Add Math Project 2010 2nd

ADDITIONAL MATHEMATICS PROJECT WORK 2 2010

ADDITIONAL

MATHEMATICS

PROJECT WORK 2010

EDWARD THOMAS

930605-12-6353

5 INOVATIF

MR. TAN LIAN KEE


APPRECATION

PART 1

PART 2

PART 3

PART 4

PART 5


FURTHER

EXPLORATION

REFLECTION

After weeks of struggle and hard work to complete assignment given to us by

our teacher, Mr. Tan Lian Kee. I finally did it within 2 weeks with satisfaction and

senses of success because I have understood more deeply about the interest and

investment more than before. I have to be grateful and thankful to all parties who

have helped me in the process of completing my 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 thankfulness once again

to all parties concerned.


Firstly, I would like to thanks my Additional Mathematics’ teacher, Mr. Tan for

patiently explained to us 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

thankfulness to my cousin and friends, who have patiently explained to me and did

this project with me in group.


PART 1

a) History of probability

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' (Latin probabilis) 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. 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 of Pierre de Fermat and Blaise Pascal (1654). Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject. Jakob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Doctrine of Chances (1718) treated the subject as a branch of mathematics. See Ian Hacking's The Emergence of Probability and James Franklin's The 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's Opera 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 probability of errors by a curve y = φ(x), x being any error and y its probability, and laid down three properties of this curve:

1. it is symmetric as to the y-axis;2. the x-axis is an asymptote, the probability of the error being 0;3. the area enclosed is 1, it being certain that an error exists.

He also gave (1781) a formula for the law of facility of error (a term due to Lagrange, 1774), but one which led to unmanageable equations. Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.

PART 1

The method of least squares is due to Adrien-Marie Legendre (1805), who introduced it in his Nouvelles méthodes pour la détermination des orbites des comètes (New Methods for Determining the Orbits of Comets). In ignorance of Legendre's contribution, an Irish-American writer, Robert Adrain, editor of "The Analyst" (1808), first deduced the law of facility of error,

h being a constant depending on precision of observation, and c a scale factor ensuring that the area under the curve equals 1. He gave two proofs, the second being essentially the same as John Herschel's (1850). Gauss gave the first proof which seems to have been known in Europe (the third after Adrain's) in 1809. Further proofs were given by Laplace (1810, 1812), Gauss (1823), James Ivory (1825, 1826), Hagen (1837), Friedrich Bessel (1838), W. F. Donkin (1844, 1856), and Morgan Crofton (1870). Other contributors were Ellis (1844), De Morgan (1864), Glaisher (1872), and Giovanni Schiaparelli (1875). Peters's (1856) formula for r, the probable error of a single observation, is well known.

In the nineteenth century authors on the general theory included Laplace, Sylvestre Lacroix (1816), Littrow (1833), Adolphe Quetelet (1853), Richard Dedekind (1860), Helmert (1872), Hermann Laurent (1873), Liagre, Didion, and Karl Pearson. Augustus De Morgan and George Boole improved the exposition of the theory.

Andrey Markov introduced the notion of Markov chains (1906) playing an important role in theory of stochastic processes and its applications.

The modern theory of probability based on the meausure theory was developed by Andrey Kolmogorov (1931).


On the geometric side (see integral geometry) contributors to The Educational Times were influential (Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin).

PART 1

a) Probability in our lives

i) Weather forecasting

Suppose you want to go on a picnic this afternoon, and the weather report says that the chance of rain is 70%? Do you ever wonder where that 70% came from?

Forecasts like these can be calculated by the people who work for the National Weather Service when they look at all other days in their historical database that have the same weather characteristics (temperature, pressure, humidity, etc.) and determine that on 70% of similar days in the past, it rained.

As we've seen, to find basic probability we divide the number of favorable outcomes by the total number of possible outcomes in our sample space. If we're looking for the chance it will rain, this will be the number of days in our database that it rained divided by the total number of similar days in our database. If our meteorologist has data for 100 days with similar weather conditions (the sample space and therefore the denominator of our fraction), and on 70 of these days it rained (a favorable outcome), the probability of rain on the next similar day is 70/100 or 70%.

Since a 50% probability means that an event is as likely to occur as not, 70%, which is greater than 50%, means that it is more likely to rain than not. But what is the probability that it won't rain? Remember that because the favorable outcomes represent all the possible ways that an event can occur, the sum of the various probabilities must equal 1 or 100%, so 100% - 70% = 30%, and the probability that it won't rain is 30%.

http://mathforum.org/dr.math/faq/faq.prob.intro.html#note

http://mathforum.org/dr.math/faq/faq.prob.intro.html


ii) Batting averages

Let's say your favorite baseball player is batting 300. What does this mean?

A batting average involves calculating the probability of a player's getting a hit. The sample space is the total number of at-bats a player has had, not including walks. A hit is a favorable outcome. Thus if in 10 at-bats a player gets 3 hits, his or her batting average is 3/10 or 30%. For baseball stats we multiply all the percentages by 10, so a 30% probability translates to a 300 batting average.

This means that when a Major Leaguer with a batting average of 300 steps up to the plate, he has only a 30% chance of getting a hit - and since most batters hit below 300, you can see how hard it is to get a hit in the Major Leagues!

PART 1

a) Introduction

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 some

mechanism (precognition, telepathy, psychokinesis, cheating, etc.) which

allowed the subject to perform better than chance would seem to permit. To

interpret the results of our RetroPsychoKinesis experiments, we'll be using the

mathematics of probability, so it's worth spending some time explaining how we

go about quantifying the consequences of chance.


PART 1

b) Difference between the Theoretical and Empirical Probabilities

The term empirical means "based on observation or experiment." An empirical probability is generally, but not always, given with a number indicating the possible percent error (e.g. 80+/-3%). A theoretical probability, however, is one that is calculated based on theory, i.e., without running any experiments.

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.

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.

Comparing Empirical and Theoretical Probabilities:

Karen and Jason roll two dice 50 times and record their results in the accompanying chart.1.) What is their empirical probability of rolling a 7?2.) What is the theoretical probability of rolling a 7?

Sum of the rolls of two dice

3, 5, 5, 4, 6, 7, 7, 5, 9, 10,


3.) How do the empirical and theoretical probabilities compare?

12, 9, 6, 5, 7, 8, 7, 4, 11, 6,

8, 8, 10, 6, 7, 4, 4, 5, 7, 9, 9, 7, 8, 11, 6, 5, 4, 7, 7, 4,3, 6, 7, 7, 7, 8, 6, 7, 8, 9

Solution: 1.) Empirical probability (experimental probability or observed probability) is 13/50 = 26%.2.) Theoretical probability (based upon what is possible when working with two dice) = 6/36 = 1/6 = 16.7% (check out the table at the right of possible sums when rolling two dice). 3.) Karen and Jason rolled more 7's than would be expected theoretically.

PART 2

a) {1, 2, 3, 4, 5, 6}

b)

2

1 (2,1)

2 (2,2)

3 (2,3)

4 (2,4)

5 (2,5)

6 (2,6)

1

1 (1,1)

2 (1,2)

3 (1,3)

4 (1,4)

5 (1,5)

6 (1,6)

3

1 (3,1)

2 (3,2)

3 (3,3)

4 (3,4)

5 (3,5)

6 (3,6)

4

1 (4,1)

2 (4,2)

3 (4,3)

4 (4,4)

5 (4,5)

6 (4,6)


PART 3

a) Table 1 show the sum of all dots on both turned-up faces when two dice are tossed simultaneously.

Sum of the dots on both turned-up faces

(x)

Possible outcomes Probability, P(x)

2 (1,1) 1/36

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

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

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

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

5

1 (5,1)

2 (5,2)

3 (5,3)

4 (5,4)

5 (5,5)

6 (5,6)

6

1 (6,1)

2 (6,2)

3 (6,3)

4 (6,4)

5 (6,5)

6 (6,6)


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

10 (4,6),(5,5),(6,4) 3/36

11 (5,6),(6,5) 2/36

12 (6,6) 1/36

Table 1

PART 3

b) Table of possible outcomes of the following events and their corresponding probabilities.

Events Possible outcomes Probability,

P(x)

A{(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) }

3036

B ø ø


CP = 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), (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) }

C = P U QC = {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) }

2236

DP = Both number are primeP = {(2,2), (2,3), (2,5), (3,3), (3,5), (5,3), (5,5)}

R = The sum of 2 numbers are evenR = {(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 ∩ RD = {(2,2), (3,3), (3,5), (5,3), (5,5)}

536

PART 4

a)

Sum of the two numbers (x)

Frequency (f ) fx fx2

2 2 4 83 4 12 364 4 16 645 9 45 2256 4 24 1447 11 77 5398 4 32 2569 6 54 48610 3 30 30011 1 11 121

12 2 24 288


∑ f = 50 ∑ fx = 329 ∑ fx2= 2467Table 2

i) Mean = x

¿ ∑ fx

∑ f

= 32950

= 6.58

ii) Variance =

= ∑ fx 2

∑ f - x2

= 2467

50 – (6.58)2

= 6.044

iii) Standard deviation = √(∑ fx2

∑ f−x2)

= √6.0436

= 2.458

PART 4

b)

Sum of the two numbers (x)

Frequency (f ) fx fx2

2 4 8 163 5 15 454 6 24 965 16 80 4006 12 72 4327 21 147 10298 10 80 6409 8 72 64810 9 90 900


11 5 55 605

12 4 48 576

∑ f = 100 ∑ fx = 691 ∑ fx2= 5387Prediction of mean = 6.91

i. Mean ¿ 691100

= 6.91

ii. Variance = ∑ fx 2

∑ f - x2

=5387100

−(6.91)2

= 6.122

iii. Standard deviation = √6.122

= 2.474

Prediction is proven.

PART 5

a)

Mean = ∑ x P(x)

= [2( 136 )+3( 1

18 )+4 ( 112 )+5( 1

9 )+6 ( 536 )+7( 1

6 )+8( 536 )+9( 1

9 )+10( 112 )+11( 1

18 )+12( 136 )]


= 7

Variance = ∑ x2

P(x) – (mean)2

=

[22( 136 )+32( 1

18 )+42( 112 )+52( 1

9 )+62( 536 )+72( 1

6 )+82( 536 )+92( 1

9 )+102( 112 )+112( 1

18 )+122( 136 )]

- (7)2

= 54.83 – 49

= 5.83

Standard deviation = √5.83

= 2.415

PART 5

b)

Part 4 Part 5

n = 50 n = 100

Mean 6.58 6.91 7.00

Variance 6.044 6.122 5.83

Standard deviation 2.458 2.474 2.415

We can see that, the mean, variance and standard deviation that we obtained through experiment in part 4 are different but close to the theoretical value in part 5.


For mean, when the number of trial increased from n=50 to n=100, its value get closer (from 6.58 to 6.91) to the theoretical value. This is in accordance to the Law of Large Number. We will discuss Law of Large Number in next section.

Nevertheless, the empirical variance and empirical standard deviation that we obtained i part 4 get further from the theoretical value in part 5. This violates the Law of Large Number. This is probably due to

a. The sample (n=100) is not large enough to see the change of value of mean, variance and standard deviation.

b. Law of Large Number is not an absolute law. Violation of this law is still possible though the probability is relative low.

In conclusion, the empirical mean, variance and standard deviation can be different from the theoretical value. When the number of trial (number of sample) getting bigger, the empirical value should get closer to the theoretical value. However, violation of this rule is still possible, especially when the number of trial (or sample) is not large enough.

PART 5

c)

The range of the mean

Conjecture: As the number of toss, n, increases, the mean will get closer to 7. 7 is the theoretical mean.

Image below support this conjecture where we can see that, after 500 toss, the theoretical mean


become very close to the theoretical mean, which is 3.5. (Take note that this is experiment of tossing 1 die, but not 2 dice as what we do in our experiment)

FURTHER EXPLORATION

In probability theory, the law of large numbers (LLN) is a theorem that describes the

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

the average of the results obtained from a large number of trials should be close to the expected

value, and will tend to become closer as more trials are performed.

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

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

http://en.wikipedia.org/wiki/Probability

http://en.wikipedia.org/wiki/Dice

http://en.wikipedia.org/wiki/Expected_value

http://en.wikipedia.org/wiki/Expected_value

http://en.wikipedia.org/wiki/Average

http://en.wikipedia.org/wiki/Theorem

http://en.wikipedia.org/wiki/Probability_theory


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

of their values (sometimes called the sample mean) is likely to be close to 3.5, with the accuracy

increasing as more dice are rolled.

Similarly, when a fair coin is flipped once, the expected value of the number of heads is

equal to one half. Therefore, according to the law of large numbers, the proportion of heads in a

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

after n flips will almost surely converge to one half as napproaches infinity.

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

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

becomes large. That is, the probability that the absolute difference is a small number approaches

zero as number of flips becomes large. Also, almost surely the ratio of the 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 of the roulette wheel, its earnings

will tend towards a predictable percentage over a large number of spins. Any winning streak by a

player will eventually be overcome by the parameters of the game. It is important to remember

that the LLN only applies (as the name indicates) when a large number of observations are

considered. There is no principle that a small number of observations will converge to the

expected value or that a streak of one value will immediately be "balanced" by the others.

REFLECTION

While I was conducting the project, I had learned many moral values that I practice.

This project work had taught me to be more confident when doing something especially the

homework given by the teacher. I also learned to be a disciplined type of student which is always

sharp on time while doing some work, complete the work by myself and researching the

informations from the internet. I also felt very enjoy when making this project during the school

holidays.

http://en.wikipedia.org/wiki/Roulette

http://en.wikipedia.org/wiki/Almost_surely

http://en.wikipedia.org/wiki/Limit_of_a_sequence

http://en.wikipedia.org/wiki/Almost_surely

http://en.wikipedia.org/wiki/Fair_coin

http://en.wikipedia.org/wiki/Sample_mean

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