Some Probability Teaching Activities
Louis J. Gross
Departments of Mathematics and
Ecology and Evolutionary Biology
University of Tennessee
Knoxville, TN 37996-1300
gross@math.utk.edu
- Below are some activities originally designed for an Elementary School
math enrichment program (for 3-5th graders). I have successfully used some
of these with undergraduates and graduate students as well (suitably
modified of course). For some code to simulate some of the below in MATLAB,
see the page describing my MATLAB codes at
http://www.math.utk.edu/~gross/matlabfiles/quant.lifesci.matlab.html
and particularly the codes binomial.m, levelcrs.m, level2.m, ordersta.m,
levelopt.m, and urn.m
Objectives:
1. Provide opportunities for students to make hypotheses and evaluate them
with data they collect, thus illustrating the role of hypothesis formulation
and testing in science and mathematics.
2. Provide opportunities for individual experimentation related to
mathematical concepts, both within the structure of a class session, and at
home for students who wish to pursue the subjects further.
3. To structure the activities as games, involving both individual and group
effort.
Procedures:
1. I bring to class an "Hypothesis Machine" - a contraption that allows
students to pour ball bearings down a central chute and watch how they
disperse among 20 side chutes. This illustrates unpredictability, since they
cannot predict exactly what chute the balls will fall in, as well as the
concept of a histogram as a summary of lawful behavior within many-particle
systems. Students can control the ball distribution to some extent by placing
blocking pins in various locations, and experiment to see where they wind up.
I leave the machine in a class for 1-2 weeks for students to play with when
appropriate.
2. Coin tossing experiments. Here each student will make guesses as to what
will happen when they toss a coin once, and then 10 times. We will summarize
the data from the whole class by making histograms of Head/Tail frequencies,
as well as histograms for the number of heads in the 10 tosses. This set of
experiments is perhaps more appropriate for 3rd rather than 4th grade.
3. Urn experiments. Here the class is broken into groups of 5 students, and
each group carries out experiments by drawing marbles of two different types
from an urn. In the first case, there is replacement, each urn contains 5
balls of each color, the group draws first one ball with replacement and
records their results (repeating 10 times) then draws 2 balls with replacement
(repeating 10 times). We will summarize the class results. Then, starting with
just 2 balls in each urn, each group draws a ball at random, replaces it along
with one of the same color, then repeats until there are 20 balls in the urn.
We then count the number which are of each type and summarize differences
between urns by different groups. This also provides some experience with
fractions. In the limit here, each urn will approach its own fraction of balls
of each type - the fractions will not be the same in each urn.
For a wonderful paper on this subject with biological applications, and accessible
to undergraduate life science students, see
Cohen, Joel E. 1976. Irreproducible Results and the Breeding of Pigs (or
Nondegenerate Limit Random Variables in Biology). BioScience 26:391-394.
4. Let's Make a Deal game. There are 3 identical boxes with a wnning coin hidden
in one of the boxes. Each student gets a turn to try to pick the box with the
winning coin - if they pick the winning box they get their choice of a large
candy bar, if not, they get a small candy bar. The game is played by first
having each student choose from a hat a card that says "SWITCH" or "DON'T
SWITCH", which they keep hidden from the class. They then choose one of the 3
boxes, and I then show them another box which is not the winner, leaving 2
boxes to choose from. They then either switch or don't switch, and we keep
track of who got the winning box and who didn't. The class then compares the
two "strategies" of switch and don't switch based upon our data.