add math project 2010
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SMK PUCHONG BATU 1447100 PUCHONG
SELANGOR DARUL EHSAN.
PROJECT WORK FORADDITIONAL MATHEMATICS 2010
PROBABILITY IN OUR DAILY LIFE
NAME : MUHAMMAD HARIZ FADHILLA BIN SULAIMAN
CLASS : 5 AMANAH
IC NUMBER : 930805-10-5119
SUBJECT TEACHER : MR PHANG CHIA CHEN
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[ CONTENT ]
INTRODUCTION…………………………………………..1
AIM………………………………………………………….2
PART 1…………………………………………………...3 - 6
PART 2……………………………………………….…..7 - 8
PART 3………………………………………………….9 - 12
CONCLUSION………………………………………...13 - 14
ACKNOWLEDGEMENT…………………………………..15
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[ INTRODUCTION ]
Probability theory is the branch of mathematics concerned with
analysis of random phenomena. The central objects of probability theory
are random variables, stochastic processes, and events: mathematical
abstractions of non-deterministic events or measured quantities that may
either be single occurrences or evolve over time in an apparently random
fashion. Although an individual coin toss or the roll of a die is a random
event, if repeated many times the sequence of random events will exhibit
certain statistical patterns, which can be studied and predicted. Two
representative mathematical results describing such patterns are the law of
large numbers and the central limit theorem.
As a mathematical foundation for statistics, probability theory is
essential to many human activities that involve quantitative analysis of large
sets of data. Methods of probability theory also apply to descriptions of
complex systems given only partial knowledge of their state, as in statistical
mechanics. A great discovery of twentieth century physics was the
probabilistic nature of physical phenomena at atomic scales, described
in quantum mechanics.
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[ AIM ]
The aims of carrying out this project work are :
i. to apply and adapt a variety of problem-solving strategies to solve problem;
ii. to improve thinking skills;
iii. to promote effective mathematical communication;
iv. to develop mathematical knowledge through problem solving in a way that increases students’ interest and confidence;
v. to use the language of mathematics to express mathematical ideas precisely;
vi. to provide learning environment that stimulates and enhances effective learning;
vii. to develop positive attitude towards mathematics.
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PROJECT WORK
[ PART 1 ]
(A) History:
The mathematical theory of probability has its roots in attempts to
analyze games of chance by Gerolamo Cardano in the sixteenth century,
and by Pierre de Fermat and Blaise Pascal in the seventeenth century
(for example the "problem of points"). Christiaan Huygens published a
book on the subject in 1657.
Initially, probability theory mainly considered discrete events, and its
methods were mainly combinatorial.
Eventually, analytical considerations compelled the incorporation
of continuous variables into the theory.
This culminated in modern probability theory, the foundations of
which were laid by Andrey Nikolaevich Kolmogorov. Kolmogorov
combined the notion of sample space, introduced by Richard von Mises,
and measure theory and presented hisaxiom system for probability
theory in 1933. Fairly quickly this became the undisputed axiomatic
basis for modern probability theory.
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(B) Two categories of PROBABILITY
The difference between empirical and theoretical probability is
an important part of our ability to apply probability to a real world set of
data. What is the difference between empirical and theoretical
probability? Give two to three examples speculative of professions where
probability could be used. Explain your answer
The difference between empirical, means observation or
experience and theoretical probability or speculative are as clear as night
and day. Empirical probability is the data that has been proven through
trial and error such as the statics on the accidents that involve driving
while under the influence. Even the proven data for deaths that are
smoking related. The theoretical probability is like guessing and taking a
chance you are right much like playing a game of cards you are taking
that chance you have the better hand. Insurance policies are made
possible by empirical probability. We know the amount of accidents,
and we know the amount of times something happens without error.
Based on that, it can be calculated what the chance (and thus the cost) is
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of a certain event. (professional) Gambling is about theoretical
probability. One can assume that all the chips, cards, tables or whatever
are completely fair (or even calculate the unfairness, based on the
method of shuffling), so one can calculate the odds of a certain set of
cards coming up, before they ever have.
Dangerous medical procedures can also have empirical probability
playing as a factor. There is always a chance that someone dies under the
knife, or that someone cures on their own. Based on those odds, a doctor
could advise for or against certain procedures. Those odds are based on
other patients who have gone through the same thing.
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[ PART 2 ]
(A) When we playing the monopoly, we have to toss the die once to find
who is going to start the game first. The possible outcomes when we toss
the die is 1,2,3,4,5, and 6. This is because a die has 6 surface as shown
in Figure 1.
Figure 1.
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(B) When we tossed two dice simultaneously, the possible outcomes is as
shown in the Table 2.
DIE1/DIE2 1 2 3 4 5 61 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6)3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6)5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6)6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
Table 1.
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[ PART 3 ]
(A)The Table 2 below shows the probability and the possible
outcomes to get the sum of the dots on both turned-up faces.
Sum of the dots on both
Possible outcomesProbability,
P(x)turned-up faces
(x)2 (1,1) 1/363 (1,2),(2,1) 1/184 (1,3),(2,2),(3,1) 1/125 (1,4),(2,3),(4,1),(3,2) 1/96 (5,1),(4,2),(3,3),(2,4),(1,5) 5/36
7(6,1),(5,2),(4,3),(3,4),(2,5),
(1,6) 6/368 (6,2),(5,3),(4,4),(3,5),(2,6) 5/369 (6,3),(5,4),(4,5),(3,6) 1/9
10 (6,4),(4,6),(5,5) 1/1211 (6,5),(5,6) 1/1812 (6,6) 1/36
Table 2.
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(B) Based on the Table 2:
A ={The two numbers are not the same} ={(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,3),(2,4),(2,5),(2,6),
(3,1),(3,2),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,5),(4,6),(5,1), (5,2),(5,3),(5,4),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5)}
B = {The product of the two numbers is greater than 36}= ø
P = Both number are primeP = {(2,2),(2,3),(2,5),(3,3),(3,5),(5,3),(5,5)}
Q = Difference of 2 number is oddQ = {(1,2),(1,4),(1,6),(2,1),(3,4),(3,6),(4,1),(4,3),(4,5),(5,4),
(5,6),(6,1),(6,3),(6,5)}
C = {Both numbers are prime number or the difference between two numbers is odd}
C = {P U Q}C = {(1,2),(1,4),(1,6),(2,1),(2,2),(2,3),(2,5),(3,2),(3,3),(3,4),
(3,6),(4,1),(4,3),(4,5),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,3),(6,5)}
R = The sum of 2 numbers are even
D = {The sum of the two numbers are even and both }D = {P ∩ R}D = {(2,2),(3,3),(3,5),(5,3),(5,5)}
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[ PART 4 ]
(A) An activity has been conducted by tossing two dice simultaneously 50
times. The frequency (f) of the sum of all dots on both turned-up faces
has been recorded in Table 3 below. The value of mean, variance and
standard deviation of the data has been calculated.
Sum of the two numbers (x) Frequency(f) fx fx²2 2 4 83 5 15 454 5 20 805 1 5 256 4 24 1447 8 56 3928 8 64 5129 5 45 405
10 7 70 70011 3 33 36312 2 24 288
Total 50 360 2962Table 3.
Mean = 360÷50 = 7.2
Varience = 2962÷50-7.2² = 7.4
Standard Deviation = √7.4
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= 2.72
(B) When the number of tossed of the two dice simultaneously is increase to
100 times, the value of mean is also change.
(C)Another activity same as (A) has been conducted by tossing two dice
simultaneously 100 times.
Sum of the two numbers (x) Frequency(f) fx fx²2 2 4 83 6 18 544 10 40 1605 9 45 1446 16 96 5767 17 119 8338 12 96 7689 11 99 891
10 13 130 130011 2 22 24212 2 24 288
Total 100 693 5264Table 4.
Mean = 693÷100= 6.93
Varience = 5264÷100-6.93²= 4.6151
Standard Deviation = √4.6151= 2.148
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Therefore, the prediction that has been made in (B) is true.
The mean, varience, and standard deviation is change
although the number of tossed of the dice was increase until
100 times because when the number of tossed change the
frequency also change.
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[ PART 5 ]
(A)Based on Table 2, the actual mean, the varience and the standard
deviation of the sum of all dots on the turned-up faces by using the
formulae given was determined.
Mean = ∑ x P(x)
Sum of the two numbers (x) Frequency (f) fx fx²
2 1 2 43 2 6 184 3 12 485 4 20 1006 5 30 1807 6 42 2948 5 40 3209 4 36 324
10 3 30 30011 2 22 24212 1 12 144
total 36 252 1974Varience = ∑ x² P(x) – (mean)²
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Maen = 252÷36
= 7
Varience = 1974÷36-7²
= 5.83
Standard Deviation = √5.83
= 2.415
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(B) The mean, variance and standard deviation obtained in
Part 4 and Part 5 is not same because Part 4 was the
Empirical Probability and Part 5 was the Theoretical
Probability. For example, the empirical probability of
rolling a 7 is 4/25 = 16%. But the theoretical probability of
rolling a 7 is 6/36 = 1/6 = 16.7%. Table below show the
comparison between Mean, Variance, and Standard
Deviation.
Part 4Part 5
n= 50 n= 100
Mean 7.2 6.93 7
Variance 7.4 4.6151 5.83
Standard deviation 2.72 2.148 2.415
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(C)The range of mean of the sum of all dots on the turned-
up faces as n changes is 2≤mean≤12. This is because as the
number of toss, n, increases, the mean will get closer to 7. 7
is the theoretical mean.
FURTHER EXPLORATION
Common intuition suggests that if a fair coin is tossed many times,
then roughly half of the time it will turn up heads, and the other half it
will turn up tails. Furthermore, the more often the coin is tossed, the
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more likely it should be that the ratio of the number of heads to the
number of tails will approach unity. Modern probability provides a
formal version of this intuitive idea, known as the law of large
numbers. This law is remarkable because it is nowhere assumed in
the foundations of probability theory, but instead emerges out of
these foundations as a theorem. Since it links theoretically derived
probabilities to their actual frequency of occurrence in the real world,
the law of large numbers is considered as a pillar in the history of
statistical theory.
The law of large numbers (LLN) states that the sample
average of (independent and identically
distributed random variables with finite expectation μ) converges
towards the theoretical expectation μ.
It is in the different forms of convergence of random variables that
separates the weak and the strong law of large numbers
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It follows from LLN that if an event of probability p is observed
repeatedly during independent experiments, the ratio of the observed
frequency of that event to the total number of repetitions converges
towards p.
Putting this in terms of random variables and LLN we have
are independent Bernoulli random variables taking values 1 with
probability p and 0 with probability 1-p. E(Yi) = p for all i and it follows
from LLN that converges top almost surely.
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[ ACKNOWLEDGEMENT ]
I would like to express my gratitude and thanks to my teacher, Mr
Phang Chia Chen for his wonderful guidance for me to be able to complete
this project work. Next, is to my parents for their continuous support to me
throughout this experiment. Special thanks to my friends for their help, and
to all those who contributed directly or indirectly towards the completion of
this project work.
Throughout this project, I acquired many valuable skills, and hope
that in the years to come, those skills will be put to good use.
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