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:
 |
To help students to appreciate the power of
simulation for approximating probabilities |
|
To develop students’ intuition about
probabilistic reasoning and their ability to express
it verbally |
|
To provide experience using elementary
geometric arguments to solve probability
problems |
|
To provide experience with performing
calculations related to the bivariate normal
probability
distribution
|
|
To lead students to think about the effects of
the parameters of the bivariate normal probability
distribution on probability
calculations
| |
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:
|
The time that they agree to wait
increases |
|
The probability distribution of their arrival
times changes from uniform to normal (with same mean
and standard deviation) |
|
The means of their arrival times move farther
apart |
|
The standard deviations of their arrival times
become smaller | |
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: