MAlTA LEVINE & RAYMOND H. ROLWING
Teachers of secondary and undergraduate college mathematics courses are frequently encouraged to stress applications of mathematics and to involve the students in the learning process beyond the traditional lecture-discussion setting. Although these goals are widely accepted, they are not always easy to achieve. The content of elementary probability and statistics, however, lends itself particularly well to the use of experiments and projects to stimulate creativity, provide for the pursuit of individual interests, and demonstrate a wide variety of illustrations and applications.
In this paper we would like to share some experiences which we have had with probability and statistics classes.
The assignment, which we shall describe, has been given to four different types of classes, several times. Among the types of students in the various courses were freshmen with a limited mathematics background, juniors in the College of Engineering, pre-service and in-service high school teachers, and high school seniors. The classes varied a great deal in terms of size, mathematical maturity, background, and textbooks used in the course. The presentation of the assignment, however, varied only in the length of time spent to develop the basic concepts. The solutions, of course, varied extensively in terms of mathematical creativity and sophistication.
In each case, after a one to four week introduction to the concepts of probability, a set of problems was distributed. Some examples were chosen merely because they were interesting, nonintuitive, or entertaining illustrations of probability theory. Others were selected as examples of applications of probability to the natural, social, or behavioral sciences.
Among the examples were the following problems:
1. A certain cook can prepare 2 cereals, Lumpies and Soggies, but sometimes she burns them. In fact, when she cooks Lumpies, her probability of burning it is 0.1. When she cooks Soggies, her probability of burning it is 0.4. Whenever she burns Lumpies she cooks Soggies the next day. However, she really doesn’t like Soggies very well even when it isn’t burned. Consequently, after cooking it one day, she always goes back to Lumpies. Begin with Lumpies. Record 30 days. What percent of the time does the family eat Lumpies? What percent of the time does the family eat burned cereal?
2. Put two black marbles in an urn labeled X and two white marbles in an urn labeled Y. Draw a marble from each urn. Put the one from X into Y and put the one from Y into X. Repeat, each time recording the number of black marbles in urn X. Continue until you have a sequence with 30 entries. What is the probability that the 30th entry will be 1? What is the probability that the booth entry will be 1?
3. Of the subjects of a certain king, not all are truthful. In fact, if a subject is selected at random, the probability that he always tells the truth is 2/3. The probability that he never tells the truth is 1/3. The king of this country is trying to decide whom to marry. There are only two possible choices, Princess Anne and Princess Barbara. One day the king whispers to one of his subjects that his choice is Anne. This confidant hastily whispers to another person "The king has chosen ______". Which name he says depends, of course, on whether or not he is truthful. So it goes. Each person, when he hears the rumor, whispers either the name he hears or the other to someone who has not heard. Eventually, 12 people have heard the rumor. What is the probability that the 12th person has heard the truth?
4. An urn has 9 white balls and 11 black balls. A ball is drawn, and replaced. If it is white, you win 5 cents; if black, you lose 5 cents. You have a dollar to gamble with and your opponent has 50cents. If you keep on playing until one of you loses all his money, what is the probability that you will lose your dollar?
5. Twenty-five ballots are cast, 14 for A, 11 for B. What is the probability that as the votes were counted, A was always leading?
After a discussion of techniques that could be used to simulate an experiment, three class hours were devoted to gathering data for the problems. Students were then encouraged to make conjectures and consider possible analytical techniques for solving the problems.
Finally, the following assignment was given: Present a complete or partial solution for each problem. You may consult reference books and you may work in groups. Each of your solutions must explain how you arrived at your conclusion and what you learned about probability as you attempted to solve this problem.
Students approached the problems in a variety of
ways. For example, some students simulated the experiment in Problem 1,
using slips of paper marked Lumpies and Soggies. Others carried
out the simulation on a computer or a programmable calculator. A number
of students used a tree diagram to approach the problem analytically. Although
Markov chains had not been discussed in class, many students used library
resources to learn about the Markov process and then applied matrix computations
to the solution of the problem. Problem 3 provided another exercise in
which students’ approaches varied. Again, the less sophisticated students
merely simulated the experiment. More sophisticated students realized that
the formula for a binomial distribution was applicable. Some groups drew
the first few steps in a tree diagram and tried to discern a pattern. One
group discovered that the probability that the n-th person has heard the
truth could be represented by the formula P(n) = 
and commented that "this goes to 1/2 very quickly". And other students
viewed the exercise as a Markov experiment and used a programmable calculator
to carry out the associated matrix multiplications. For all the examples,
the diverse ability and mathematical background of the students were evident
in the solutions which they submitted. In most cases, students found hints
or solutions to at least one of the problems in a textbook or recreational
mathematics book.
Although the students seemed almost unanimous in expressing their enjoyment of the project, the engineering students and the high school seniors exhibited the greatest enthusiasm, the most cooperation and efficiency in working in small groups, and the highest degree of ingenuity in simulating experiments and solving problems.
There was no formal, objective evaluation of the assignment. Its success, however, was reflected in the comments submitted along with the solutions. The following statements are representative of the students’ opinions:
"I also learned how handy a programmable calculator can be and that solutions can be approximated very closely by experiment."
"Some of the problems were kind of fun to solve."
"I learned the value of working in groups …After co-operating a few times, I know the engineer in the field is not on his own, but is a member of a team." "I enjoyed solving the problems because we weren’t given a direction in which to go."
"I learned more from the project than from anything
else all quarter … Even though the project was time consuming and at times
frustrating, I’m glad I had the opportunity to do it. It gave me confidence."
University of Cincinatti
References
2. School Mathematics Study Group, Introduction to Probability, Parts I and II, A. C. Vroman, Inc., Pasadena, Calif., 1966.