Objective: To illustrate the use of
statistical simulation to find a rule which will lead to the
prediction of probabilities.Level:
General math classes in high school or college,
elementary probability courses, or probability courses for math
majors.
Prerequisites: Basic probability concepts
and properties of circles and squares.
Platform:
Any computing environment, which supports
random number generation including calculators. An Excel program and a
Matlab program are included. A reference is included for using
calculators to perform this demo.
Instructor's
Notes:
A common game at carnivals and fund raisers involves
coin tossing. There are various forms of such games. The game we
investigate in this demo consists of a board with a grid of uniform
squares. The objective of the game is toss a coin onto the board. If the
coin lands entirely within a square then you win a prize, otherwise you
lose your coin. In Figure 1 the red circles indicate winners.
(The carnival board is usually much larger than shown here.)
Figure 1. |
A player soon realizes that the
chances of winning are related to the size of the squares and the size
of coin used. So the design of the board and the type of coin used are
important to entice players to participate, yet have the game operator
not give away too many prizes. Determining the design of the game can
be investigated in two ways:
<> Fix the size of the coin, but
vary the size of the squares.
<> Fix the size of squares but
vary the diameter of the coin.
In our development we will fix the size of squares
and vary the diameter of the coin. Since countries have different size
coins for various monetary denominations we employ rather generic
dimensions. The discussion can be specialized for various sizes of
coins.We present two approaches. Approach 1
concentrates on finding a formula for determining the probability of a
winning toss by a simple construction. Approach 2 uses a game based
approach of tossing experiments to get an idea of the approximate
probability of a winning toss.
Approach 1. Suppose a single square on the board
is
while the coin has radius r so its diameter d = 2r. The
crucial question is
What constraint should be imposed on the
location of the center
of the circle to be declared a winning toss?
To aid in this regard, experiment by constructing a
square 10 units by 10 units on construction paper or poster board. Next
cut a circle of radius 2.5 units out of cardboard or poster board. (See Figure 2.) Through the center of the circle insert your
pencil and move the circle around on the 10 by 10 square in such a way that you
always have a winner. For each winning position mark the location of the center with the
pencil. Do this lots of times to try to determine the constraint on
the centers that will guarantee winners. In Figure 3 we have simulated
moving the circle into winning positions and leaving a "track" of the
centers.
Figure 2. |
Figure 3. |
Use your picture to carefully estimate the ratio
of the area of the region of winning positions to the area of
the square. If you want to figure this out for yourself go to the next
paragraph; for a substantial hint click here
to see a picture where we have simulated the drawing of the winning
center positions using a computer
routine.
We simulated the coin toss game with s = 10
and r = 2.5 for various numbers of tosses. The results are
displayed in Table 1. Compute the ratio of winners to tosses in each
case. Compare these results with the ratio of the area of the region of
winners to the area of the square from your experiment as
illustrated in Figures 2 and 3.
Table 1. |
Assuming that your comparison of ratios
are fairly close as the number of tossed increased make a conjecture (a
guess based on limited evidence) about the probability of a win in the
coin toss game when s = 10 and r = 2.5. (In the event the
comparisons are not close, use the hint mentioned below Figure 3 by
clicking here and then revise your ratio of
the area of the region of winners to the area of the square.)
Next develop a formula in terms of letters s
and r for the probability of tossing a winner in the case when s = 10 and r
= 2.5 so that it gives the answer from your conjecture. Check your
formula by substituting in the values 10 and 2.5. If the formula works
for the special case of s = 10 and r = 2.5, then we want
to see if it generalizes to other cases. When s = 10 and r = 1, the
probability of tossing a winner is about 0.64. In the formula you
developed substitute s = 10 and r = 1 and see if you get
approximately 0.64. If you are having difficulty developing the formula,
click here for help.
Once you have checked your formula as described in
the previous paragraph use it to complete the Table 2 by entering the
probability of a winning toss.
Table 2. |
If groups are used to investigate the probability of
a winning toss you the ideas outlined above, then the size of the square
can be chosen differently for various groups.
After students have completed the investigation a
follow up using an Excel file whose screen is shown in Figure 4 can be
employed. Click here or on Figure to execute
or download the Excel file.
Figure 4. |
Approach 2. This approach emphasizes
statistical simulation. Using the Excel program, whose screen is shown
in Figure 4, have students (or groups of students) complete a set of
experiments for each of the choices for a radius. (Click
here to execute or download the Excel
program.) Table 3 can be used for recording the results of the
experiments. After compiling the information requested the students (or
groups) should estimate the probability of tossing a winner for each of
the choices of a radius by computing the ratio of winners to the
number of tosses. Then the students (or groups) should compare their
results before a conjecture (a guess based on limited evidence)
is made for each case of the radius.
Table 3. |
Remind students to type GO in then
yellow box and that holding down the F9 key will generate
successive tosses. The execution is quite rapid with F9 key
depressed. Even if they slightly exceed the number of tosses in an
experiment they need not repeat it, but just record the actual numbers
tallied by the program. It is recommended that for each selection of
the number of tosses the experiment should be started at the beginning
instead of "piggy backing" on the previous number.
Auxiliary Resources:
1. At
http://www.hpcalc.org/details.php?id=693 is a description and a
zipped file for implementation of the coin toss game on an HP38G
calculator. A nice extension of the the statistical experimentation is
suggested. Use the experimental data, say for 500 tosses, for each
choice r of the radius to estimate p, the probability of
a win. To the data pairs (r, p) generate a least squares
quadratic polynomial that gives an approximate formula in terms of
r to predict p. For an example, click
here.
2. As a web exercise, have students
search for coin toss games similar to that described here that have a
different configuration to the board or some other way for players to
win.
3. A Matlab program, which can be
downloaded by clicking here, functions
differently than the Excel program. The images of the circles
representing the coin tosses are retained with winners appearing in
red. The radius can be any value in the interval (0,4.9). To see an
animation of this program click here.
4. Other statistical demos in the
collection Demos with Positive Impact:
<> A collection of probability
demos:
http://astro.temple.edu/~dhill001/elemprob/elemprob.html
<> Monte Carlo area simulations:
http://astro.ocis.temple.edu/~dhill001/montecarlo/monte.html