Hello everyone and welcome back to our advanced risk management series. In the last post, we began exploring some of the tools and techniques that can be used for advanced risk management. We examined scenario analysis and sensitivity analysis. You can review the post here. In this post, we would kick off right from where we left off in the last post by exploring probability distributions.

Probability Distributions

So far, the tools that we have examined gave us an indication of various scenarios and the sensitivity of individual variables on an outcome. However, none of these tools actually consider the likelihood of occurrence of a particular value over the other. So what is the likelihood of the best case or worst case scenario? Another thing to consider when analyzing scenarios is that the higher the number of samples/scenarios, the more accurate the estimation of an unknown variable would be.

So if we know that in a thousand possible occurrences, we can have different subscription costs varying from 300 to 500 and we want to make a forecast, how do we decide how to distribute this range of values? This is where probability distributions come in. Probability distributions, as the name suggests, are used to define the probabilistic behavior of a particular behavior over multiple iterations. In simple terms, probability distributions are used to describe how the changes in certain unknowns in a project are spread (distributed) over range of values.

Now, probability distributions can get fairly complex from a mathematical perspective but you do not need to worry so much about the finer details of the Mathematics. For the purpose of this article, we would describe three examples of probability distributions.

  1. Triangular distribution: Triangular distribution is used when the variables are continuous over a particular set of values and are most likely to occur at a particular value. In other words, when you have a best case, a worst case and a most likely case, then you can use triangular distribution. Again, this concept is not foreign to project management. In fact, the PERT average estimation uses a basic form triangular distribution to compute the averages. You would recall that the pert average is (o + 4m + p)/6 where o is optimistic, m is most likely and p is pessimistic. In more complex analysis of various variables, the triangular distribution can be used to improve the quality of the outputs of the computations. We would consider an example later in this article. A diagrammatic representation of a triangular distribution is shown below;

    Figure 1: Triangular distribution has minimum, maximum and most likely values.

  2. Normal Distribution: The normal distribution is the most popular kind of probability distribution. The concept of the normal distribution is that values are continuously distributed around a central limit (and this limit is called the mean). Now, although there is a fairly complex mathematical theory to support normal distribution, the key thing to remember in this case is that the normal distribution shows that most of the occurrences of a variable would be slightly deviated around the mean. The mathematical formula for normal distribution is:

    I know the formula seems complex with all the mysterious Greek symbols. But the key things to remember here are that µ is the mean and σ is the standard deviation. These two values (mean and standard deviation) are also used to define the Tolerance Limits of a normal distribution.

    The concept of Tolerance limits is fairly simple. It just states that in a normal distribution, about 68% of the occurrences occur in one standard deviation of the mean and about 95% occur in 2 standard deviations of the mean. When we allow a margin of error of three standard deviations, then we can safely assume that about 99.7% of the occurrences would happen in that range. These values are very useful to a project manager in making quick estimations based on historical information about the mean and standard deviation.

    The tolerance levels can be represented with the diagram shown below;

    Figure 2: Tolerance Levels in a normal Distribution

  3. Uniform Distribution: This kind of distribution is used when there is an EQUAL probability of occurrence of all values within a range. If the cost of a variable can range between $500 and $550, and the probability of any of the values in between that range is the same, then the probability distribution is uniform.

    Figure 3: Uniform Distribution of values between two points a and b

On their own, these distributions might be fascinating (if you love complex mathematical analysesJ), but the real strength of the probability distribution lies in their usefulness in analyzing a complex web of scenarios. One of these kinds of analyses is the Monte Carlo Simulations. We would spend the rest of this article discussing Monte Carlo Simulation.

Monte Carlo Simulations

Monte Carlo Simulation is a popular class of algorithms that is based on random sampling of multiple variables to obtain an unknown variable. Although Monte Carlo analyses have a great range of applications, we would only focus on its use as a risk analysis tool. Again, like some of the other tools that we have explored here, there is a huge mathematical computation in the analysis but I would spare you the horror since you can have all that done in software.

To make things clearer, let us consider the NPV example from the last article (http://resources.intenseschool.com/advanced-risk-management-part-1/). I have reposted the summary table of the scenario analysis here;

Figure 4: Summary of Scenario Analysis from previous article.

By using Monte Carlo Simulation to compute the values of the NPV, we can factor in random occurrence of the key variables within the defined ranges.

Thankfully, we do not have to compute this manually. There are many tools that can be used to perform these kinds of simulation. The tool for this was used for this analysis is the @Risk software, which is a part of a software suite developed by Palisade Corporation. You can download a trial here (http://www.palisade.com/trials.asp). It uses Monte Carlo simulation to generate different values of the NPV using random values of the changing variable within a defined distribution range. For this exercise, since the values of the best case, worst case and base cases were known, the triangular distribution was used to define the distributions. Assigning the NPV as the output, a Monte Carlo simulation of 1000 iterations of random values gives the output shown in the figure below;

Figure 5: Output of Monte Carlo Analysis using @Risk Software.

From the results of the analysis, 90% of the NPV falls between -£118,624 and £44,065, while about 5% falls above £44,065 and another 5% falls below -£118,624. The mean of the distribution is -£34,428. This shows that on the average, an investment in the business would yield a negative net present value.

Although it is significantly more difficult to compute, when compared to the tools described in the previous article (scenario analysis and sensitivity analysis), the Monte Carlo simulation provides the most useful information because it gives an indication of the probabilities, making it more realistic than the What-if analyses. Furthermore, by combining multiple iterations of random samples of all the variables, the results of the Monte-Carlo analysis are closer to the real world expectations than the best and worst case scenario analyses.


The tools we have discussed in this series might not be used on every project but they come in handy for a project manager when complex decisions have to be made based on analysis of uncertain variables. Although these tools do not directly eliminate the risk, they provide more quantitative information that can serve as the basis for making decisions. Another great use of the tools is for communication. Since project managers live (and die) by communication (or lack of it), any tool that can present complex information in a quantifiable manner is potentially viable to a project manager.

As usual, I would encourage you to test these tools out for yourself. Install excel and try out simple computations of scenario and sensitivity analyses. Also download third-party software and try some more complex Monte Carlo Simulations.

Do not forget to drop your questions and comments using the comments box and I look forward to discussing more project management concepts soon! Until then, keep having fun managing projects!

Image Sources: Wikipedia