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
Here we go........
I. Random Toss of a single
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:
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.
Probability Percentage of Tossing Heads
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).
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
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!
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