Probability Functions in Decision Trees

You might be asking: What about delays other than 20 days, or why 20 days? The project manager and project team may be able to estimate much finer segments than 20 days. There really is no limit to how many individual discrete estimates could be made and summed at a node. It is required that the sum of all probabilities equal 1. So as more discrete estimates are made, say for 1, 2, 5, 10, or other days of delay, the individual probabilities must be made individually less so that the total summation of the "p"s equaling

1 is honored.

Now you may recognize this discussion as similar to the discussion in the chapter on statistics regarding the morphing of the discrete probability distribution into the continuous probability function. Obviously, the values at the input of the summing node are the values from the discrete probability function of the random variable or event that feeds into the summing node. There is no reason that the random variable could not be continuous rather than discrete. There is no reason that the discrete probability function cannot be replaced with a single continuous probability function for the event.

Suppose the inputs to the summing nodes are replaced with continuous probability functions for a continuous random variable. Figure 4-7 shows such a case. Now comes the task of summing these continuous functions. We know we can sum the expected values, and if they are independent random variables, we know we can sum the variances with simple arithmetic. However, to sum the distribution functions to arrive at a distribution function at the decision square is another matter. The mathematics for such a summing task is complex. The best approach is to use a Monte Carlo simulation in the decision tree. For such a simulation you will need software tools, but they are available.

Figure 4-7: Decision Tree with Continuous Distribution.

t3]Schuyler, John, Risk and Decision Analysis in Projects, Second Edition, Project Management Institute, Newtown Square, PA, 2001, chap. 5, p. 59.

t4]The Delphi technique refers to an approach to bottom-up estimating whereby independent teams evaluate the same data, each team comes to an estimate, and then the project manager synthesizes a final estimate from the inputs of all teams.

Summing node -

Summing node -

Team LiB

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