Probability, more or less

Give Me a Number, please!

Every executive wants to know the completion date for his / her project.

But suppose you developed a curve for one of the following, all with the same “most likely” completion span (e.g. 15 months). How would you differentiate between them, when asked by the exec, to properly respond?

Chart Narrow SYM1

Chart Wide SYM1

chart unsym high1

Chart unsym low1

One response might be: “Please give me a few minutes to talk about probability (risk) curves, and how we derived them for this project”.


Practical Background



Excerpts from an article by Dr. Sam Savage, of Stanford:

(Using probability curves and Monte Carlo simulations) can enhance our view of uncertainty and risk as dramatically as X-rays enhance our view and treatment of broken bones. Pioneering efforts to evaluate uncertainty using computer simulation were used in the study of physics during the Manhattan Project and then moved to Wall Street in the 1970s and ’80s. By the early ’90s, these techniques found acceptance among technical personnel on desktop computers in industries as diverse as finance, energy, and the environment. The way we think about uncertainty is being reshaped along with the prospects for business success. .

There’s a common misconception that business projections based on average assumptions are, on average, correct. I refer to this phenomenon as the flaw of averages, and it gums up well-intentioned plans with alarming regularity in all lines of work.

An analyst at a petroleum company once described how he got his boss to understand the implications of uncertainty on their plans. He took the boss’ own spreadsheet, replaced several cells with uncertainties, and ran a Monte Carlo simulation on it. In this way, his boss was able to gain a better understanding of his own model.

We will run sample Monte Carlo’s on spreadsheets, as suggested above, in the Practical Applications section, below. But the studies require that we input distributions, so a little on those, first…

1. The Normal (Gaussian) Distribution


The most famous distribution is the Normal, or Gaussian. It’s very close to what you will find when you flip lots of coins, or collect people’s IQ’s or heights or weights. In other words, it fits lots of well dispersed and totally independent events.

Part of its beauty – you can describe the curve with only two variables: its mean and deviation, and from that gets lots of information. The downside: it fits with reality (especially regards independence) much less frequently than its use would suggest.


In fine print, in the curve to the left of Gauss on the Ten-Deutsche Mark note above, is the following formula:

The general formula for the probability density function of the normal distribution is:

gauss formula1

where mu is the location parameter (mean) and sigma is the scale parameter (deviation). (you can see how these factors affect the curve by clicking on the link to an applet, the blue curve on black next to Gauss, above).

Also, here (from the ten deutsche mark curve):

Gauss Detail

The 68-95-99.7% Rule

2. Curve Fit your own Distribution based on Data

If you have applicable data, that’s best. There are plenty of good curve-fitting programs out there; I like John Gilmore, and have been using his software, CurveFit .


(from an excel spreadsheet with 20,000 trials)

Model graph1

model eq

model data


3. Use a “Most likely, Minimum, Maximum” Distribution (by Experts)

Unfortunately, there is not always data for a curve fit, nor enough independence to assume a Gaussian distribution.

That means, typically, we overlay the “best” schedule or estimate with an expert analysis on the probabilities for each independent segment. The easiest, most intuitive method is based on an assessment for each segment of the “most likely“, “minimum” and “maximum ” values.

How to distribute these?

(a) The easiest, but least sophisticated, would be to assume that every probability within the min-max range is the same . This is not often used.

uniform 0

(b) The more common, yet also easy to calculate distribution is the triangular distribution.

uniform 1 uniform 1-2

It has been published that the problem with the triangular distribution is that it may not properly model especially the “tails” .


(c) While more complicated, the “PERT” distribution typically better reflects real situations, and is often my distribution of choice.

uniform 2 uniform 2-1

However, the PERT is more difficult to calculate (see the equations below). I have recently been trying out the software “RiskAmp” which performs these distributions and runs Monte Carlo analyses on them. Crystal Ball can also be configured this way, and I suspect that @Risk can, as well.

beta density

beta density note

4. Concluding Remarks on Distribution Functions

Sam Savage says it well: “You may have heard that a simulation is only as accurate as the distributions fed into it. I disagree. Before you climb on a ladder to paint your house, you shake it to test its stability. The random forces imparted when you shake the ladder are quite different from the random forces imparted when you climb on it. So are you going to stop shaking ladders because you discover that you’ve been using the wrong distribution all these years? Monte Carlo simulation provides a way to “shake” your plan to test its stability.”


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