This article is an excerpt from the Book -

**I Want To Be A RMP**.

It is from

*Chapter – 8: Quantitative Risk Analysis.*For partial index of the book, refer:

**Book Index - I Want To Be A RMP**.

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**Probability Distribution in Risk Management**

Probability distribution is a foundational area to understand if you want to use the Monte Carlo or Latin Hypercube simulation techniques. This is also foundational with respect to Program Evaluation and Review Technique (or PERT). Let us first understand what statistical distribution means.

*Distribution represents a range of numbers and their chances (or probability) of occurrence.*- Minimum (or Optimistic) duration = 1 hour
- Most likely duration = 2 hours
- Maximum (or Pessimistic) duration = 3 hours

Now that you have three possibilities, which one is correct? Can you conclusively say about it? No. Because any one of them can come out to be correct as there are uncertainties involved, such as traffic conditions! So, you do not know for sure.

But when we estimate we take the most likely one or the best guess, i.e., 2 hours. By doing this, a lot of additional information is lost. We do the same thing for schedule or cost estimation – we take the most likely or most possible estimate, but forget to take the best case and/or worst case scenario.

With risk analysis via quantitative risk analysis (QTRA), we can use this extra information – other possible scenarios, not ONLY the most likely scenario. In other words, instead of saying an activity in a project is going to take “X” amount of days, we also can add other days using a distribution. For each number, there is a probability of occurrence available.

By doing that we will be modelling the activities (and hence overall project) more accurately. This will help to build a more realistic plan – both from schedule and cost perspectives. This also helps in answering the questions, which are foundational in QTRA as we saw before, i.e.,

- What is the chance that this project will finish on time or within budget?
- Assuming that there is a 50% chance for the previous question, next question can be – what is probability we can meet the planned project finish date?
- Which project activities are most likely to delay the project?
- And so on.

By adding other possible duration (optimistic, pessimistic) other than most likely, we are factoring the uncertainties for an activity. In other words, we are considering the risks involved. Hence probability distributions give you the needed lever to factor in the risks of the activities and overall project.

Let us see what are the possible types of probability distributions.

**8.7.1. Triangular Distribution**

This is the most common type of distribution used. This is called triangular because the shape of the curve comes as triangular. This is used when there is no pre-existing data, but only expert opinions.

**Symmetrical Triangular Distribution**

By looking at the graph above, you can say:

“There is approximately a 30% chance of the duration being 6 days, approximately a full chance of the duration being 8 days’ days and there is also a 30% chance of the duration being 10 days.”

Putting it differently, we can say this:

“The task is most likely to have a duration of 8 days. There is small chance that the task will take 6 days and also a chance that it will take 10 days. However, realistically speaking, it will be somewhere between 6 days to 10 days.”

**Asymmetrical Triangular Distribution**

Do note that the triangle shown need not be symmetric. Asymmetrical ones are shown below.

From here, you can calculate the duration with respective chances or probabilities.

Let us take another example. Here I am using the Primavera Risk Analysis software to visualize the possible distributions. We have a task – “PRD Preparation” in a project. PRD stands for ‘Product Requirement Documentation’. It has been estimated to be 5 days (again it is the most likely estimate, but we do not have the minimum and maximum value).

If you use the triangle or triangular probability distribution, it will come as shown below.

The duration can be 4 or 5 or 6 days – shown in the X-axis. Also shown are the minimum value (around 10% chance), most likely value (100% chance) and maximum value of 6 (around 10%) chance. The chances (or probabilities) are shown on the Y-axis.

You would be thinking where these values are coming from. As noted before, you can use Expert Judgement and/or enter those values while modelling the tasks. For example:

- Minimum duration = 75% of the planned (or remaining) duration of the activity.
- Likely duration = 100% of the planned (or remaining) duration of the activity.
- Maximum duration = 125% of the planned (or remaining) duration of the activity.

**8.7.2. Rectangular (Or Uniform) Distribution**

In this distribution, you can use a maximum value and a minimum value, but not any most likely value. In the below example, we have a uniform or rectangular distribution.

Looking at it, you can say the task has a minimum duration of 4 days, but maximum duration of 12 days.

You can use Uniform Probability Distributions when you specify the extremes of uncertainty of the activity under consideration and when the intermediate values have equal chances of occurring. It is also possible when you can NOT draw any inference on the possible distribution shape.

Taking our previous example of “PRD Preparation” which was estimated to be 5 days, using Primavera, we have these values for Uniform distribution.

**8.7.3. Beta Distribution**

Beta distribution, like triangular distribution has also 3 possible values – worst case, most likely, and best case. Like triangular, it also gives more weightage to the most likely case.

However, there are differences. In case of beta distribution, there can be many values (not just one) which are close to the most likely value. Unlike triangular, the shape for beta distribution is smoother. Also, unlike Triangular, the tails in Beta distribution the extremes tail off more quickly.

Below is an example of Beta Distribution.

Beta distribution can also be symmetric or asymmetric in shapes. The notations happen like Beta (6, 8, 10). As you can see above, there can be many values close to the most likely value.

Using the Primavera tool, for our task “PRD Preparation”, BetaPert distribution will come as below.

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Beta distribution is further elaborated in the book. Along with it, other distributions covered are:

- 8.7.4. Normal Distribution
- 8.7.5. Lognormal Distribution
- 8.7.6. Discrete Distribution

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