teacher in the same environments, their marks in any test would be normally distributed. About 68% of the students would gain around the average, a few would well above it and a few well below. In a recently conducted test, the average score was 80 with a standard deviation of 6. It means that 68% of them got marks between 74 & 86. There were few exceptions. A few students were well above 86 while a few were much below 74.

With this back ground, we can find probability of a student getting 95 marks. It gives a z-value of 2.5 which means there is hardly 1% chance of scoring 95%. What about chances of 87%? The probability would increase to 12%

In our estimates for construction of the yatch, average time was 28 days with a probability of 50%. If we increase the project duration, the probability will increase and vice versa. When a sponsor asks for say 99% probability, the duration can be extended accordingly. In this case, if the construction team is given 32 days, there is hardly 1% chance that they would disappoint the sponsors. It does not mean that they would necessarily take that much time; maybe they complete it in 28 days or even less. But one should not expect a miracle.

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Comments 37 comments





Fahad Khan 7 years ago

Thank You so much for sharing this case study. It really helped me in clarifying certain aspects of the topic.


Fahad Khan

Zubair khan 7 years ago

This case study simply show the remarkable knowledge and expertise you have on Project management i think you have to represent pakistan on different forum of world. you are simply the assest of the nation.

Best Regards,

Zubair Khan

hafeezrm profile image

hafeezrm 7 years ago from Pakistan Author

Dear Kashan,

There are three time estimates. Of these, one is "normal time". Since things go normally, the statisticians have assigned it a weight of 4, to others one each. So the weighted average can be calculated by using the formula.

What is normal? Normalancy differs from place to place. If most of the people come late and one hour out of eight are wasted, the normal hours would be seven. If there are strikes due to which two months are lost, the normal year would be of 10 months.

Hope it clears the point.

Sana Rao 7 years ago

Thank you sir itz really helpful but i want to know that where we ill apply it to get an experience about it,i mean will u please gimme an example for these.

SHERAZ KHAN MBA4-A 7 years ago



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Fahad_Khan 7 years ago from Karachi

Thank You so much for another case study. Easy and Simple to understand. It clarified everything with the help of diagrams.


Muhammad Fahad Khan

Noor ul Ain Azhar 7 years ago

The case helped me have a clear understanding of the topic. specially the example explaining the normal distrbution of students' scores in a class. being a student myself, i could identify with and thus understand the concept given in the example. however, there is one little confusion yet to be clarified, sir; in practical PM scenarios, is it the duration of the project that helps predict the probability of project completion?or does the required probability of success lead to the calculation of the project duration? i would be thankful for your assistance in this regard.

hafeezrm profile image

hafeezrm 7 years ago from Pakistan Author

Thanks for your comments.

Facts come first.  These are observed and used for future prediction and probability.  A person would observe that how long projects, of a certain type say spinning mills, have taken time in completion in the past.  He would then analyse why some projects took so much time and why some projects were completed much earlier.  Maybe he observed that the spinning projects undertaken by well-established groups were completed much earlier than those undertaken by the new entrants. So keeping the past trends and present conditions in mind, one can make any prediction.

In my article, the probability was given by z-score.  Critical time was calculated by weighted average.  If all activities are performed at their optimistic time, it may take just 17 days but its probability would be much below zero, but in pessimistic time when it takes 39 days, the probability would be 99.9999999%.

Thanks for an intelligent question?

Munesh kumar 7 years ago

thank u sir for giving us such a important information. what i am understanding from this case study that how can we manage the time during project and how much time will we have 2 spend on the small parts of the projects. with the hepl of probababilty we will find that k what is the probability of completing the project in given time. Am i right sir?

but sir i didn't understant that ehy we find variance?

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hafeezrm 7 years ago from Pakistan Author

Thanks Zubair for your compliments

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hafeezrm 7 years ago from Pakistan Author

Dear Munesh,

You are right about probability.

Variance and Probability have a link. As stated in the article, in order to find probability, you need (i) Critical Time, (ii) Desired Time, (iii) Standard Deviation. With the help of these parameters, you would calculate z-score which would indicate you its corresponding probability via z-table. Now perhaps you have realized that Standard Deviation can only be calculated if you know Variance.

Variance and Standard Deviation both are indicators of risk or uncertainty. Higher the Standard Deviation, higher the uncertainty and hence risk. The spread of any normal curve shows the effect of Standard Deviation or risk. High risk, high return is based on the same theme.

Noor ul Ain Azhar 7 years ago

ok, i understand the concept much better now. Thanks a lot for your kind assistance.

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mariamgazdar 7 years ago

Thanks sir, it is easy and simple to understand.

khalid AKhter profile image

khalid AKhter 7 years ago from KARACHI

the case study is really amazing and really very informative for me i have already told u that i have alredy studied this but the way u eloborate it that is mind blowing...each step is like a crystal and there is no need of further studing this ...this TIME ESTIMATION method is fixed in my mind dur to ur xplaniation...thanx for sharing this wid us ....

usman alvi 7 years ago

this case study clear my concepts of PERT and CPM thank u.

Mubin Ashraf 7 years ago

You are bringing more information to us. As you are teaching us through this pages also. Thanks.

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nadp 6 years ago from WEST PALM BEACH, FLORIDA

I enjoyed this hub. I teach Discrete Mathematics, and PERT is one of the topics I sometimes cover. I also teach statistics. It is interesting to read how these subjects are used in management.

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hafeezrm 6 years ago from Pakistan Author

Thanks @nadp for visiting my hub and for your comments.

barabas 6 years ago

igawa mu nga ng case study

uswege mwaipungu 6 years ago

am learning a lot on calculations of PERT in probability style with very thanks to those who prepare that usefuly material

jim young 5 years ago

presumably Most Likely has a weight of (4 or some 68%) since this means two SDs either side of normal?

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hafeezrm 5 years ago from Pakistan Author

Dear Jim Young,

You are very correct. Thank for your comments.

Kishan Chand 5 years ago

Thank you sir for giving us such a important information in simple words i am easily understand. I am understanding from this case study that how can we manage the time during project and how much time will we have to spend on the small parts of the projects. with the help of probability we will find that what is the probability of completing the project in given time.

Vig 5 years ago

Y have the weights been assigned as 1:4:1

hafeezrm profile image

hafeezrm 5 years ago from Pakistan Author

Dear @Vig,

These weights are probabilities and part of the formula based on calculus.

Sharif Allan 5 years ago

Thanks a lot for this case study ..


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hafeezrm 5 years ago from Pakistan Author

Thanks Sharif Allan for visiting my page.

Eyad Saadeh 5 years ago

Dear Sir,

after 3 days searching I reached this blog

Thanks for the clarification

but for me still some missing links, I need your help

1- why you choose Normal distribution? why not triangle or uniform or...? when we need each on of them? do you have recommended reading for that?

2- I understand how you calculate the mean and SD but what is z-table? what is the area under the curve? and how you reach the 68% or 86%, 1%, 12% or any %??

3- I understand the example of class marks, I know the relation between the mean and 1 or 2 or 3 sigma, but again how we calculate the confidence level??

4- do you have examples for Montecarlo simulation

I know it too many questions! maybe because I have IT background not statistics, do you have some recommended materials to cover the gap

Thanks again for your time and effort

hafeezrm profile image

hafeezrm 5 years ago from Pakistan Author

Dear Eyad Saadeh,

These are elementary topics. I suggest you go through a book of Business Statistics.

However, I would try to give answers to your questions.

Normal Distribution: It is a natural distribution. It deals with standard behavior like size oranges from the same garden or marks of students from the same class.

Normal Probabilities and z score are reciprocal of each other. Confidence and probabilities are the same. If a house take 180 days on the average, you have only 50% confidence or probability. If you want to be 99% sure than you would have to increase the duration which would depend upon z-score of 99%, average 180 days and standard deviation of the normal distribution. (You can find z-table or z-score from any book on statistics. It is two dimensional, one z-score and the other probability.

You can go to: for normal curve

You can use RAND() for montecarlo simulation. It just shows chance number. If you have 100 chips in a box each number from 1 to 100, the chance of finding any particular number say 35 is 1/100. Some, planner go for simulation ( trying all possible blends like price, variable cost, fixed cost and quantity to find out what are the out-come if simulation is done 50 times. If it shows for 30 times there would be a profit, it may be a good risk.

Please read about these terms from Internet and ask a nice question.

zohaib noor ullah 4 years ago

dear sir Thank You so much for sharing this case study. It really helped me in clarifying certain point of the topic.


zohaib noor ullah

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hafeezrm 4 years ago from Pakistan Author

Thanks Zohaib for your comments.

Ryan 4 years ago

Dear Sir,

Thanks for your case study, but I got a question here. Some one would like to use Te=(a+3m+2b)/6 as the pessimistic formula to calculate the mean, so we will get a different curve compared with the standard one. However, how could we use these two curve to calculate the probability? For example, if I got 30 days with 90% confidence in standard formula but 20 days in pessimistic formula, how can I combine them?

hafeezrm profile image

hafeezrm 4 years ago from Pakistan Author

Dear Ryan,

Knowledge is dynamic. New techniques are discovered everyday. If some one has suggested the new formula, maybe it is based on his or her observations that Lucky time (optimistic days) is less, only 1/6 or 17%, most likely 50% and the remaining are unlucky (pessimistic)33%. If so, the two formulas should NOT be mixed but one of them selected and used consistently.

How you have calculated 30 days and 20 days? It would be of interest to me.

N.Youssouf 4 years ago

thank so much sir! I'm a student of higher institute of agriculture and animal husbandry(isae Rwanda)department of agribusiness your publication helped me so much in my work

Anon 4 years ago

Good stuff, but what is a yatch?

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hafeezrm 4 years ago from Pakistan Author

A yatch is a luxury boat. You may like to go this link to learn more about it.

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