Additional Ma Thematic

8/9/2019 Additional Ma Thematic

1/36

PROJECT WORK FOR

ADDITIONAL MATHEMATHICS-2010-

Probability and their

Application in Our Daily Life

Name : Siti Noor Albaiyah Binti Yahaya

Class : 5 Science

I/C Number : 930812-01

Teacher : Miss Sim Yew Choo

School : SMK Puteri Wangsa


2/36

CONTENT

Objective

InroductionPart 1

Part 2

Part 3

Part4Part5

Further Exploration

Reflection


3/36

ObjectiveThe aims carrying out this project work are:

i. To apply and adapt a variety of problem-solving strategies to solve problems;

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.


4/36

INTRODUCTION

What is Probability

Probability is a way of expressing knowledge or belief that anevent will occur or 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, andphilosophy

to draw conclusions about the likelihood of potential events and the underlying mechanics of complex

systems.


5/36

PART

1


6/36

Theory of Probability

History of Probability

Probability has a dual aspect: on the one hand the probability or likelihood of hypotheses given the

evidence for them, and on the other hand the behavior ofstochastic processes such as the throwing of

dice or coins. The study of the former is historically older in, for example, the law of evidence, while

the mathematical treatment of dice began with the work ofPascal and Fermat in the 1650s.

Probability is distinguished from statistics. While statistics deals with data and inferences from it,

(stochastic) probability deals with the stochastic (random) processes which lie behind data or

outcomes.

Some highlight in the history of probability are:

18th century: Jacob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham deMoivre's The

Doctrine of Chances (1718) put probability on a sound mathematicalfooting, showing how to calculate

a wide range of complex probabilities. Bernoulli proved a version of the fundamental law of large

numbers, which states that in a large number of trials, the average of the outcomes is likely to be very

close to the expected value - for example, in 1000 throws of a fair coin, it is likely that there are close

to 500 heads (and the larger the number of throws, the closer to half-and-half the proportion is

likely to be).

19th century: The power of probabilistic methods in dealing with uncertainty was shown by Gauss's

determination of the orbit ofCeres from a few observations. The theory of errors used the method of

least squares to correct error-prone observations, especially in astronomy, based on the assumption

of a normal distribution of errors to determine the most likely true value.


7/36

Towards the end of the nineteenth century, a major success of explanation in terms of probabilities

was the Statistical mechanics ofLudwig Boltzmann and J. WillardGibbs which explained properties of

gases such as temperature in terms of the randommotions of large numbers of particles.

The field of the history of probability itself was established by Isaac Todhunter's monumental History of

the Mathematical Theory ofProbability from the Time ofPascal to that of Lagrange(1865).

20th century: Probability and statistics became closely connected through the work on hypothesis

testing ofR. A. Fisherand Jerzy Neyman, which is now widely applied in biological and psychological

experiments and in clinical trials of drugs. A hypothesis, for example that a drug is usually effective,

gives rise to a probability distribution that would be observed if the hypothesis is true. If observations

approximately agree with the hypothesis, it is confirmed, if not, the hypothesis is rejected.

The theory of stochastic processes broadened into such areas as Markovprocesses and Brownian

motion, the random movement of tiny particles suspended in a fluid. That provided a model for the

study of random fluctuations in stock markets.

Application of Probability in Daily life

Two major applications of probability theory in everyday life are in risk assessment and in trade on

commodity markets. Governments typically apply probabilistic methods in environmental regulation

where it is called "pathway analysis", often measuring well-being using methods that are stochastic in

nature, and choosing projects to undertake based on statistical analyses of their probable effect on

the population as a whole. A good example is the effect of the 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 trader that a war is more likely vs. less likely sends prices up or down,

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


8/36

nor necessarily very rationally. The theory ofbehavioralfinance emerged to describe the effect of such

groupthink on pricing, on policy, and onpeace and conflict.

It can reasonably be said that the discovery of rigorous methods to assess and combine 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 auto mobiles and consumer electronics, utilize reliabilitytheory in the design of the

product in order to reduce the probability of failure. Theprobability of failure may be closely associated

with the product's warranty.


9/36

Theorical Probabilities and Empirical Probabilities

Theorical Probabilities:

Probability theory is the branch ofmathematics concerned with analysis ofrandom phenomena.The

central objects of probability theory are random variables, stochastic processes, and events:

mathematical abstractions ofnon- deterministic events or measured quantities that may either be

single occurrences orevolve 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 largenumbers and the central limit

theorem.

As a mathematical foundation forstatistics, 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.

Empirical Probabilities

Empirical probability, also known as relative frequency, or experimental probability, is the ratio of the

number favorable outcomes to the total number of trials, not in a sample space but in an actual

sequence of experiments. In a more general sense, empirical probability estimates probabilities from

experience and observation. The phrase a posteriori probability has also been used as an alternative

to empirical probability or relative frequency.This unusual usage of the phrase is not

directly related to Bayesian inference and not to be confused with its equally occasional use to refer to

posterior probability, which is something else.


10/36

In statistical terms, the empirical probability is an estimate of a probability. If modelling using a

binomial distribution is appropriate, it is the maximum likelihood estimate. It is the Bayesian estimate

for the same case if certain assumptions are made for the priordistribution of the probability.

An advantage of estimating probabilities using empirical probabilities is that this procedure is relatively

free of assumptions. For example, consider estimating the probability among a population of men that

they satisfy two conditions: (i) that they are over 6 feet in height; (ii) that they prefer strawberry jam to

raspberry jam. A direct estimate could be found by counting the number of men who satisfy both

conditions to give the empirical probability the combined condition. An alternative estimate could be

found by multiplying the proportion of men who are over 6 feet in height with the proportion of men

who prefer strawberry jam to raspberry jam, but this estimate relies on the assumption that the two

conditions are statistically independent.

A disadvantage in using empirical probabilities arises in estimating probabilities which are either very

close to zero, or very close to one. In these cases very large sample sizes would be needed in order to

estimate such probabilities to a good standard of relative accuracy. Here statistical models can help,

depending on the context, and in general one can hope that such models would provide improvements

in accuracy compared to empirical probabilities, provided that the assumptions involved actually do

hold. For example, consider estimating the probability that the lowest of the daily- maximum

temperatures at a site in February in any one year is less zero degrees Celsius. A record of such

temperatures in past years could be used to estimate this probability. A model-based alternative would

be to select of family ofprobabilitydistributions and fit it to the dataset contain past yearly values: the

fitted distribution would provide an alternative estimate of the required probability. This alternative

method can provide an estimate of the probability even if all values in the record are

greater than zero.


11/36

Difference between 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.


12/36

PART2


13/36

PART 2

Question:

a) Suppose you are playing monopoly game with two of your friends. To

start the game, each player will have to toss the dice once. The player who

obtain number will start the game. List all the possible outcomes when the

dice is tossed once.


14/36

Solution :

There are three player, considered as P1,P2, and P3. The total side of the

die which is cube is six, and the number of dots on the dice is 1, 2, 3, 4, 5

and 6 respectively.

Thus, the possible outcomes are:

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

Question:

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 two dice are tossed simultaneously

and 2 appears on one dice and 3 on the other, the outcome of the toss

is (2,3). Hence, the player shall move the token 5 spaces. Notes: The

events (2,3) and (3,2) should be treated as two different events.

List all the possible outcomes when two dice are tossed simultaneously. Organize and present your

list clearly. Consider the use of table, chart or even diagram.

Solution :

By tossing two dice, the total possible outcomes are:

{(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)}

OR


15/36

By using table, the possible outcomes when two dice are tossed can be

listed.

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)

The total possible outcomes from the tossing of the two dice is 36, or

n(S)=6X6=36, which are applied from the multiplication rule.

OR


16/36


17/36

PART

3


18/36

PART 3

Question:

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

simultaneously.a) Complete the table 1 by listing all possible outcomes and their corresponding probabilities.

Sum of the dots onboth turned up faces

(x)

Possible outcomes Probability,p(X)

1 - 0

2 (1,1) 1/36

3 (1,2),(2,1) 1/184 (1,3),(2,2),(3,1) 1/12

5 (1,4),(2,3),(3,2),(4,1) 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) 1/68 (2,6),(3,5),(4,4),(5,3),(6,2) 5/36

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

10 (4,6),(5,5),(6,4) 1/12

11 (5,6),(6,5) 1/18

12 (6,6) 1/36Total 36 1

b) Based on Table 1 that you have competed, list all the possible

outcomes of the following events and hence find their corresponding

probabilities:

c) A= {The two numbers are not the 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}


19/36

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)

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)}

P(A)=??

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

P(A)=1/6

As P(A)=P(A)=1/6, thusP( A) =1- 1/6

=5/6

B={},as the maximum product is 6X6=36. This event is impossible to occur.

Thus,P(B)=0

Prime number(below six):2,3,5

Odd number(below six):1,3,5

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), (3,5), (3,6),

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

=23/36

D = P R

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

P(D) =5/36


20/36

Answers:

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)}

P(A)= 5/6

B={}

P(B)=0

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

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

P(C)= 23/36

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

P(D) =5/36


21/36

PART

4


22/36

Part 4

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.

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

Mean Variance: and Standard deviation Of the data


23/36

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

-the number of tosses is increased double, the mean will slightly change, maybe will inducted

by 2.


24/36

c) test your prediction in (b) 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 daviation of the new data

Was your prediction proved?


25/36


26/36

PARt

5


27/36

Part 5When 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:

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 formula given.

Compare the mean, variance and standard deviation obtained in Part 4 and Part 5. What can

you say aboutthe values? Explain in your words your interpretationand your understanding of

the values that you have obtained and relate your answers to the Theorical and Empirical

Probabilities If n is the number of times of two dice are tossed simultaneously, what is the range

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

your conjucture.


28/36


29/36


30/36

The mean, variance and the standard deviation of data in Part 4 and Part 5 are totally different. Mean,

variance, and standard deviation of the data in Part 5 exceeds the mean, variance, and standard

deviation of the data in Part 4 by o.44, 0.0857, and 0.0179 respectively. The values are different

because there are two different method used to identify the mean, variance, and standard deviation

which are by conducting an experiment as conducted in Part 4 and by using formulae in Part 5. In Part

4, the values may varies as the result from the tossing of the dice are always different. The probability

to always get the same number are very small, which is 1/36. Thus, it affect the values of the mean,

variance, and standard deviation of the data. The method used in Part 4 to obtain these values also

known as Empirical Probabilities experiment. Theoretical probabilities are used in identifying those

data in Part 5. The data are obtained from the formula and the data will be constant as it is only

theoretical.

2 mean 12

Conjecture: As the number of n increases, the mean will become closer to the theoretical mean, which

are 7.00. Support and proof From the part 4 experiment, it is obvious that when the number of n

increases, which are 100, the mean become closer to 7 than when the value of n 50.


31/36

FUTHER

EXPLORATION


32/36

Further ExplorationIn probability theory, the Law of Large Numbers (LNN) is a theorem that describes the result of

performing the same experiment a large number of times. Conduct a research using the internet to

find out the theory of LLN. When you have finished with your research, discuss and write about your

findings. Relate the experiment that you have done in this project to the LLN.


33/36

Answer:

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 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 aftern

flips will almost surely converge to one half asn 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.

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 roul ette 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


34/36

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.

An illustration of the Law of Large Numbers using die rolls. As the number of die rolls increases, the

average of the values of all the rolls approaches 3.5.

Same goes to the project, as the tosses increases to 100 times, the mean become nearer to 7, which

the actual value of mean. If the experiment is continue until 200 times of tossing, the mean will

become closer to 7.


35/36

REFLECTION


36/36

Reflection

While I conducting the project, I had learned some moral values that I practice. This project had taught

me to responsible on the works that are given to me to be completed. This project also had make me

felt more confidence to do works and not to give easily when we could not find the solution for the

question. I also learned to be more discipline on time, which I was given about a month to complete

these project and pass up to my teacher just in time. I also enjoy doing this project during my school

holiday as I spend my time with friends to complete this project and it had tighten our friendship.

Documents

Additional Ma Thematic