Addmaths Project 2010

8/8/2019 Addmaths Project 2010

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Appreciation Part 1 Part 2 Part 3 Part 4 Part 5 Further Exploration Reflection


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After many weeks of a tough struggle and some hard work to complete the

assignment given to us by our teacher, Mrs. Sharifah Nur Afizah, I finally manage to

complete it within 2 weeks with satisfaction and senses of success because I have understood

more thoroughly about the interest and investment more than before. I am more than grateful

to all parties who have helped me in the process of completing this assignment. It was a great

experience for me as I have learnt to be more independent and to work as group. For this, I

would like to take this opportunity to express my gratitude once again to all partiesconcerned.

Firstly, I would like to thanks my Additional Mathematics teacher, Mrs. Sharifah Nur

Afizah for patiently explaining and guiding us step by step the proper and precise way to

complete this assignment. With her help and guidance, many problems I have encountered

had been solved.

Beside that, I would like to thanks my parents for all their support and encouragement

they have given to me. In addition, my parents had given me guidance on the methods to

account for investment which have greatly enhanced my knowledge on particular area. Last

but not least, I would like to express my gratitude to my friends, who have patiently

explained to me and did this project with me in group.


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Most experimental searches for paranormal phenomena are statistical in nature. A subject

repeatedly attempts a task with a known probability of success due to chance, then the

number of actual successes is compared to the chance expectation. If a subject scores

consistently higher or lower than the chance expectation after a large number of attempts, one

can calculate the probability of such a score due purely to chance, and then argue, if the

chance probability is sufficiently small, that the results are evidence for the existence of somemechanism (precognition, telepathy, psychokinesis, cheating, etc.) which allowed the subject

to perform better than chance would seem to permit.

Claims of evidence for the paranormal are usually based upon statistics which diverge

so far from the expectation due to chance that some other mechanism seems necessary to

explain the experimental results. To interpret the results of our RetroPsychoKinesis

experiments, we'll be using the mathematics of probability and statistics, so it's worth

spending some time explaining how we go about quantifying the consequences of chance.


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TASK 1

(a)

HISTORY OF PROBABILITY

Probability is a way of expressing knowledge or belief that an event will occur or has

occurred. In mathematics the concept has been given an exact meaning inprobability theory,

that is used extensively in such area of study as mathematics, statistics, finance, gambling,

science, and phisolophy to draw conclusions about the likelihood of potential events and

underlying mechanics of complex systems.

The scientific study of probability is a modern development. Gambling shows that

there has been an interest in quantifying the ideas of probability for millennia, but exact

mathematical descriptions of use in those problems only arose much later.

According to Richard Jeffrey, "Before the middle of the seventeenth century, the term

'probable' (Latinprobabilis) meant approvable, and was applied in that sense, univocally, to

opinion and to action. A probable action or opinion was one such as sensible people would

undertake or hold, in the circumstances."[4]

However, in legal contexts especially, 'probable'

could also apply to propositions for which there was good evidence.

Aside from some elementary considerations made by Girolamo Cardano in the 16th

century, the doctrine of probabilities dates to the correspondence ofPierre de

Fermat and Blaise Pascal (1654). Christiaan Huygens (1657) gave the earliest known

scientific treatment of the subject. Jakob Bernoulli'sArs Conjectandi (posthumous, 1713)

and Abraham de Moivre'sDoctrine of Chances (1718) treated the subject as a branch of

mathematics. See Ian Hacking'sThe Emergence of Probability and James Franklin'sThe

Science of Conjecture for histories of the early development of the very concept of

mathematical probability.

The theory of errors may be traced back to Roger Cotes'sOpera

Miscellanea (posthumous, 1722), but a memoir prepared by Thomas Simpson in 1755

(printed 1756) first applied the theory to the discussion of errors of observation. The reprint

(1757) of this memoir lays down the axioms that positive and negative errors are equally

probable, and that there are certain assignable limits within which all errors may be supposed

to fall; continuous errors are discussed and a probability curve is given.

Pierre-Simon Laplace (1774) made the first attempt to deduce a rule for the combination of

observations from the principles of the theory of probabilities. He represented the law of


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probabilit of errors by a curve y = (x), x bei any error and yits probability, and laid down

t ree properties oft is curve:

1. itis symmetric as to t e y-axis;2. t e x-axis is an asymptote, t e probability oft e error being 0;3. t e area enclosed is 1, it being certain t at an error exists.

He also gave (1781) a formula fort e law of facility of error (a term due to Lagrange, 1774),

but one which led to unmanageable equations.DanielBernoulli (1778) introduced the

principle ofthe maximum product ofthe probabilities of a system of concurrent errors.

PR

BAB

E

R

Li e othertheories, the theory of probabilityis a representation of probabilistic

concepts in formaltermsthatis, in terms that can be considered separately from theirmeaning. These formalterms are manipulated by the rules of mathematics and logic, and any

results are then interpreted ortranslated backinto the problem domain.

There have been atleasttwo successful attempts to formali e probability, namely

the Kolmogorov formulation and the Cox formulation. In Kolmogorov's formulation

(seeprobability space), sets are interpreted as events and probability itself as a measure on a

class of sets. In Cox's theorem, probability is taken as a primitive (thatis, not further

analyzed) and the emphasis is on constructing a consistent assignment of probability values

to propositions. In both cases, the laws of probability are the same, except fortechnical

details.

There are other methods for quantifying uncertainty, such as theDempster-Shafer

theory orpossibility theory, butthose are essentially different and not compatible with the

laws of probability as they are usually understood.

APPLICA ION

Two major applications of probability theory in everyday life are in riskassessment

and in trade on commodity markets. Governments typically apply probabilistic methods

inenvironmental regulation where itis called "pathway analysis", often measuring well-

being using methods that are stochasticin nature, and choosing projects to undertake based

on statistical analyses oftheir probable effect on the population as a whole.

A good example is the effect ofthe perceived probability of any widespread Middle

East conflict on oil prices - which have ripple effects in the economy as a whole. An

assessment by a commodity traderthat a waris more likely vs. less likely sends prices upor


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down, and signals other traders of that opinion. Accordingly, the probabilities are not

assessed independently nor necessarily very rationally. The theory ofbehavioral

finance emerged to describe the effect of such groupthinkon pricing, on policy, and on peace

and conflict.

It can reasonably be said that the discovery of rigorous methods to assess andcombine probability assessments has had a profound effect on modern society. Accordingly,

it may be of some importance to most citizens to understand how odds and probability

assessments are made, and how they contribute to reputations and to decisions, especially in

a democracy.

Another significant application of probability theory in everyday life is reliability.

Many consumer products, such as automobiles and consumer electronics, utilize reliability

theory in the design of the product in order to reduce the probability of failure. The

probability of failure may be closely associated with the product's warranty.

(b)

CATEGORIES OF PROBABILITY

Empirical Probability of an event is an "estimate" that the event will happen based on how

often the event occurs after collecting data or running an experiment (in a large number of

trials). It is based specifically on direct observations or experiences.

Empirical Probability Formula

P(E) = probability that an event,E, will occur.

top = number of ways the specific event occurs.bottom = number of ways the experiment could

occur.

Theoretical Probability of an event is the number of ways that the event can occur, divided

by the total number of outcomes. It is finding the probability of events that come from a

sample space of known equally likely outcomes.


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Theoretical Probability Formula

P(E) = probability that an event,E, will occur.n(E) = number of equally likely outcomes ofE.

n(S) = number of equally likely outcomes of samplespace S.

Comparing Empirical and Theoretical Probabilities:

Empirical probability is the probability a person calculates from many different trials.

For example someone can flip a coin 100 times and then record how many times it came up

heads and how many times it came up tails. The number of recorded heads divided by 100 is

the empirical probability that one gets heads.

The theoretical probability is the result that one should get if an infinite number of

trials were done. One would expect the probability of heads to be 0.5 and the probability of

tails to be 0.5 for a fair coin.


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(a)

Suppose you are playing the Monopoly game with two of your friends. To start the game,

each player will have to toss the die once. The player who obtains the highest number will

start the game. List all the possible outcomes when the die is tossed once.

={1,2,3,4,5,6}

(b)

Chart

Die 2

6 (1,6) (2,6) (3,6) (4,6) (5,6) (6,6)

5 (1,5) (2,5) (3,5) (4,5) (5,5) (6,5)

4 (1,4) (2,4) (3,4) (4,4) (5,4) (6,4)

3 (1,3) (2,3) (3,3) (4,3) (5,3) (6,3)

2 (1,2) (2,2) (3,2) (4,2) (5,2) (6,2)

1 (1,1) (2,1) (3,1) (4,1) (5,1) (6,1)

Die 1

0 1 2 3 4 5 6


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Table

1 2 3 4 5 6

1 (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)


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(a)

(b)

A = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6)

(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)

(3,1), (3,2), (3,3), (3,4), (3,5), (3,6)

(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)

(5,1), (5,2), (5,3), (5,4), (5,5), (5,6)

(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

B =

P = Both number are prime

= {(2,2), (2,3), (2,5), (3,3), (3,5), (5,3), (5,5)}

Q = Difference of 2 number is odd

= { (1,2), (1,4), (1,6), (2,1), (2,3), (2,5), (3,2), (3,4), (3,6), (4,1), (4,3), (4,5), (5,2),

(5,4), (5,6), (6,1), (6,3), (6,5) }

Sum of the dots on both

turned faces(x)

Possible outcomes Probability, P(x)

2 (1,1) 1/36

3 (1,2)(2,1) 2/36=1/18

4 (1,3)(2,2)(3,1) 3/36=1/12

5 (1,4)(2,3)(3,2)(4,1) 4/36=1/9

6 (1,5)(2,4)(3,3)(4,2)(5,1) 5/36

7 (1,6)(2,5)(3,4)(4,3)(5,2)(6,1) 6/36

8 (2,6)(3,5)(4,4)(5,3)(6,2) 5/36

9 (3,6)(4,5)(5,4)(6,3) 4/36=1/9

10 (4,6)(5,5)(6,4) 3/36=1/12

11 (5,6)(6,5) 2/36=1/18

12 (6,6) 1/36


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C = P U Q

= {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

= {(1,1), (1,3), (1,5), (2,2), (2,4), (2,6), (3,1), (3,3), (3,5), (4,2), (4,4), (4,6), (5,1), (5,3),

(5,5), (6,2(, (6,4), (6,6)}

D = P R

= {(2,2), (3,3), (3,5), (5,3), (5,5)}


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(a)

x f fx fx2

2 2 4 6

3 4 12 364 4 16 64

5 9 45 225

4 24 144

7 11 77 539

8 4 32 256

9 6 54 486

10 3 30 300

11 1 11 121

12 2 24 128

77 50 329 2305

From the table,

f = 50

fx = 329

fx = 2467

(i)

(ii)


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(iii)

(b)

x f fx fx2

2 4 8 16

3 5 15 45

4 6 24 96

5 16 80 4006 12 72 432

7 21 147 1029

8 10 80 640

9 8 72 648

10 9 90 900

11 5 55 605

12 4 48 576

=100 = 91 =5387

From the table,


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(i)

(ii)

(iii)


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(a)

x 2 3 4 5 7 8 9 10 11 12

P(x) 1/36 1/18 1/12 1/9 1/36 1/6 1/36 1/9 1/12 1/18 1/36


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(b)

Part 4

Part 5

n=50 n=100

Mean 6.58 6.91 7.00

Variance 6.0436 6.1219 5.83

Standard Deviation 2.458 2.474 2.415

For n = 50, mean=6.58

For n = 100, mean=6.91

Actual mean=7

Hence, we get different mean for different number of experiment. As the number of experiments getting bigger, the empirical (experimental) mean will

tend to be close to the theoretical(actual) mean.

The same will goes with the variance and standard deviation

(c)

0 < mean 7

n becomes n becomes

smaller bigger


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Inprobability theory, the law oflarge numbers (LLN) is a theoremthat describes the result

of performing the same experiment a large number oftimes. According to the law,

the average ofthe results obtained from a large number oftrials should be close to

the expected value, and willtend to become closer as more trials are performed.

For example, a single roll of a six-sided die produces one ofthe numbers 1, 2, 3, 4, 5, 6, each

with equalprobability. Therefore, the expected value of a single die rollis

According to the law oflarge numbers, if a large number of dice are rolled, the

average oftheir values (sometimes called thesample mean) is likely to be close to 3.5, with

the accuracy increasing as more dice are rolled.

Similarly, when a fair coinis flipped once, the expected value ofthe number of heads

is equalto one half. Therefore, according to the law oflarge numbers, the proportion of heads

in a large number of coin flips should be roughly one half. In particular, the proportion of

heads aftern flips willalmost surelyconvergeto one half as n approaches infinity.

Though the proportion of heads (and tails) approaches half,almost surelythe absolute

(nominal) difference in the number of heads and tails will become large as the number of

flips becomes large. Thatis, the probability thatthe absolute difference is a small number

approaches zero as number of flips becomes large. Also, almost surely the ratio ofthe

absolute difference to number of flips will approach zero. Intuitively, expected absolute

difference grows, but at a slower rate than the number of flips, as the number of flips grows.

The LLN is important because it"guarantees" stable long-term results for random

events. For example, while a casino may lose money in a single spin oftheroulette wheel, its

earnings willtend towards a predictable percentage over a large number of spins. Anywinning streak by a player will eventually be overcome by the parameters ofthe game. Itis

importantto rememberthatthe LLN only applies (as the name indicates) when al

numberof observations are considered. There is no principle that a small number of

observations will converge to the expected value orthat a streak of one value will

immediately be "balanced" by the others.


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While I was conducting the project, I have learned many moral values that I

could practice in my daily life. I was very enthusiastic and excited to start project during the

school holidays. This project work had taught me to be more independent when doing

something especially the homework given by the teacher. I also learned to be a disciplined

and dedicated type of student which is always sharp on time while handing in work,

completing the work by oneself and researching the information from the internet.


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Addmaths Project 2010

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