Mathematics SBA

1. 1
This project is based on applying mathematical concepts, skills and procedures to understand,
describe or explain a real world phenomenon. The project is experiment based and involves the
collection of data.

2. 2
Table of Contents
Topic Page
Acknowledgement…………………………………………………………………..3
Project Title…………………………………………………………………………4
Purpose of Project/Problem Statement….…………………………………………..4
Introduction…………………………………………………………………………5
Method of Data Collection………………………………………………………….6
Presentation of Data…………………………………………………………...……7
Interpretation/Analysis of Data……………………………………………………..9
Conclusion………………………………………………………………………….10
Bibliography………………………………………………………………………..11

3. 3
Acknowledgement
With God all things are possible and so firstly I would liketo thank God for the strength and
guidance he has given me towards the completion of this project.
I would like to express gratitude to my mother for her constant motivation, and also to my
teachers.

4. 4
Project Title
The use of simple experiments and mathematical principles to determine the fairness of a coin
tossed.
Purpose of Project/Problem Statement
The students of 5th
form group 8 Biology class were placed in groups as they were about to
conduct a debate. In order to decide the team that would go first in the debate, our teacher did a
coin toss. However, my friend Shelly disapproved of this because she believed that a coin toss
was unfair since in her opinion the coin always lands with heads uppermost. She also believed
that Miss could flick it in a way that it lands on her preferred symbol. In trying to solve the
problem that has arisen in the class, I decided to do a study to determine the fairness of a coin
toss by conducting a research.
This project aims to carry out simple experiments to prove or disprove Shelly’s belief that the
symbol of the coin that lands uppermost is influenced by the way in which the coin is tossed.

5. 5
Introduction
Probability may be defined as the measure of how likely an event is to occur. The probability of
an event occurring is measured on a scale of 0 to 1, that is the probability is a number between 0
(the impossible event) and 1 (the certain event).
In situations where several equally likely outcomes are possible, the probability of a particular
event is measured by:
The number of favourable outcomes
The total number of possible outcomes
After a coin is tossed, the theoretical probability of any of the two possible symbols landing with
a definite face upwards is
1
2
. That is, either the coin lands with heads on top or vice versa.
Coin toss is a mutually exclusive event in that the two symbols cannot land uppermost at the
same time; it’s either one or the other.

6. 6
Method of data collection
A simple experiment was carried out with the use of an ordinary twenty dollar coin which was
thrown 50 times for each of the following situations in order to determine the fairness of a coin
tossed:
1. Placing the coin in a plastic cup, shaking it around for an adequate amount of time before
throwing it onto a carpeted surface.
2. Shaking the coin in the palm of the hands for an adequate amount of time before it was
thrown onto a carpeted surface.
3. Placing the coin in a plastic cup, shaking it around for an adequate amount of time before
it was thrown onto a tiled surface.
4. Holding the coin in the palms of the hand, shaking it around for an adequate amount of
time before throwing it onto a tiled surface.
5. Shaking the coin in a plastic cup for an adequate amount of time before it was thrown
onto a stony surface.
6. Holding the coin in the palms of the hand, shaking it around for an adequate amount of
time before throwing it onto a stony surface.
The frequencies of the number of landings uppermost were recorded in a table after the
experiment was conducted.
Every effort was taken in trying to avoid bias in the experiment as the number of times the coin
was shaken before it was thrown was not predetermined and the experimenter did not try to
throw the coin in any specific way.

7. 7
Presentation of data
Table showing the frequencies of each uppermost landing that occurred during
the experiment that was carried out
A Comparative Bar Chart illustrating a comparison between symbols landing
uppermost throughout the experiment
0
5
10
15
20
25
30
35
Situation 1 Situation 2 Situation 3 Situation 4 Situation 5 Situation 6
Frequencies
Situations throughout the experiment
Comparative Bar Chart
Heads
Tails
Symbol Frequencies Total
FrequenciesSituation
1
Situation
2
Situation
3
Situation
4
Situation
5
Situation
6
Heads 24 21 29 31 25 18 148
Tails 26 29 21 19 25 32 152

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Interpretation of data
The comparative chart indicates the frequency at which the coin was tossed and
resulted in either heads or tail. Situation 1, 2, 3, and 5 all show the probability
spread of frequencies between 20 and 31 which is relatively objective and does not
reveal any attempt in skewing the outcome either way. There is evidence however
to suggest that when the hand is being used the results have greater disparity as is
seen in situation 2, 4 and 6.
On the other hand when a more objective media is used in the case of a cup the results are almost
equal as in the cases of situation 1, 3, and 5 which is perfect. We need to take note of the fact
that the situation will change even more drastically as the environment changes in the case of
situation 6 where the coin was tossed outside in the midst of rocks. We must remember that this
will not be necessary in the classroom as no rocks will be there so the disparity will never be that
wide.
A Pie Chart depicting a comparison of the total frequencies of the symbols
This pie chart is a clear indication that the findings show that the probability of landing heads or tail is
relatively equal.
49%
51%
Pie Chart showing the comparison between the total frequency of
the symbols landing uppermost throughout the experiment
Heads
Tails

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Table showing the probability of the heads landing uppermost in the experiment
that was carried out and also the theoretical probability
Situation 1 2 3 4 5 6
Relative
Probability
24
50
= 0.48
21
50
= 0.42
29
50
= 0.58
31
50
= 0.62
25
50
= 0.5
18
50
= 0.36
TheoreticalPr
obability
1
2
= 0.5
1
2
= 0.5
1
2
= 0.5
1
2
= 0.5
1
2
= 0.5
1
2
= 0.5
Table showing the probability of the tails landing uppermost in the experiment
that was carried out and also the theoretical probability
Situation 1 2 3 4 5 6
Relative
Probability
26
50
= 0.52
29
50
= 0.58
21
50
= 0.42
19
50
= 0.38
25
25
= 0.5
32
50
= 0.64
Theoretical
Probability
1
2
= 0.5
1
2
= 0.5
1
2
= 0.5
1
2
= 0.5
1
2
= 0.5
1
2
= 0.5
Interpretation and Analysis of data
The Pie chart shows at a glance how close the outcomes are. Tails was slightly higher with 51%
as opposed to heads with 49%. This difference is not significant as it is very low.
In situation one (1), the relative probability of heads landing uppermost in the experiment differs
from the theoretical probability by a minus 0.2. In situation 2, 3, 4 and 6, the relative
probabilities differ by a -0.8, +0.8, +1.2 and a -1.4. In situation 5, the relative probability of
heads landing uppermost in the experiment was the same as the theoretical probability of heads
landing uppermost in a typical coin toss. If all the differences are to be added, this will show a
relative difference of only -0.2. This tells us that the coin is fairly unbiased in a typical coin toss
since there is such a small difference between the relative probability and the theoretical
probability.

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Conclusion
In analysing the results it was found that the relative probability of each symbol landing
uppermost was rather close the theoretical probability and would have been closer if the number
of throws were greater. This proves that a coin toss is fairly unbiased since there is an equal
probability of either symbol landing uppermost.
A coin toss is based on chance and therefore, there is equal possibility of either symbols landing
uppermost in any instance. And so, in solving the problem faced in the class by the students,
Miss may use the probability theory or this experiment as a guide to explain the concept of
probability, and help the students to see that there is no bias in a coin toss once it is done
properly. By using this method the child is assured based on the experiment that was done, that
this is a fair and just way of selecting who goes first and there is no favouritism.
It is recommended that in future related areas that persons conducting a coin toss should do a
series of throws then find the average in order to improve the accuracy of the results. Placing the
coin in an object such as a cup will reduce biasness.

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Bibliography
 www.google.com
 Developing Mathematical Minds for CSEC Additional Mathematics by Fayad W. Ali and
Shereene A. Khan