 align=middle width=80% Student's version HTML Format Word Format Computer Supplements Random_Rendezvous.MPJ

Random Rendezvous

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: Letting M denote Mary’s arrival time and T denote Tom’s, M-T follows a normal distribution with mean 30-30=0 and standard deviation .  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.

Students are then asked to relax the assumption of independence and investigate what happens as the correlation coefficient between Tom’s and Mary’s arrival time changes.  Again they use simulation first and then theoretical calculations based on the normal distribution, determining the standard deviation of M-T as .  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: 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".