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elements of uncertainty—the weather, number of attendees, and the outcome of

games of chance. We simpliﬁed the problem analysis by assuming
deterministic

values
(speciﬁc and unchanging) for these uncertainties. In particular, we consid-

ered only a single result for each of the uncertain values, for example
rainy
weather

as the weather condition. We also reduced uncertainty to a value determined as an

average, for example the winning odds for the game of chance, WOD. In doing so,

we fashioned the analysis to focus on various scenarios we
expected
to occur. On

the face of it, this is not a bad approach for analysis. We have scenarios in which

we can be relatively secure that the deterministic values represent what Fr. Eﬁa will

experience, conditional on the speciﬁc weather condition being investigated. This

provides a simpliﬁed picture of the event and it can be quite useful in decision mak-

ing, but in doing so, we may miss the richness of all the possible outcomes, due to

the
condensation
of uncertainty that we have imposed.

What if we have a problem in which we desire a greater degree of accuracy,

and a more complete view of possible outcomes? How can we create a model to

allow simulation of such a problem and how do we conceptualize such a form of

analysis? To answer these questions, let me remind you of something we discussed

earlier in Chap. 6—sampling. As you recall, we use sampling when it is difﬁcult, or

impossible, to investigate every possible outcome in a population. If Fr. Eﬁa had 12

uncertain elements in his problem and if each element had 10 possible outcomes,

how many distinct outcomes are possible; that is, if we want to consider
all
com-

binations of the uncertain outcomes, how many will Fr. Eﬁa face? The answer is

10
12

etc.) possible outcomes, which is a whopping 1 trillion

(1,000,000,000,000). For complex problems, 12 elements that are uncertain with

10 or more possible outcome values each are not at all unusual. In fact, this is a

relatively small problem. Determining 1 trillion distinct combinations of possible

outcome values is a daunting task, and I further suggest that it may be impossible.

This is where sampling comes to our rescue. If we can perform carefully planned

sampling, we can arrive at a reasonably good estimate of the variety of outcomes

we face: not a complete view, but one that is useful and manageable. By this, I mean

that we can determine enough outcomes to produce a reasonably complete proﬁle of

the entire set of outcomes. This proﬁle will become one of our most important tools

for analysis and decision making. We call it a
risk proﬁle
of the problem outcomes,

but more on this later. Now, how can we organize our efforts to accomplish efﬁcient

and accurate sampling?

(10

×

10

×

10

...

8.3 The Monte Carlo Sampling Methodology

In the 1940s Stanislaw Ulam, working with famed mathematician John von

Neumann and other scientists, formalized a methodology for arriving at approximate

solutions to difﬁcult quantitative problems which came to be called Monte Carlo

methods. Monte Carlo methods are based on
stochastic processes
, or the study of

mathematical probability and the resolution of uncertainty. The reference to Monte

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