Objective:
To
illustrate a network flow situation using a probability experiment employing
the Monte Carlo technique. (For a general description of the Monte
Carlo technique go to the
Elementary
Probability Demo Collection.)
Level: High
school, precalculus, or probability courses for math majors.
Prerequisites:
Basic
probability concepts; knowledge of equally likely chance and the chance
of winning a lottery.
Platform: A
0 -1 spinner device or a calculator or computer
which has a function that randomly generates zeros and ones. There
are two accompanying JAVA
applets which perform simulations discussed in this demo:
javapumpsim
and javapumpexperimenter.
Instructor's
Notes:
Two of the most fundamental notions of probability
are those of equally likely outcomes and independent events. Certainly
most students have seen spinners in board games. A 0 -1 spinner provides
a easy illustration of these two important notions. (See Figure 1. ) In
the elementary grades 0 - 1 spinners are often used as teaching devices
for very basic probability concepts. Here we provide a demo suitable for
high school and collegiate liberal arts math courses that employ such spinners
in for a simulation. (In place of a "physical " spinner a "digital" spinner
in the form of a random number generator on a calculator or computer can
be used.) Of course this can be considered a Monte Carlo simulation in
which the random numbers generated to model the behavior of a system are
just zero or one.
Figure 1.
Our simulation can be described as follows. A section
of a city has aging water mains and five pumping stations as shown in Figure
2. Assume that at a particular time each pumping station, numbered 1 through
5, has probability 1/2 that it will fail. We want to estimate the probability
that water will flow from A to B by some path as depicted in Figure 2.
(The line segments with arrows show the water mains or conduits.)
Figure 2.
One way to use this demo is in a collaborative
group setting where the
class is divided
into 5 groups. Each group has its own 0 - 1 spinner (or some equivalent
technology for generating zeros and ones randomly). Each of the groups
represents a pumping station, and uses its random number generator for
zeros and ones to determine if its pump fails; 0 indicates the pump fails
and 1 indicates the pump is operational. After each group has used its
spinner, the result is analyzed to see if there is a path for the water
to go from A to B. For instance if the random values in the following table
are generated, then water can flow from A to B through
pump station 5 to station 4 and reach B. This path is shown in Figure 3.
Figure 3.
The following animation illustrates the
procedure described above. This animation was created from the MATLAB routine
pumpsim.m and can be downloaded by clicking on pumpfiles.ZIP.
See also the JAVA applet
javapumpsim
which has the same functionality.
Students can construct a chart to keep track
of how many times the water successfully flows from A to B, compared
to the total number of trials. The probability of water successfully flowing
from A to B at any particular time can be estimated by the ratio of the
number of successes to the total number of trials only after a large
number of trials. A natural question is, how many trials should be used?
If you get this question, your students have tuned in to this modeling
problem and are ready for more ideas in probability theory.
The following table gives some sample probability
estimates for a successful flow of the water from A to B. If you
ran a series of the experiments with the same number of trials your results
may vary. (The estimated probabilities have been rounded to two decimal
places.)
Based on this set of tabular data you might
suspect the exact probability is close to 0.59 and in fact it can be shown
that the exact probability in this situation is 19/32 which is about 0.59375.
Hence of simulation produces reasonable estimates. (A MATLAB routine pumpdemo.m
can be downloaded to generate data like that in the preceding table; click
on pumpfiles.ZIP. For a description of
this routine click on pumpdescription. In
addition the JAVA applet
javapumpexperimenter
can be used for such experiments.)
By using this demo students can grasp the nature
of random phenomena and gain some experience with a powerful technique
that can be used to analyze many real-world problems. They will have engaged
in mathematical prediction and estimation, two important aspects in mathematics.
This simulation can be modified in a variety of ways.
For instance, vary the probability that a pumping station fails. Remove
or add a pumping station. Assign different probabilities to each pumping
station; maybe one station is significantly older than another. These modifications
open up a variety of modeling simulations that students can explore. If
your students propose such changes to the original problem you have really
enticed them to think about the underlying modeling process and should
consider the use of this demonstration a success.
Credits:
This demo was adapted from
E. Silver and J.P. Smith, "Random
Digits and Simulation", in Teaching Statistics and Probability, 1981
Yearbook of the National Council of Teachers of Mathematics, pp.70-73,
Reston, Va.: The Council, 1981.
by David R. Hill.
We also acknowledge contributions by Un
Jung Sin , student at Temple University. The MATLAB routines were written
by David R. Hill. The Java applets were developed by Philip Nicastro, a
student at Temple University.
