PROJECT WORK FORADDITIONAL MATHEMATHICS

-2010-

Probability and their Application in Our Daily Life

Name Ahmad Firdaus b. Shariffudin

Class 5 Jaya

I/C

Teacher Miss Suriani

School Sekolah Menengah Sains Alam Shah

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CONTENT

Objective

Part 1

Part 2

Part 3

Part4

Part5

Further Exploration

Reflection2

Objective

The 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.

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INTRODUCTION

What is Probability

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

has occurred. In mathematicsthe 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.

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PART 1

Theory of Probability

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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

of stochastic 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 of Pascal 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 de

Moivre's The Doctrine of Chances (1718) put probability on a sound mathematical

footing, 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.

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

probabilities was theStatistical mechanics of Ludwig Boltzmann and J. Willard

Gibbs which explained properties of gases such as temperature in terms of the random

motions of large numbers of particles.

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

monumental History of the Mathematical Theory of Probability from the Time of Pascal

to that of Lagrange (1865).

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

on hypothesis testing of R. A. Fisher andJerzy 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.[5]

The theory of stochastic processes broadened into such areas as Markov processes 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 nor necessarily very rationally. The theory of behavioral

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

peace 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.

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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.

Theorical Probabilities and Empirical Probabilities

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Theorical Probabilities:

Probability theory is the branch of mathematics concerned with analysis

of random phenomena.[1] 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 instatistical 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,[1]

[2] 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.[3] The phrase a posteriori probability has also been used as an alternative to

empirical probability or relative frequency.[4] 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.

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

using a binomial distribution is appropriate, it is themaximum likelihood estimate. It is

the Bayesian estimate for the same case if certain assumptions are made for the prior

distribution of the probability

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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. Herestatistical 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 ofprobability

distributions 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.

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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.

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PART 2

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Part 2

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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.

SolutionThere 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

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

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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

OR

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PART 3

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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 Table 1 by listing all possible outcomes and their corresponding probabilities.Sum of the dots on both turned up faces(x)

Possible outcomes Probability,p(x)

1 - 02 (1,1) 1/363 (1,2), (2,1) 1/184 (1,3), (2,2), (3,1) 1/125 (1,4), (2,3), (3,2), (4,1) 1/96 (1,5), (2,4), (3,3), (4,2), 5,1) 5/367 (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) 1/68 (2,6), (3,5), (4,4), (5,3), (6,2) 5/369 (3,6), (4,5), (5,4), (6,3) 1/910 (4,6), (5,5), (6,4) 1/1211 (5,6), (6,5) 1/1812 (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: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}

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Solution

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’)=1/6As P(A’)=P’(A)=1/6, thus P(A)=1-1/6 =5/6B={},as the maximum product is 6X6=36. This event is impossible to occur.Thus,P(B)=0Prime number(below six):2,3,5Odd number(below six):1,3,5C = P U QC={(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)}=23/36D = P ∩ RD={ (2,2), (3,3), (3,5), (5,3), (5,5)}

P(D) =5/36

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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

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PART 4

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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.

Sum of the two numbers(x)

Frequency( )

2 1 2 4

3 2 6 18

4 5 20 80

5 3 15 75

6 6 36 216

7 10 70 490

8 8 64 512

9 6 54 486

10 6 60 600

11 2 22 242

12 1 12 144

Total 50 361 2867

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

i) Meanii) Variance: and

iii) Standard deviation

Of the data

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Solution,

From the table,

i)

mean, =

ii)

variance,

23

=

=5.2116

iii)

Standard deviation,

=

= 2.2829

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.

New mean,

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

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ii)variance: and

iii)standard deviation

of the new data.

Was your prediction proved?

Solution:

Sum of the two numbers(x)

Frequency( )

2 5 10 20

3 5 15 45

4 10 40 160

5 9 45 225

6 15 90 540

7 16 112 784

8 14 112 896

9 13 117 1053

10 6 60 600

11 5 55 605

12 2 24 288

Total 100 680 5216

Solution,

From the table,

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i) mean, =

ii) variance,

=

=5.92

iii) Standard deviation,

=

= 2.4331

The prediction is wrong. The new mean is 6.8, which 0.42 lesser than the original mean.

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PART 5

Part 5

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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 formula given.

(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 words your interpretation and your understanding of the values that you have obtained and relate your answers to the Theorical and Empirical Probabilities

(c) 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.

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Solution:

Sum of the dots on both turned up faces(x)

Possible outcomes Probability,p(x)

1 - 02 (1,1) 1/363 (1,2), (2,1) 1/184 (1,3), (2,2), (3,1) 1/125 (1,4), (2,3), (3,2), (4,1) 1/96 (1,5), (2,4), (3,3), (4,2), 5,1) 5/367 (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) 1/68 (2,6), (3,5), (4,4), (5,3), (6,2) 5/369 (3,6), (4,5), (5,4), (6,3) 1/910 (4,6), (5,5), (6,4) 1/1211 (5,6), (6,5) 1/1812 (6,6) 1/36Total 36 1

(a) i) Mean=

+12

=7

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ii) Variance=

+144 ]-

=54.8333-49

=5.8333

iii) standard deviation,

=

=2.4152

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

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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.

(c)

Conjecture: As the number of n increases, the mean will become closer to the theoretical mean, which are 7.00.

Support and proof

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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.

FURTHER

EXPLORATION

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Further Exploration

In 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.

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Answer:

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

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

theaverage 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.

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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 n 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 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.

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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.

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REFLECTION

Reflection

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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.

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