Exploring Probability
Write up #2
By Nancee Klaff
October 3, 2001

Defining the Problem:

I set out to discover patterns of the probability of flipping a coin and getting "heads."    I set up many different samples to explore how probability varies.  Each time, I made a guess of what I thought would occur, discovered the actual behavior, and in some cases, I made conclusions based on my findings.    I examined the following samples:
1.  How the number of samples (tosses) effects probability
2.  How the number of coins per toss effects the probability
3.  How changing the probability of each possible result effects
probability

Here we go........

Discussion of Procedures and Results:

I.  Random Toss of a single coin.

When a coin, one side "heads" and one side "tails," is tossed, the expected results are either heads (1)  or tails (0).  Before setting up a random generation of numbers between 1 and 0, rounded to the nearest whole number, I predicted that the probability of getting a (1) or "heads" would be 1/2, or 50%. Then I set up the random generation on an Excel Spreadsheet of 100 tosses.  I made additional columns for the total number of "heads" out of the 100 tosses and for the percentage.

Here's what I found:  I found that the result was not always 1/2, or 50%.  I found that it varied.  For example, my first set of results produced 38% heads.  So, I considered the possible factors that might have caused this result. I reasoned that my sample of 100 was too small to produce accurate results. To prove this theory, I then generated a total of 3500 samples and compared the results:

 Heads per 100 Sample Tosses Percentage Heads per 3500 Sample Tosses Percentage 36 45 48 29 42 36% 45% 48% 29% 42% 1720 1704 1632 1840 1781 49% 48.7% 46.6% 53% 51%

From these results I was able to conclude that probability results become more accurate as the number of samples increases (conclusion #1).

II.  Random Toss of 2 Coins

Next, I set up a spreadsheet page with 2 columns of random numbers between 1 and 0, which represented either "heads" or "tails."    I set out to discover how adding a second coin per toss would change the probability of getting a heads.

In this toss, there are 4 possible outcomes:  Heads/Heads, Head/Tails, Tails/Heads, and Tails/Tails........therefore, I predicted that the probability of getting no heads was 25%, the probability of getting 2 heads was 25%, and the probability of getting one heads was 50%.

The results were much as I expected.  I used a sample of 130 tosses in my experiment.  I then made a chart summarizing the both the one coin toss and the two coins toss.

So, when compared to one coin, the probability of getting at least one heads when 2 coins are tossed is greater.   From that, I drew the following conclusion:If the ways in which a certain result can be attained increases, the probability of achieving that certain result also increases (conclusion #2).

 A Direct Proportion Increase the number of ways to get "heads"  then the probability of getting "heads" increases.

III.  Many coins per Toss

In order to explore whether or not this conjecture is actually true, I decide to test it by increasing the number of coins used per toss several times.  If my conjecture is true, the probability of getting heads should increase as the number of possible ways to get a heads in a toss increases. In fact, that is what happened when I set up columns for 3 coin tosses and 5 coin tosses.  Here are the results summed up in a chart:

Probability Percentages for Many Coins per Toss

As the chart summarized, my findings were consistant with the conjecture from above.  These findings, however, were based on a sample size of 130 tosses.  Based on the first conclusion (see first conclusion), these results may not be very reliable because the sample size is so small.  After considering that, I went back and changed all the sample sizes to 2000 and the results were similar.  I was then able to conclude that the above chart was a fair representation of the true probability of tossing coins.

IV.  Using Coins with Varied Individual Probabilities:

Another possibility to explore is if the probability of getting heads to tails is changed from 1:1 to 2:1, how does this change the probability of all the examples already explored?

In order to set this up, I expanded the range of the randomly generated number to from 0 to 2.  I considered both 0 and 1 to represent heads and 2 represented tails.  I then generated columns and created a chart of results.

The expected results would be that the probability for getting heads would be greater now.  This is because the possible ways increased.  Simply put, with one coin the possible outcomes were:  h/h, h/t, t/h, t/t.  With the skewed coins, the possible outcomes are: h1/h1, h1/h2, h1/t, h2/h1, t/h1, h2/h2, t/h2, h2/t, t/t.  How many of these new outcomes have heads?  8/9, or 88.9%.  How many of the previous outcomes have heads?  3/4, or 75%.  Thus, in the following chart of new results, we can expect the probabilities to be greater.

Probability of Coins with Varied Individual Probabilities

Compare these results to the results on the chart with non-varied individual probabilities.

The second conclusion made earlier holds true for these results also.  (See second conclusion.)    So, after examining the many examples of coin probability, I was able to conclude the following:
1.  Greater sample sizes produce more accurate probabilities
2.  If the possible outcomes in which an element is present is increased, the probability of that element also increases.

Implications for Instruction

This particular project does not require very much background knowledge of probability properties.  A student can complete this assignment having very little knowledge of probability, provided they understand how to set up a spreadsheet and generate random numbers.  The probability of flipping one coin is basic enough to introduce before any other probability topics are covered.
This project is ideal in introducing the idea of varying probabilities to students.  It allows students to see the results on their own and make their own conclusions about the behavior of each set of coins.  The conclusions are rather easy to develop and few hints should be necessary to guide them.  If a teacher is looking for a good way to help students help them selves to some knowledge, this is it!