In diagrams like Figures 3 and 4
have students draw radii of the circles through points A and
B. This may give them a feel for the idea discussed below.
To see animations for drawing radii similar to situations depicted in
Figures 3 and 4 click on the following:
Animation for Figure 3.
Animation for Figure 4.
Now we need a way to determine from
measurements if all the points along the circumference of the
circle through points A and B are entirely within the
unit circle. Recall that all such points must be a distance less than
1 from the origin. It is not practical to check the distance from the
origin to each point on the circle from A to B, there
are just too many. Our focus is on the distance from the origin so an
alternative is to imagine that we have a rod extending the origin to
the midpoint of the segment from A to B (see the black
segment in Figure 5) and a radius
of the circle through A and B that pivot around the end
of the rod. As the radius pivots about the end of the rod the circle
through A and B is traced. Since all the points on the
circle through A and B are a distance of 1/2 times the
length of segment through A and B from the midpoint we can observe the
following:
If the distance from the origin
to the midpoint of the segment from A to B (namely
the length of the rod) + the radius of the circle through A
and B is less than 1, then entire circle through A
and B is within the unit circle.
This is illustrated in Figure 5.
Have students construct an algebraic test to determine if the circle
through A and B lies within the unit circle.
Figure 5. |
The level of experience of students can be used
by an instructor to gage
how much the instructor has to contribute to obtain algebraic expressions to
answer (a) - (d). Before any programs can be constructed to
estimate the probability of a success for either Problem 1 or Problem 2 the
foundations indicated above must be established. With the appropriate
algebraic ideas developed from the geometry models,
beginning students in a programming course who have had some practice in
constructing programs with loops should be able to generate code to estimate
the probabilities involved. If graphics to illustrate the geometry of the
two problems is to included, then a discussion of how to easily draw a
circle should be included. For the unit circle, plotting the points (cos(t),
sin(t)) as t goes from 0 to 2p
will give the unit circle. To obtain a circle of radius R centered at (x,y),
plot the points (x+Rcos(t), y+Rsin(t)) as t goes from 0 to 2p.
Software Available:
We have developed software that
illustrates the geometry involved with each problem and implements the
Monte Carlo technique to estimate the probabilities of a success for each
problem in both Excel and MATLAB. Figures 6 and 7 show thumbnails of the
screens for the Excel programs that illustrate geometry involved. Click on
these figures to see the full screen. You can click on the figure label to
either run or download the corresponding Excel files. Figure 6 corresponds
to Problem 1 and Figure 7 corresponds to Problem 2.
Figure 6.
Figure 7.
Figures 8 and 9 show thumbnails of
Excel programs that estimate the probability of a success for Problems 1
and 2 respectively. In both routines the estimates for a variety of a
number valid trials are displayed. You can click on the figure label to
either run or download the corresponding Excel files.
Figure 8.
Figure 9.
Each of the problems discussed in this
demo can be used to illustrate the use of some basic mathematical topics
that play a fundamental role in the estimation of the solution of problems
that require advanced mathematics to "solve" exactly. Determination of the
exact probabilities for these problem use techniques from probability that
are beyond the intended scope of this demo. For information on the
theoretical probability of a success for the two problems see the
Auxiliary Resources below.
To download the MATLAB routines, which
merge the graphics and estimation of t he probabilities of a success click
here for a zipped file.
Auxiliary Resources:
1. A collection of other probability
demos that use Monte Carlo simulation is available at
http://mathdemos.org/elemprob/elemprob.html . (This demo is part
of the collection.) The other problems addressed in this collection are
fairly simple to state, but the simulations are more complicated to
construct. They accompanying software is visually appealing and can be
used at a variety of levels.
2. Another demo which focuses on area
estimation using Monte Carlo simulation is available at
http://mathdemos.org/montecarlo/monte.html . The mathematical
concepts that are modeled are more sophisticated than ideas in this demo.
3. The basic ideas for the two problems
discussed in this demo appeared as problems in mathematics journals.
For Problem 1, see Problem 178
(proposed by Roger L. Creech) in The Two-Year College Mathematics
Journal, 1980, vol. 11 No. 5, p.336; a solution appeared in The Two-Year
College Mathematics Journal, 1982, vol. 13 No. 2, p.151.
For Problem 2, see the following
related material; Problem 1092 (proposed by Roger L. Creech) in The
Mathematics Magazine, 1980, vol. 53 No. 1, p.49; a solution appeared in
Mathematics Magazine, 1981, vol. 54 No. 2, p.87.
A resource that discusses both of
these problems and includes an analytic proof of probabilities of a
success appears in an article by Joseph E. Chance and Pearl W. Brazier,
"Two Problems That Illustrate the Techniques of Computer Simulation",
Mathematics Teacher, 1986, Vol. 79, No. 9, pp.726 - 731. The exact
answer for problem 1 is (1 - 3(3)^{1/2})/(4p)
which is about 0.58650 and the exact answer for Problem 2 is 2/3.
4. A variation of Problem 2 is to
require that the two points in the unit circle also be chosen so that the
distance between them is less than 1. The Excel program or MATLAB program
can be easily modified to estimate the probability of a success. A good
question for students is, will the probability of a success in this case
be larger or smaller than that for Problem 2?
5. For a brief history of the Monte
Carlo method go to
http://www.geocities.com/CollegePark/Quad/2435/history.html .