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.