Allan J. Rossman and
Beth L. Chance
California Polytechnic State University
San Luis Obispo, CA 93407
Statistics Teaching and
Resource Library, November 5, 2001
© 2001 by
Allan J. Rossman and Beth L.
Chance,
all rights reserved. This text may be freely shared among
individuals, but it may not be republished in any medium without express
written consent from the author and advance notification of the editor.
This activity leads students to
appreciate the usefulness of simulations for approximating probabilities.
It also provides them with experience calculating probabilities based on
geometric arguments and using the bivariate normal distribution. We have
used it in courses in probability and mathematical statistics, as well as
in an introductory statistics course at the post-calculus level.
The scenario of the activity is easy to state and to understand: Tom and
Mary agree to meet for lunch at a certain restaurant, but their arrival
times are random variables. Furthermore, they agree to wait only for a
certain length of time for the other to arrive. The goal is to calculate
the probability that they meet. Students are first told to assume
independent arrival times uniformly distributed between noon and one
o’clock. They are first to approximate this probability through simulation
and then to calculate it exactly using geometric arguments. Then the
assumption about arrival times is relaxed to allow for different types of
distributions, including normal. Next the assumption of independence is
relaxed, providing students with experience performing bivariate normal
probability calculations.
In each case students are expected to approximate the solution through
simulation before solving it exactly. They are also expected to employ
graphical as well as algebraic problem-solving strategies, in addition to
their simulation analyses. Finally, students are asked to explain
intuitively why it makes sense for the probabilities to change as they do.
Key words: simulation, probability, geometry, independence, bivariate
normal distribution
Objectives
The objectives of this activity are:
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To help
students to appreciate the power of simulation for
approximating probabilities |
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To develop
students’ intuition about probabilistic reasoning
and their ability to express it verbally |
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To provide
experience using elementary geometric arguments to
solve probability problems
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To provide
experience with performing calculations related to
the bivariate normal probability distribution
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To lead
students to think about the effects of the
parameters of the bivariate normal probability
distribution on probability calculations
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Activity Description
Students need access to software for
conducting simulations. Minitab instructions are provided in the prototype
activity, but any package could be used. A graphing calculator could be
used with fewer repetitions of each simulation.
This activity is designed to be completed in a 75-minute class period. The
first part of the activity, involving analysis of uniform arrival times,
can be completed in a 50-minute class period.
The simulation analysis of the case of independent, uniform arrival times
should yield a scatterplot such as the following, where solid circles
represent cases where Tom and Mary meet, open circles where they do not
meet (15 minute wait time):
The distributions of the differences in
arrival times and absolute differences, for one simulation of 1000
repetitions, are:
Because these arrival distributions are
independent and uniform, one can determine the probability of meeting as
the area of the region where they meet divided by the total area of the
square. It is easier to find the probability of not meeting, since those
two regions are each triangles of area .5(45)(45), so the total area of
the region where they do not meet is (45)(45) = 2025. The probability of
meeting is therefore 1-(45)(45)/(60)(60) = 7/16 = .4375. In a
calculus-based course, an instructor could also have students evaluate
this probability as a double integral over this region. One could also ask
students to first find the distribution of the difference between two
independent uniform distributions and then integrate that probability
density function.
When one extends the waiting time to 30
minutes, the probability of meeting rises to 1-(30/60)2 = 3/4 =
0.75. For a waiting time of m minutes, the
probability of meeting is 1 - [(60-m)/60]2, a sketch of which
appears below:
When the distributions of arrival times
are assumed to be independently normal with mean 30 minutes after noon and
standard deviation 10 minutes for each person, a simulation analysis
produces results such as the following:
As most students expect, the probability
of their meeting increases because they are both more likely to arrive
near 12:30. Students can also find that the simulation is consistent with
the theoretical result that the difference in arrival times follows a
normal distribution:
Assessment
A key to assessing whether students
learn what they should from this activity is to ask them to compare their
findings from one situation to the next and to explain why it makes sense
for the probabilities to change as they do. Asking questions about the
effects on the meeting probability of further alterations of the
probability distributions or parameters can further test students’
understanding. Examples include asking students to explain what happens to
the probability of meeting as:
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The time that
they agree to wait increases |
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The
probability distribution of their arrival
times changes from uniform to normal (with
same mean and standard deviation)
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The means of
their arrival times move farther apart |
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The standard
deviations of their arrival times become
smaller |
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One can also assess students’ ability to
perform simulations in this problem-solving setting by changing the
probability distribution yet again, perhaps to beta distributions. Such an
exercise would be assigned either for homework or on an exam, if one had
access to computers during exams or was willing to include a take-home
exam component.
Another type of assessment follow-up to this activity could involve exact
calculations rather than simulation analyses. Specifically, students could
be told that the arrival times are normal and be expected to determine the
probability of meeting by using the result that the difference of two
normal distributions is itself normal. Moreover, students who have studied
multivariable calculus could be given a bivariate probability density
function for the arrival times and asked to use double integration to
calculate the probability of meeting.
One helpful built-in feature of this activity is that preceding the
theoretical analyses with simulations provides students with an immediate
way of checking the reasonableness of their analyses.
Teacher notes
This activity lends itself to use with a
variety of student audiences. We have used it with introductory students
at the post-calculus level, but many phases of it are also appropriate for
an introductory, algebra-based course. The parts involving uniform
distributions and independent normal distributions are appropriate for all
introductory students, but the bivariate normal analysis with a non-zero
correlation is best reserved for more mathematically inclined students.
In terms of where this activity “fits” into one’s syllabus, we believe
that it can be most effective as an early introduction to understanding
probability as long-term relative frequency and to the usefulness of
simulation. While it may be helpful for students to have some prior
experience with statistical software before engaging in this activity,
that experience is not essential.
We have used this activity as an in-class experience for students,
encouraging them to work in pairs. We believe that it could also work well
as an out-of-class assignment, with class time then devoted to students’
presenting and discussing their results and findings.
A further extension of the activity, appropriate for courses at the
post-calculus level, might use a more “generic” bivariate probability
density function and have students apply double integrals to calculate
probabilities. Another extension would be to use more unusual arrival time
distributions where simulation would provide a much more efficient
problem-solving strategy than integration.
Acknowledgements
This activity was developed under grant
#9950476 from the National Science Foundation. It is part of a project to
develop curricular materials for a data-oriented, active learning approach
to introductory statistics at the post-calculus level. The materials will
be published by Duxbury Press.
Editor's note:
Before 11-6-01, the "student's version" of an activity was called the
"prototype".
. Thus, the probability of their meeting is
Pr(-15<M-T<15), which standardizes to Pr(-1.06<Z<1.06) = .7108 and is indeed higher than the .4375 probability with
independent uniform arrival times.
. They discover that the probability of meeting
increases as the correlation coefficient between their arrival times
increases. Simulation results for a correlation coefficient of .6 are
shown below: