rabbito's addmath

8/9/2019 RABBITO's ADDMATH

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`Name: Rabiatul Adawiyah bt. Abdul Rahman

`Class: 5 Alamanda

`School: SMK. Bandaraya Kota Kinabalu, Sabah

`Subject s teacher: Sir Chia Mun Meng


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No. Title Page

1. Appreciation 3

2. Part 1 ( Introduction ) 4

3. Part 2 5

4. Part 3 6

5. Part 4 8

6. Part 5 11

8. Futher Exploration- The Law of Large Numbers ( LLN )

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9. Reflection 16


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

teacher, Sir Chia Mun Meng. 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, Sir Chia Mun

Meng 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. Besides 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 my group.


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

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

in probability theory, that is used extensively in such areas of

study as mathematics, statistics, finance, gambling, science, and philosophy to

draw conclusions about the likelihood of potential events and the underlying

mechanics of complex systems.

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. Although an individual coin toss or the roll of

a die is a random event, if repeated many times the sequenceof 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. The probability

theory can be divided into two categories, which are;

Theoretical Probabilities and Empirical Probabilities.

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.

P
http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Statistical_randomnesshttp://en.wikipedia.org/wiki/Random_variablehttp://en.wikipedia.org/wiki/Stochastic_processhttp://en.wikipedia.org/wiki/Event_(probability_theory)http://en.wikipedia.org/wiki/Dicehttp://en.wikipedia.org/wiki/Law_of_large_numbershttp://en.wikipedia.org/wiki/Law_of_large_numbershttp://en.wikipedia.org/wiki/Central_limit_theoremhttp://en.wikipedia.org/wiki/Central_limit_theoremhttp://en.wikipedia.org/wiki/Law_of_large_numbershttp://en.wikipedia.org/wiki/Law_of_large_numbershttp://en.wikipedia.org/wiki/Dicehttp://en.wikipedia.org/wiki/Event_(probability_theory)http://en.wikipedia.org/wiki/Stochastic_processhttp://en.wikipedia.org/wiki/Random_variablehttp://en.wikipedia.org/wiki/Statistical_randomnesshttp://en.wikipedia.org/wiki/Mathematics


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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.

Answer: { 1 , 2 , 3 , 4 , 5 , 6 }

(b) Instead of one die, two dice can also be tossed simultaneously by each player. The

player will move the token according to the sum of all dots on both turned-up faces.

For example, if the two dice are tossed simultaneously and 2 appears on one die

and 3 appears on the other, the outcome of the toss is (2,3). Hence, the player shall

move the token 5 spaces. List all the possible outcomes when twoi dice are tossed

simultaneously. Organize and present your list clearly. Consider the use of table,

chart or even tree diagram

Answer:

1 2 3 4 5 6

1 1,1 2,1 3,1 4,1 5,1 6,1

2 1,2 2,2 3,2 4,2 5,2 6,2

3 1,3 2,3 3,3 4,3 5,3 6,3

4 1,4 2,4 3,4 4,4 5,4 6,4

5 1,5 2,5 3,5 4,5 5,5 6,5

6 1,6 2,5 3,6 4,6 5,6 6,6


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Table 1 shows the sum of all dots on both turned-up faces when two dice are tossed

simultaneously.

(a) Complete Table 1 by listing all possible outcomes and their corresponding

probabilities.

Answer:

(Table 1)


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(b) Based on Table 1 that you have completed, list all the possible outcomes of the

following events and hence find their corresponding probabilities.

A = { The two numbers are not same }

B = { The product of the two numbers is greater than 36 }

C = { Both numbers are prime or the difference between two numbers is odd }

D = { The sum of the two numbers are even and both numbers are prime }

Answer:


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(a) Conduct an activity by tossing two dice simultaneously 50 times. Observe the sum of

all dots on both turned-up faces. Complete the frequency table below.

Answer:


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Based on the Table 2 that you have completed, determine the value of:

(i) Mean

(ii) Variance; and

(iii) Standard deviation

of the data.

Answer:

(b) Predict the value of the mean if the number of tosses is increased to 100 times.

Answer:

Prediction of the mean = 6.91


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(c) Test your prediction by continuing Activity 3 (a) until the total number of tosses is

100 times. Then, determine the value of:

(i) Mean;

(ii) Variance; and

(iii) Standard deviation

of the new data.

Was your prediction proven?

Answer:


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When two dice are tossed simultaneously, the actual mean and variance of the sum of all

dots on the turned-up faces can be determined by using the formulae below:

Mean =

Variance =

(a) Based on Table 1, determine the actual mean, the variance and the standard deviation

of the sum of all dots on the turned-up faces by using the formulae given.

Answer:


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(b) Compare the mean, variance and standard deviation obtained in Part 4 and Part 5.

What can you say about the values? Explain in your own words, your own

interpretation and your understanding of the values that you have obtained and realte

your answers to the Theoretical and Empirical Probabilities.

Answer:

The mean, variance and standard deviation that I obtained through experiment in Part

4 are different but close to the theoretical value in Part 5.

For mean, when the number of trial is increased from n=50 to n=100, its value gets

closer (from 6.58 to 6.91) to the theoretical value. This is obeyed the Law of Large

Number in the next section.

Nevertheless, the empirical variance and empirical standard deviation that I obtained

in Part 4 get further from the theoretical value in Part 5. This is violated the Law of

Large Number. This is probably due to:

The sample (n=100) is not large enough to see the change of value of mean,

variance and standard deviation.

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) is getting

bigger, the empirical value should get closer to the theoretical value. However, the

violation of this rule is still possible, especially when the number of trial (or sample)

is not large enough.


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(c) If n is the number of times two dice are tossed simultaneously, what is the range of the

mean of the sum of all dots on the turned-up faces as n changes? Make your

conjecture and support your conjecture.

Answer:

The range of the mean :

2 mean 12

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, which is 3.5. (Take note that

this is the experiment of tossing 1 die,but not 2 dice as we do in our experiment )
http://en.wikipedia.org/wiki/File:Largenumbers.svg


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n 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, eachwith equal probability. Therefore, the expected value of a single die roll is:

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 approaches 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.

I
http://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Theoremhttp://en.wikipedia.org/wiki/Averagehttp://en.wikipedia.org/wiki/Expected_valuehttp://en.wikipedia.org/wiki/Dicehttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Dicehttp://en.wikipedia.org/wiki/Expected_valuehttp://en.wikipedia.org/wiki/Averagehttp://en.wikipedia.org/wiki/Theoremhttp://en.wikipedia.org/wiki/Probability_theory


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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 information from the internet. I also felt very enjoy when making this

project during school holidays.

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