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Monte Carlo
refers to a widely used approach for solving
complex problems using computer algorithms
to simulate the variables in the problem.
Typically an algorithm is developed to
"model" the problem, and then the algorithm
is run many times (from a few hundred up to
millions) in order to develop a statistical
data set for how the model behaves.
For the simplest
example, consider the basic coin toss.
This is a process which has two possible
outcomes (heads or tails), each with a 50%
probability. In a million coin tosses,
roughly half will be "heads" and half will
be "tails". No complex math is
required to know this.
A simple
Monte Carlo simulation would prove the same
result. If you were to develop a
spreadsheet with a random number generator
resulting in 0 for "heads" and 1 for "tails"
(which we will do later in the tutorial),
then cause the spreadsheet to recalculate a
million times, each time recording the
results in a database, you could then run a
report on the database which would reveal
that very close to 50% of the recalculations
resulted in 0 or "heads" and the other 50%
in 1 or "tails".
Taking this a
step further, suppose you wanted to know the
likelihood of getting "heads" 7 times out of
10 coin tosses. Again here,
statistical mathematical equations provide
an accurate answer without the need for
Monte Carlo. But a Monte Carlo
spreadsheet could simulate a series of ten
coin tosses as easy as a single coin toss,
and could keep a tally of how many times 7
"heads" were returned from 10 tosses after
running a few thousand iterations of the
model. Analyzing the resulting data
should give an answer very close to the
mathematical statistical probability of
getting 7 "heads".
Now go one step
further. Suppose your commute to work
consists of the following:
-
Drive 2
miles on a highway, with 90% probability
you will be able to average 65 MPH the
whole way, but with a 10% probability
that a traffic jam will result in
average speed of 20 MPH.
-
Come to
an intersection with a traffic light
that is red for 90 seconds, then green
for 30 seconds.
-
Travel 2
more miles on a surface street,
averaging 30 MPH with a standard
deviation of 10 MPH.
You want
to know how much time to allow for the
commute in order to have a 75%
probability of arriving at work on time.
And, when you have an important meeting,
you want to know how early you need to
leave the house in order to have a 99.5%
probability of arriving on time.
Though
you could conceivably solve this problem
using mathematical equations, a problem
with this degree of complexity becomes
more efficient to solve using a Monte
Carlo simulation.
The basic
approach would be to build a spreadsheet
with random number generators which
replicate the probabilities in the
problem, and which calculate the total
commute time for a particular drive to
work. Each time you recalculate
the spreadsheet, the drive times and
traffic light delays are generated
afresh according to the actual
probabilities you have programmed.
By simulating 10,000 trips to the office
in a few seconds on your spreadsheet,
you can then easily see what the 75th
percentile and 99.5th percentile are for
your drive time.
That's
essentially all there is to it.
The more complex the problem, the better
it lends itself to Monte Carlo
simulations over mathematical solving.
Many times the problem may involve some
guessing at how a particular variable
behaves (is it a normal distribution
curve, or a linear?), but by using Excel
spreadsheets to build your models, it's
easy to change the assumptions behind
each variable to study the sensitivity
the results on each of your assumptions.
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