# Coin Toss Game

Coin Toss Simulation

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. M. Haruta, MFlaherty, J. McGiveney, and R. McGiveney, "Coin Tossing", Mathematics Teacher, Vol. 89, No. 8, November 1996, p. 642-645.

5. Other statistical demos in the collection Demos with Positive Impact:

<> A collection of probability demos:
http://mathdemos.org/mathdemos/elemprob/elemprob.html

<> Monte Carlo area simulations:
http://mathdemos.org/mathdemos/montecarlo/monte.html

Comment: Interest in the coin toss game came from an inquiry from a computer science student in England who had an assignment to write a Matlab program that computed the probability of a winning toss. He had seen the demos at the URLs listed above, but was having some difficulty formulating the strategy for the simulation. We exchanged a few emails and I referred him to the site in Resource 1.

Credits:  This demo was developed by  David R. Hill ,Department of Mathematics, Temple University and is included in Demos with Positive Impact with his permission.

DRH 5/23/2005     Last updated 9/15/2010 DRH

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