Bayes Theorem for project managers

Introduction

Projects are fraught with uncertainty, so it is no surprise that the language and tools of probability are making their way into project management practice. A good example of this is the use of Monte Carlo methods to estimate project variables.  Such tools enable the project manager to present estimates in terms of probabilities  (e.g. there’s a 90% chance that a project will finish on time) rather than illusory certainties. Now, it often happens that we want to find  the probability of an event occurring given that another event has occurred. For example, one might want to find the probability that a project will finish on time given that a major scope change has already  occurred.  Such conditional probabilities,  as they are referred to in statistics,  can be evaluated using Bayes Theorem. This post is a discussion of Bayes Theorem using an example from project management.

Bayes theorem by example

All project managers want to know whether the projects they’re working on will finish on time. So, as our example, we’ll assume that a project manager asks the question: what’s the probability that my project will finish on time? There are only two possibilties here: either the project finishes on (or before) time or it doesn’t.  Let’s express this formally.  Denoting the event the project finishes on (or before) time by , the event the project does not finish on (or before) time by and the probabilities of the two by and respectively, we have:

……(1),

Equation (1) is simply a statement of the fact that the sum of the probabilities of all possible outcomes must equal 1.

Fig 1.  is a pictorial representation of the two events and how they relate to the entire universe of projects done by the organisation our project manager works in. The rectangular areas  and   represent the on time and not on time projects, and the sum of the two areas, , represents all projects that have been carried out by the organisation.

Fig 1: On Time and Not on Time projects

In terms of areas, the probabilities quoted above can be expressed as:

……(2),

and

……(3).

This also makes explicit the fact that the sum of the two probabilities must add up to one.

Now, there are several variables that can affect project completion time. Let’s look at just one of them: scope change. Let’s denote the event “there is a major change of scope”  by and the complementary event (that there is no major change of scope) by .

Again, since the two possibilities cover the entire spectrum of outcomes, we have:

……(4).

Fig 2. is a pictorial representation of by and .

Fig 2: "Major Change" and "No Major Change" projects

The rectangular areas and represent the projects that have undergone major scope changes and those that haven’t respectively.

……(5),

and

……(6).

Clearly we also have since the number of projects completed is a fixed number, regardless of how it is arrived at.

Now things get interesting.  One could ask the question: What is the probability of finishing on time given that there has been a major scope change?  This is a conditional probability because it represents the likelihood that something will happen (on-time completion) on the condition that something else has already happened (scope change).

As a first step to answering the question posed in the previous paragraph, let’s combine the two events graphically.  Fig 3 is a combination of Figs 1 and 2. It shows four possible events:

1. On Time with Major Change (, ) – denoted by the rectangular area in Fig 3.
2. On Time with No Major Change (, ) – denoted by the rectangular area in Fig 3.
3. Not On Time with Major Change (, ) – denoted by the rectangular area in Fig 3.
4. Not On Time with No Major Change (, $\tilde latex C$) – denoted by the rectangular area in Fig 3.

Fig 3: Combination of events shown in Figs 1 and 2

We’re interested in the probability that the project finishes on time  given that it has suffered a major change in scope. In the notation of conditional probability, this is denoted by .   In terms of areas, this can be expressed as

……(7) ,

since (or equivalently )  represent all projects that have undergone a major scope change.

Similarly, the conditional probability that a project has undergone a major change given that it has come in on time, , can be written as:

……(8) ,

since .

Now, what I’m about to do next may seem like pointless algebraic jugglery, but bear with me…

Consider the ratio of the area   to the big outer rectangle (whose area is ) . This ratio can be expressed as follows:

……(9).

This is simply multiplying and dividing by the same factor ( in the second expression and in the third.

Written in the notation of conditional probabilities, the second and third expressions in (9) are:

……(10),

which is Bayes theorem.

From the above discussion, it should be clear that Bayes theorem follows from the definition of conditional probability.

We can rewrite Bayes theorem in several equivalent ways:

……(11),

or

……(12),

where the denominator in (12)  follows from the fact that a project that undergoes a major change will either be on time or will not be on time (there is no other possibility).

A numerical example

To complete the discussion, let’s look at a numerical example.

Assume our project manager has historical data on projects that have been carried out within  the organisation.  On analyzing the data, the PM  finds that 60% of all projects finished on time. This implies:

……(13),

and

……(13),

Let us assume that our organisation also tracks major changes made to projects in progress.  Say 50% of all historical projects are found to have major changes. This implies:

……(15).

Finally, let us assume that our project manager has access to detailed data on successful projects, and that an analysis of this data shows that 30% on time projects have undergone at least one major scope change. This gives:

……(16).

Equations (13) through (16) give us the numbers we need to calculated    using Bayes Theorem.  Plugging the numbers in equation (11), we get:

……(16)

So, in this organisation, if a project undergoes a major change then there’s a 36% probability that it will finish on time.  Compare this to the 60% (unconditional) probability of finishing on time.  Bayes theorem enables the project manager to quantify the impact of change in scope on project completion time, providing the relevant historical data is available. The italicised bit in the previous sentence is important; I’ll have more to say about it in the concluding section.

In closing this section I should emphasise that although my discussion of Bayes theorem is couched in terms of project completion times and scope changes,  the arguments used are general.  Bayes theorem holds for any pair of events.

Concluding remarks

It should be clear that the probability calculated in the previous section is an extrapolation based on past experience. In this sense, Bayes Theorem is a formal statement of the belief that one can predict the future based on past events. This goes beyond probability theory;  it is  an assumption that underlies much of science.  It is important to emphasise that the prediction is based on enumeration, not analysis: it is solely based on ratios of the number of projects in one category versus the other; there is no attempt at finding a causal connection between the events of interest. In other words, Bayes theorem suggests there is a correlation between major changes in scope and delays, but it  does not tell us why. The latter question can be answered only via a detailed study which might culminate in a theory  that explains the causal connection between changes in scope and completion times.

It is also important to emphasise that data used in calculations should be based on events that akin to the one at hand. In the case of the example, I have assumed that historical data is for projects that resemble the one the project manager is working on.  This assumption must be validated because there could be situations in which  a major change in scope actually reduces completion time (when the project is “scoped-down”, for instance). In such cases, one would need to ensure that the numbers that go into Bayes theorem are based on historical data for “scoped-down” projects only.

To sum up: Bayes theorem expresses a fundamental relationship between conditional probabilities of two events. Its main utility is that it enables us to make probabilistic predictions based on past events; something that a project manager needs to do quite often. In this post I’ve attempted to provide a straightforward explanation of Bayes theorem – how it comes about and what its good for. I hope I’ve succeeded in doing so. But if you’ve found my explanation confusing, I can do no better than to direct you to a couple of excellent references.

Recommended Reading

1. An Intuitive (and short) explanation of Bayes Theorem – this is an excellent and concise explanation by Kalid Azad of Better Explained.
2. An intuitive explanation of Bayes Theorem – this article by Eliezer Yudkowsky is the best explanation of Bayes theorem I’ve come across. However, it is very long, even by my verbose standards!