SMK PUCHONG BATU 1447100 PUCHONG

SELANGOR DARUL EHSAN.

PROJECT WORK FORADDITIONAL MATHEMATICS 2010

PROBABILITY IN OUR DAILY LIFE

NAME : MUHAMMAD HARIZ FADHILLA BIN SULAIMAN

CLASS : 5 AMANAH

IC NUMBER : 930805-10-5119

SUBJECT TEACHER : MR PHANG CHIA CHEN

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[ CONTENT ]

INTRODUCTION…………………………………………..1

AIM………………………………………………………….2

PART 1…………………………………………………...3 - 6

PART 2……………………………………………….…..7 - 8

PART 3………………………………………………….9 - 12

CONCLUSION………………………………………...13 - 14

ACKNOWLEDGEMENT…………………………………..15

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[ INTRODUCTION ]

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: 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 in statistical

mechanics. A great discovery of twentieth century physics was the

probabilistic nature of physical phenomena at atomic scales, described

in quantum mechanics.

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[ AIM ]

The aims of carrying out this project work are :

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

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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PROJECT WORK

[ PART 1 ]

(A) History:

The mathematical theory of probability has its roots in attempts to

analyze games of chance by Gerolamo Cardano in the sixteenth century,

and by Pierre de Fermat and Blaise Pascal in the seventeenth century

(for example the "problem of points"). Christiaan Huygens published a

book on the subject in 1657.

Initially, probability theory mainly considered discrete events, and its

methods were mainly combinatorial.

Eventually, analytical considerations compelled the incorporation

of continuous variables into the theory.

This culminated in modern probability theory, the foundations of

which were laid by Andrey Nikolaevich Kolmogorov. Kolmogorov

combined the notion of sample space, introduced by Richard von Mises,

and measure theory and presented hisaxiom system for probability

theory in 1933. Fairly quickly this became the undisputed axiomatic

basis for modern probability theory.

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(B) Two categories of PROBABILITY

The difference between empirical and theoretical probability is

an important part of our ability to apply probability to a real world set of

data. What is the difference between empirical and theoretical

probability? Give two to three examples speculative of professions where

probability could be used. Explain your answer

The difference between empirical, means observation or

experience and theoretical probability or speculative are as clear as night

and day. Empirical probability is the data that has been proven through

trial and error such as the statics on the accidents that involve driving

while under the influence. Even the proven data for deaths that are

smoking related. The theoretical probability is like guessing and taking a

chance you are right much like playing a game of cards you are taking

that chance you have the better hand. Insurance policies are made

possible by empirical probability. We know the amount of accidents,

and we know the amount of times something happens without error.

Based on that, it can be calculated what the chance (and thus the cost) is

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of a certain event. (professional) Gambling is about theoretical

probability. One can assume that all the chips, cards, tables or whatever

are completely fair (or even calculate the unfairness, based on the

method of shuffling), so one can calculate the odds of a certain set of

cards coming up, before they ever have.

Dangerous medical procedures can also have empirical probability

playing as a factor. There is always a chance that someone dies under the

knife, or that someone cures on their own. Based on those odds, a doctor

could advise for or against certain procedures. Those odds are based on

other patients who have gone through the same thing.

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[ PART 2 ]

(A) When we playing the monopoly, we have to toss the die once to find

who is going to start the game first. The possible outcomes when we toss

the die is 1,2,3,4,5, and 6. This is because a die has 6 surface as shown

in Figure 1.

Figure 1.

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(B) When we tossed two dice simultaneously, the possible outcomes is as

shown in the Table 2.

DIE1/DIE2 1 2 3 4 5 61 (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)

Table 1.

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[ PART 3 ]

(A)The Table 2 below shows the probability and the possible

outcomes to get the sum of the dots on both turned-up faces.

Sum of the dots on both

Possible outcomesProbability,

P(x)turned-up faces

(x)2 (1,1) 1/363 (1,2),(2,1) 1/184 (1,3),(2,2),(3,1) 1/125 (1,4),(2,3),(4,1),(3,2) 1/96 (5,1),(4,2),(3,3),(2,4),(1,5) 5/36

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

(1,6) 6/368 (6,2),(5,3),(4,4),(3,5),(2,6) 5/369 (6,3),(5,4),(4,5),(3,6) 1/9

10 (6,4),(4,6),(5,5) 1/1211 (6,5),(5,6) 1/1812 (6,6) 1/36

Table 2.

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(B) Based on the Table 2:

A ={The two numbers are not the same} ={(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)}

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

P = 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),(3,4),(3,6),(4,1),(4,3),(4,5),(5,4),

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

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

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),

(3,6),(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

D = {The sum of the two numbers are even and both }D = {P ∩ R}D = {(2,2),(3,3),(3,5),(5,3),(5,5)}

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[ PART 4 ]

(A) An activity has been conducted by tossing two dice simultaneously 50

times. The frequency (f) of the sum of all dots on both turned-up faces

has been recorded in Table 3 below. The value of mean, variance and

standard deviation of the data has been calculated.

Sum of the two numbers (x) Frequency(f) fx fx²2 2 4 83 5 15 454 5 20 805 1 5 256 4 24 1447 8 56 3928 8 64 5129 5 45 405

10 7 70 70011 3 33 36312 2 24 288

Total 50 360 2962Table 3.

Mean = 360÷50 = 7.2

Varience = 2962÷50-7.2² = 7.4

Standard Deviation = √7.4

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= 2.72

(B) When the number of tossed of the two dice simultaneously is increase to

100 times, the value of mean is also change.

(C)Another activity same as (A) has been conducted by tossing two dice

simultaneously 100 times.

Sum of the two numbers (x) Frequency(f) fx fx²2 2 4 83 6 18 544 10 40 1605 9 45 1446 16 96 5767 17 119 8338 12 96 7689 11 99 891

10 13 130 130011 2 22 24212 2 24 288

Total 100 693 5264Table 4.

Mean = 693÷100= 6.93

Varience = 5264÷100-6.93²= 4.6151

Standard Deviation = √4.6151= 2.148

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Therefore, the prediction that has been made in (B) is true.

The mean, varience, and standard deviation is change

although the number of tossed of the dice was increase until

100 times because when the number of tossed change the

frequency also change.

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[ PART 5 ]

(A)Based on Table 2, the actual mean, the varience and the standard

deviation of the sum of all dots on the turned-up faces by using the

formulae given was determined.

Mean = ∑ x P(x)

Sum of the two numbers (x) Frequency (f) fx fx²

2 1 2 43 2 6 184 3 12 485 4 20 1006 5 30 1807 6 42 2948 5 40 3209 4 36 324

10 3 30 30011 2 22 24212 1 12 144

total 36 252 1974Varience = ∑ x² P(x) – (mean)²

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Maen = 252÷36

= 7

Varience = 1974÷36-7²

= 5.83

Standard Deviation = √5.83

= 2.415

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(B) The mean, variance and standard deviation obtained in

Part 4 and Part 5 is not same because Part 4 was the

Empirical Probability and Part 5 was the Theoretical

Probability. For example, the empirical probability of

rolling a 7 is 4/25 = 16%. But the theoretical probability of

rolling a 7 is 6/36 = 1/6 = 16.7%. Table below show the

comparison between Mean, Variance, and Standard

Deviation.

Part 4Part 5

n= 50 n= 100

Mean 7.2 6.93 7

Variance 7.4 4.6151 5.83

Standard deviation 2.72 2.148 2.415

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(C)The range of mean of the sum of all dots on the turned-

up faces as n changes is 2≤mean≤12. This is because as the

number of toss, n, increases, the mean will get closer to 7. 7

is the theoretical mean.

FURTHER EXPLORATION

Common intuition suggests that if a fair coin is tossed many times,

then roughly half of the time it will turn up heads, and the other half it

will turn up tails. Furthermore, the more often the coin is tossed, the

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more likely it should be that the ratio of the number of heads to the

number of tails will approach unity. Modern probability provides a

formal version of this intuitive idea, known as the law of large

numbers. This law is remarkable because it is nowhere assumed in

the foundations of probability theory, but instead emerges out of

these foundations as a theorem. Since it links theoretically derived

probabilities to their actual frequency of occurrence in the real world,

the law of large numbers is considered as a pillar in the history of

statistical theory.

The law of large numbers (LLN) states that the sample

average   of   (independent and identically

distributed random variables with finite expectation μ) converges

towards the theoretical expectation μ.

It is in the different forms of convergence of random variables that

separates the weak and the strong law of large numbers

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It follows from LLN that if an event of probability p is observed

repeatedly during independent experiments, the ratio of the observed

frequency of that event to the total number of repetitions converges

towards p.

Putting this in terms of random variables and LLN we have   

are independent Bernoulli random variables taking values 1 with

probability p and 0 with probability 1-p. E(Yi) = p for all i and it follows

from LLN that   converges top almost surely.

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[ ACKNOWLEDGEMENT ]

I would like to express my gratitude and thanks to my teacher, Mr

Phang Chia Chen for his wonderful guidance for me to be able to complete

this project work. Next, is to my parents for their continuous support to me

throughout this experiment. Special thanks to my friends for their help, and

to all those who contributed directly or indirectly towards the completion of

this project work.

Throughout this project, I acquired many valuable skills, and hope

that in the years to come, those skills will be put to good use.

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