Additional Math Project Work

8/9/2019 Additional Math Project Work

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Additional Mathematics

Project work 2009 2/2010

Theory of probability

NAME : CHARLENE ANDREA

SCHOOL : SMK.ST JOHN TUARAN

FORM : 5UM

NO.IC :930519-12-5186

TEACHER S NAME :MDM.NOR AZWATI MAT NAWI


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Contents

CONTENTS

PART 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

PART 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. PART 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

PART 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

P a r t 5


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Part 1

Introduction

Probability theory is the branch ofmathematics concerned with analysis

ofrandom phenomena.[1]

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 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 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.

Probaility history

The mathematical theory ofprobability 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.[2]

Initially, probability theory mainly considered discrete events, and its methods were

mainly combinatorial. Eventually, analytical considerations compelled the incorporation

ofcontinuous variables into the theory.

This culminated in modern probability theory, the foundations of which were laid byAndrey

Nikolaevich Kolmogorov. Kolmogorov combined the notion ofsample space, introduced by Richard

von Mises, and measure theory and presented his axiom system for probability theory in 1933. Fairly

quickly this became the undisputed axiomatic basis for modern probability theory.[3]


4/16

He

i

P

i, L

, teillegiti

te

il

f

i

,

t

emati

all

giftedlawyer, whowasa friendofLeonardoda Vinci. Inhis

autobiography,

ardanoclaimed that hismotherhadattempted toaborthim. Shortlybeforehisbirth,

hismotherhad tomove from

ilan toPavia toescape theplague;her threeotherchildrendied from

thedisease.

In

0, heentered theUniversityof Paviaand laterinPaduastudiedmedicine. Hiseccentricand

confrontational styledidnot earnhimmany friendsandhehadadifficult time findingworkafterhis

studieshadended. In

,

ardanorepeatedlyapplied to the

ollegeof Physicians in

ilan, but

wasnot admitteddue tohisreputationand illegitimatebirth.

Eventually, hemanaged todevelopaconsiderablereputationasaphysicianandhisserviceswere

highlyvaluedat thecourts. Hewas the first todescribetyphoid fever.

oday, he isbest known forhisachievements inalgebra. Hepublished thesolutions to

thecubicand!uarticequations inhis

4

bookArs Magna."hesolution tooneparticularcaseof

thecubic,x#

+ ax= b$

inmodernnotation), wascommunicated tohimby%

iccol&ontana

"artaglia

$

who laterclaimed that'

ardanohadswornnot toreveal it, andengaged'

ardano ina

decade-long fight), and thequarticwassolvedby'

ardano'sstudent Lodovico&errari. Bothwere

acknowledged in the forewordof thebook, aswell as inseveral placeswithin itsbody. Inhis

exposition, heacknowledged theexistenceofwhat arenowcalledimaginarynumbers, althoughhe

didnot understand theirproperties$

(athematical field theorywasdevelopedcenturies later). InOpus

novum de proportionibushe introduced thebinomial coefficientsand thebinomial theorem.

'

ardanowasnotoriouslyshort ofmoneyandkept himselfsolvent bybeinganaccomplishedgambler

andchessplayer. Hisbookabout gamesofchance,Liber de ludo aleae$

)Bookon

0amesof

'

hance") , written in1 2

3 4

, but not publisheduntil1

4 4

5, contains the first systematic treatment

ofprobability, aswell asasectiononeffectivecheatingmethods.


5/16

Portrait of6ardanoondisplayat the School of

7athematicsand Statistics,Universityof St Andrews.

8ardano inventedseveral mechanical devices including thecombination lock, thegimbalconsistingof

threeconcentricringsallowingasupportedcompassorgyroscope torotate freely, and the8

ardan

shaftwithuniversal joints, whichallows the transmissionofrotarymotionat variousanglesand is

used invehicles to thisday. Hestudiedhypocycloids, published inde proportionibus9 @ A

0.B

he

generatingcirclesof thesehypocycloidswere laternamedC

ardanocirclesorcardaniccirclesand

wereused for theconstructionof the first high-speedprintingpresses. Hemadeseveral contributions

tohydrodynamicsandheld thatperpetual motion is impossible, except incelestial bodies. He

published twoencyclopediasofnatural sciencewhichcontainawidevarietyof inventions, facts, and

occult superstitions. Healso introduced theC

ardangrille, acryptographic tool, in9 @ @

0.

Someonealsoassumed to

C

ardano thecredit forthe inventionof thesocalledCardano's Rings, also

calledC

hineseD

ings, but it isveryprobable that theyaremoreancient thanC

ardano.

Significantly, in thehistoryofDeafeducation, hesaid that deafpeoplewerecapableofusing their

minds, argued forthe importanceof teaching them, andwasoneof the first tostate thatdeafpeople

could learn toreadandwritewithout learninghow tospeak first. Hewas familiarwithareport

byD

udolph Agricolaabout adeafmutewhohad learned towrite.

C

ardano'seldest and favoritesonwasexecuted in9 @

E

0 afterheconfessed to

havingpoisonedhiscuckoldingwife. Hisothersonwasagambler, whostolemoney fromhim. He

allegedlycropped theearsofoneofhissons.C

ardanohimselfwasaccusedofheresy in9 @ A

0

becausehehadcomputedandpublished thehoroscopeofJesus in9 @ @

4. Apparently, hisownson

contributed to theprosecution, bribedbyB

artaglia. Hewasarrested, hadtospendseveral months in

prisonandwas forced toabjurehisprofessorship. Hemoved toDome, receiveda

lifetimeannuity fromPopeFregory XIII

G

afterfirst havingbeenrejectedbyPope Pius V)and finished

hisautobiography.


6/16

Jacob BH

I P

oulliQ

alsoknownasJamH R

orJacques)Q

S

T

DecemberU V

W

4 U V

AugustU T

0W

)was

oneof themanyprominent mathematicians in theBernoulli family.

Jacob Bernoulli wasborn inBasel, SwitX

erland.Y

ollowinghis father'swish, hestudiedtheologyand

entered theministry. But contrary to thedesiresofhisparents, healso

studiedmathematicsandastronomy. He traveled throughout Europe fromU V T V

toU V

8S

, learning

about the latest discoveries inmathematicsand thesciences.

his included theworkofa

obert

Boyleanda

obert Hooke.

Jacob Bernoulli 'sgrave.

Hebecame familiarwithcalculus throughacorrespondencewithb

ottfried LeibniX

, thencollaborated

withhisbrotherJohannonvariousapplications, notablypublishingpapersontranscendental


7/16

curvesc

d

e f e

)andisoperimetryc

d g00,

d g0

d). In

d

e f

0, Jacob Bernoulli became the first person to

develop the technique forsolvingseparabledifferential equations.

Uponreturning toBasel ind

e

8h

, he foundedaschool formathematicsand thesciences. Hewas

appointedprofessorofmathematicsat theUniversityof Basel ind

e

8g

, remaining in thisposition for

therest ofhis life.

Jacob Bernoulli isbest known fortheworkArs Conjectandii

p

he Art ofq

onjecture), publishedeight

yearsafterhisdeath inr

s

r tbyhisnephew

uicholas. In thiswork, hedescribed theknownresults in

probability theoryand inenumeration, oftenprovidingalternativeproofsofknownresults.p

hiswork

also includes theapplicationofprobability theory togamesofchanceandhis introductionof the

theoremknownas thelawof largenumbers.p

he termsBernoulli trialandBernoulli numbersresult

from thiswork.p

he lunarcraterBernoulli isalsonamedafterhim jointlywithhisbrotherJohann.

Bernoulli chosea figureofalogarithmicspiraland themottoEadem mutata resurgo("Changed and

yet the same, I rise again") forhisgravestone; thespiral executedby thestonemasonswas, however,

anArchimedeanspiral.[

v

], [Jacques Bernoulli] wrote that the logarithmicspiral maybeusedasa

symbol, eitherof fortitudeandconstancy inadversity, orof thehumanbody, whichafterallits

changes, evenafterdeath, will berestored to itsexact andperfect self.iLivio

w

00w

:r r x

). Jacobhad

fivedaughtersand threesons.

This article is about the French scientist and philosopher. For the programming language, seePascal

(programming language).

Bly

is

Pascal


8/16

Blaise Pascal

Full name Blaise Pascal

B

orn June 19, 1623

Clermont-Ferrand, France

Died August 19, 1662 (aged 39)

Paris, France

Era 17th-century philosophy

Region Western Philosophy

S

hool

Continental Philosophy, precursor to existentialism

Main

interests

Theology, Mathematics

Notable ideas Pascal's Wager, Pascal's triangle,Pascal's law, Pascal's

theorem

Influen

ed by

Blaise Pascal (French pronunciation: [blz paskal]; une 19, 1623, Clermont-Ferrand August 19,

1662, Paris) was a Frenchmathematician, physicist,inventor, writerand Catholicphilosopher. He was

a child prodigy who was educated by his father, a Tax Collector in Rouen. Pascal's earliest work was

in the natural and applied sciences where he made important contributions to the study offluids, and

clarified the concepts ofpressure and vacuum by generalizing the work ofEvangelista Torricelli.

Pascal also wrote in defense of the scientific method.

In 16

2, while still a teenager, he started some pioneering work on calculating machines, and after

three years of effort and 5

prototypes[1]

he invented themechanical calculator[2][3]

. He built twenty of

these machines (called the Pascaline) in the following ten years[ ]. Pascal was a mathematician of the

first order. He helped create two major new areas of research. He wrote a significant treatise on the

subject ofprojective geometry at the age of sixteen, and later corresponded with Pierre de

Fermat on probability theory, strongly influencing the development of modern economics and social

science. FollowingGalileo and Torricelli, in 16

6 he refutedAristotle's followers who insisted

that nature abhors a vacuum. His results caused many disputes before being accepted.


9/16

In 16

6, he and his sisteracqueline identified with the religious movement within Catholicism known

by its detractors asansenism.

[5]His father died in 1651. Following a mystical experience in late 165

,

he had his "second conversion", abandoned his scientific work, and devoted himself to philosophy

andtheology. His two most famous works date from this period: the Lett

es

i

i

les and

the Penses

, the former set in the conflict between

ansenists and

esuits. In this year, he also wrotean important treatise on the arithmetical triangle. Between 165

and 1659 he wrote on the cycloid and

its use in calculating the volume of solids.

Pascal had poorhealth especially after his eighteenth year and his death came just two months after

his 39th birthday.[

b)The difference between Theoretical Probability and

Empirical Probability

Theoretical probability is the branch of probability

concerned with the theory. There is no concrete proof

and all results are based only on calculation.

Empirical probability, as its name suggests, is based on

experiments and the result.

y Genetics and other stochastic models in Application biology

y Information theory and signal processing

y Communication networks

y Application Stochastic models in operations research

The probability theory

The theoretical probability, or probability, P(E), of an event E is thefraction of times we

expect E to occur if we repeat the sameexperiment over and over. In an experiment in

which all outcomes areequally likely, the theoretical probability of an event E is P(E)


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P(E) : number of favorable outcomes = n(E)Total number of outcomes n(s)

(The "favorable outcomes" are the outcomes in E.)

The empirical approach to determining probabilities relies on datafrom actual experiments

to determine approximate probabilitiesinstead of the assumption of equal likeliness.

Probabilities in theseexperiments are defined as the ratio of the frequency of theoccupance

of an event, f(E), to the number of trials in the experiment,n, written symbolically as P(E) =

f(E)/n. If our experiment involvesflipping a coin, the empirical probability of heads is the

number of heads divided by the total number of flips.


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Part 2

Possible outcomes when dice is tossed once

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

b) Possible outcomes when two dice are tossed simultaneously

total outcomes

(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 ) }

1 2 3 5 6

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

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

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

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

5 (5,!) (5,2) (5,3) (5, ) (5,5) (5,6)

6 (6,1) (6,2) (6,3) (6, ) (6.5) (6,6)


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6 (1,6) (2,6) (3,6) ( ,6) (5,6) (6,6)^

5 (1,5) (2.5) (3,5) ( ,5) (5,5) (6,5)

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

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

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

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

1 2 3 5 6 12

)3

Method 1`

4

5

1

6

2

3

4

56

1

2

3

4

5

1 6

2

3

4

1 5

6

2

5 4 3\

1

2

3

4

5


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Part 3

Sum of the dots on

both turned up faces(x)

Possible outcomes Probaility

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),(3,3),(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),(6,2) 6/36

8 (2,6),(3,5),(4,4),(5,3),(6,2) 5/36

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

(b)Possible outcomes of

A = { (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)}

B =

P = Both number are prime

P = {(2,2), (2,3), (2,5), (3,3), (3,5), (5,3), (5,5)


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Q = Difference of 2 number is odd

Q = { (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 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

R = {(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 R

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

PART 4

(a)

Conduct an activity by tossing two dice simultaneously 50 times

SUM OF THE TWO NUMBER (X) FREQUENCY( 4 3

9 4

16 3

25 2

36 4

49 6

64 10

81 11

100 2

121 3

144 2

Table 4.1


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Aim of carrying out this additional mathematics project work are

y To apply and adapt a variaty of problem -solving strategies to solve problems

y

y To promotes effective mathematical communication

y To develop mathematical knowledge through problems solving in a way increase students

Interest and confidence .

y To use the language of mathematics to express mathematical ideas precisely .

y To provide learning enviroment that stimulates and enchances effective learning

y To develop positive attitude toward mathematics .

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