Kerja Projcj

7/26/2019 Kerja Projcj

1/38

P a g e | 1

Acknowledge

First of all, I would like to say Alhamdulillah, for giving me

the strength and health to do this project work.

Not forgotten my parents for providingeverything, such as money, to uy anything that arerelated to this project work and their advise, which is themost needed for this project. Internet, ooks, computersand all that. !hey also supported me and encouraged meto complete this task so that I will not procrastinate in

doing it.

!hen I would like to thank my teacher, "adam #aiton forguiding me and my friends throughout this project. $ehad some di%culties in doing this task, ut she taught uspatiently until we knew what to do. &he tried and tried toteach us until we understand what we supposed to dowith the project work.

'ast ut not least, my friends who were doing thisproject with me and sharing our ideas. !hey were helpfulthat when we comined and discussed together, we hadthis task done.


2/38

P a g e | 2

(jective

!he aims of carrying out this project work are)

to apply and adapt a variety of prolem*solvingstrategies tosolve prolems+

to improve thinking skills+

to promote eective mathematical communication+

to develop mathematical knowledge through prolem

solvingIn a way that increases students- interest andcondence+

to use the language of mathematics to e/pressmathematicalideas precisely+

to provide learning environment that stimulates andenhanceseective learning+

to develop positive attitude towards mathematics.


3/38

P a g e | 3


4/38

P a g e | 4

PA0! (N1


5/38

P a g e | 5

Introduction

Proaility is a way of e/pressing knowledge or elief that an eventwill occur

or has occurred. In mathematicsthe concept has een given an e/act

meaning in proaility theory, that is used e/tensively in such areas of study

as mathematics, statistics, nance, gamling, science, and philosophyto

draw conclusions aout the likelihood of potential events and the underlying

mechanics of comple/ systems.

Proailityhas a dual aspect) on the one hand the proaility or likelihood of

hypotheses given the evidence for them and on the other hand the ehavior

of stochastic processessuch as the throwing of dice or coins. !he study of

the former is historically older in, for e/ample, the law of evidence, while the

mathematical treatment of dice egan with the work of Pascaland Fermatinthe 2345s.

Proailityis distinguished from statistics. $hile statistics deals with data

and inferences from it, 6stochastic7 proaility deals with the stochastic

6random7 processes which lie ehind data or outcomes.

8I&!(09 * (0I:IN&

Ancient and medieval law of evidencedeveloped a grading of degrees ofproof, proailities, presumptions and half*proofto deal with theuncertainties of evidence in court. In 0enaissancetimes, etting wasdiscussed in terms of odds such as ;ten to one; and maritime insurancepremiums were estimated ased on intuitive risks, ut there was no theoryon how to calculate such odds or premiums.

!he mathematical methods of proaility arose in the correspondence ofPierre de Fermatand 7 on such ?uestions as the fairdivision of the stake in an interrupted game of chance. @hristian 8uygens62347 gave a comprehensive treatment of the suject.

I"P(0!AN@1 (F P0(


6/38

P a g e | 6

Proaility theory goes eyond mathematics. It involves logic and reasoning

ailities. "arketing and politics have one thing in common, iased statistics. I

elieve since you are e/posed to so many statistics, a asic understanding of

this area allows more critical thinking. !he ook ;8ow to lie with statistics; is

a classic and still in print.

&o, while many people would proaly say that proaility theory has little

importance in their lives, perhaps in some cases if they knew more, it would

have more importance.

8istory of Proaility

Etymology

Probableand likelyand their cognates in other modern languages derivefrom medieval learned 'atinprobabilisand verisimilis, deriving from @iceroand generally applied to an opinion to meanplausibleor generally approved.

Origins

Ancient and medieval law of evidencedeveloped a grading of degrees ofproof, proailities, presumptions and half*proofto deal with theuncertainties of evidence in court. In 0enaissancetimes, etting was

discussed in terms of odds such as ;ten to one; and maritime insurancepremiums were estimated ased on intuitive risks, ut there was no theoryon how to calculate such odds or premiums.

!he mathematical methods of proaility arose in the correspondence ofPierre de Fermatand 7 on such ?uestions as the fairdivision of the stake in an interrupted game of chance. @hristiaan 8uygens62347 gave a comprehensive treatment of the suject.

18th Century

Baco


7/38

P a g e | 7

the larger the numer of throws, the closer to half*and*half the proportion islikely to e7.

19th Century!he power of proailistic methods in dealing with uncertainty was shown y:aussCs determination of the orit of @eresfrom a few oservations. !hetheory of errorsused the method of least s?uaresto correct error*proneoservations, especially in astronomy, ased on the assumption of a normaldistriutionof errors to determine the most likely true value.

!owards the end of the nineteenth century, a major success of e/planation interms of proailities was the &tatistical mechanicsof 'udwig


8/38

P a g e | 8

suspended in a Guid. !hat provided a model for the study of randomGuctuations in stock markets, leading to the use of sophisticated proailitymodels in mathematical nance, including such successes as the widely*used


9/38

P a g e | 9

Applications

!wo major applications of proaility theory in everyday life are in riskassessment and in trade on commodity markets. :overnments typicallyapply proailistic methods in environmental regulationwhere it is called;pathway analysis;, often measuring well*eingusing methods that arestochasticin nature, and choosing projects to undertake ased on statisticalanalyses of their proale eect on the population as a whole.

A good e/ample is the eect of the perceived proaility of any widespread"iddle 1ast conGict on oil prices * which have ripple eects in the economyas a whole. An assessment y a commodity trader that a war is more likelyvs. less likely sends prices up or down, and signals other traders of thatopinion. Accordingly, the proailities are not assessed independently nornecessarily very rationally. !he theory of ehavioral nanceemerged todescrie the eect of such groupthinkon pricing, on policy, and on peaceand conGict.

It can reasonaly e said that the discovery of rigorous methods to assessand comine proaility assessments has had a profound eect on modernsociety. Accordingly, it may e of some importance to most citiens tounderstand how odds and proaility assessments are made, and how theycontriute to reputations and to decisions, especially in a democracy.

Another signicant application of proaility theory in everyday life isreliaility. "any consumer products, such as automoilesand consumerelectronics, utilie reliaility theoryin the design of the product in order toreduce the proaility of failure. !he proaility of failure may e closelyassociated with the productCs warranty.
http://en.wikipedia.org/wiki/Riskhttp://en.wikipedia.org/wiki/Commodity_marketshttp://en.wikipedia.org/wiki/Environmental_regulationhttp://en.wikipedia.org/w/index.php?title=Pathway_analysis&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Pathway_analysis&action=edit&redlink=1http://en.wikipedia.org/wiki/Measuring_well-beinghttp://en.wikipedia.org/wiki/Stochastichttp://en.wikipedia.org/wiki/Behavioral_financehttp://en.wikipedia.org/wiki/Groupthinkhttp://en.wikipedia.org/wiki/Democracyhttp://en.wikipedia.org/wiki/Reliability_theory_of_aging_and_longevityhttp://en.wikipedia.org/wiki/Automobileshttp://en.wikipedia.org/wiki/Reliability_theoryhttp://en.wikipedia.org/wiki/Warrantyhttp://en.wikipedia.org/wiki/Riskhttp://en.wikipedia.org/wiki/Commodity_marketshttp://en.wikipedia.org/wiki/Environmental_regulationhttp://en.wikipedia.org/w/index.php?title=Pathway_analysis&action=edit&redlink=1http://en.wikipedia.org/wiki/Measuring_well-beinghttp://en.wikipedia.org/wiki/Stochastichttp://en.wikipedia.org/wiki/Behavioral_financehttp://en.wikipedia.org/wiki/Groupthinkhttp://en.wikipedia.org/wiki/Democracyhttp://en.wikipedia.org/wiki/Reliability_theory_of_aging_and_longevityhttp://en.wikipedia.org/wiki/Automobileshttp://en.wikipedia.org/wiki/Reliability_theoryhttp://en.wikipedia.org/wiki/Warranty


10/38

P a g e | 10

!heoretical proaility

Defnition o Theoreti!l "ro#!#ility

Proaility is a likelihood that an event will happen.

$e can nd the theoretical proaility of an event using the following ratio)

'et-s do a couple of e/amples.

$ol%e& E'!m(les on Theoreti!l "ro#!#ility

E'!m(le 1

) *e toss ! !ir oin+ *h!t is the (ro#!#ility th!t ! t!il *ill sho* u(,

$olution-

!ossing a tail is the favorale outcome here.

$hen you toss a coin there are only J possile outcomes) a 8ead or a !ail

&o the options for tossing a tail are 2 out of J.

$e can also represent proaility as a decimal or as a percent.

E'!m(le 2

. #!g ont!ins 20 m!r#les/ 15 o them !re re& !n& 5 o them !re #lue inolor/ in& the (ro#!#ility o (iing ! re& m!r#le/

'et-s rst answer a few ?uestions here)

If I am going to randomly pick a marle from the ag then what results can I have)

I-ll either pick a red marle or a lue one.

"y ne/t ?uestion is what the chances of picking a red marle are)


11/38

P a g e | 11

!here are 24 red marles and just 4 lue marles.

It-s ovious that we have three times as many red marles as lue marles.

&o, the chance of picking a red marle is more than that of the lue one.

!herefore, the proaility of picking a red marle is)

E'!m(le 3

in& the (ro#!#ility o getting ! sum o 7 *hen you roll t*o &ie/

!wo dice are eing rolled. !he possile outcomes are as follows)

'et-s use the representation 6a, 7 for the outcomes where ! num#er on &ie 1and # num#er on &ie 2.

62, 27, 62, J7, 62, D7, 62, >7, 62, 47, 62, 37,6J, 27, 6J, J7, 6J, D7, 6J, >7, 6J, 47, 6J, 37,6D, 27, 6D, J7, 6D, D7, 6D, >7, 6D, 47, 6D, 37,6>, 27, 6>, J7, 6>, D7, 6>, >7, 6>, 47, 6>, 37,64, 27, 64, J7, 64, D7, 64, >7, 64, 47, 64, 37,63, 27, 63, J7, 63, D7, 63, >7, 63, 47, 63, 37

!here are D3 possile outcomes in all.

The uestion is *hen you roll t*o &ie+ *h!t !re the h!nes o getting !sum o 7,

From the list aove identify the pairs with outcomes that add up to .

'et-s highlight them this way)

62, 27, 62, J7, 62, D7, 62, >7, 62, 47, 1+ 6+6J, 27, 6J, J7, 6J, D7, 6J, >7, 2+ 5+6J, 37,6D, 27, 6D, J7, 6D, D7, 3+ 4+6D, 47, 6D, 37,6>, 27, 6>, J7, 4+ 3, 6>, >7, 6>, 47, 6>, 37,64, 27, 5+ 2, 64, D7, 64, >7, 64, 47, 64, 37,6+ 1, 63, J7, 63, D7, 63, >7, 63, 47, 63, 37

(serve that the pairs along the main diagonal add up to . !here are 3 such pairs.

&o, the proaility of getting a sum of when we roll two dice is)


12/38

P a g e | 12

1mpirical Proaility

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 eneral sense, empirical probability estimates

probabilities from experiencean!observation.["]#he phrase a posteriori probability has also been

use! as an alternative to empirical probability or relative frequency.[$]#his unusual usae of the

phrase is not !irectly relate! to %ayesian inferencean! not to be confuse! with its equally

occasional use to refer toposterior probability,which is somethin else.

In statistical terms, the empirical probability is an estimate of a probability. If mo!elin usin a

binomial !istributionis appropriate, it is the maximum likelihoo! estimate.It is the %ayesian

estimatefor the same case if certain assumptions are ma!e for theprior !istributionof the

probability

&n a!vantae of estimatin probabilities usin empirical probabilities is that this proce!ure is

relatively free of assumptions. 'or example, consi!er estimatin the probability amon a

population of men that they satisfy two con!itions( )i* that they are over + feet in heiht )ii* that

they prefer strawberry -am to raspberry -am. & !irect estimate coul! be foun! by countin the

number of men who satisfy both con!itions to ive the empirical probability the combine!

con!ition. &n alternative estimate coul! be foun! by multiplyin the proportion of men who are

over + feet in heiht with the proportion of men who prefer strawberry -am to raspberry -am, but

this estimate relies on the assumption that the two con!itions are statistically in!epen!ent.

& !isa!vantae in usin empirical probabilities arises in estimatin probabilities which are either

very close to ero, or very close to one. In these cases very lare sample sies woul! be nee!e!

in or!er to estimate such probabilities to a oo! stan!ar! of relative accuracy. /ere statistical

mo!elscan help, !epen!in on the context, an! in eneral one can hope that such mo!els woul!

provi!e improvements in accuracy compare! to empirical probabilities, provi!e! that the

assumptions involve! actually !o hol!. 'or example, consi!er estimatin the probability that the

lowest of the !aily0maximum temperatures at a site in 'ebruary in any one year is less ero

!erees elsius. & recor! of such temperatures in past years coul! be use! to estimate this

probability. & mo!el0base! alternative woul! be to select of family ofprobability !istributions

an! fit it to the !ataset contain past yearly values( the fitte! !istribution woul! provi!e an

alternative estimate of the require! probability. #his alternative metho! can provi!e an estimate

of the probability even if all values in the recor! are reater than ero.
http://en.wikipedia.org/wiki/Frequency_(statistics)http://en.wikipedia.org/wiki/Experiencehttp://en.wikipedia.org/wiki/Experiencehttp://en.wikipedia.org/wiki/Observationhttp://en.wikipedia.org/wiki/Observationhttp://en.wikipedia.org/wiki/Bayesian_inferencehttp://en.wikipedia.org/wiki/Posterior_probabilityhttp://en.wikipedia.org/wiki/Posterior_probabilityhttp://en.wikipedia.org/wiki/Binomial_distributionhttp://en.wikipedia.org/wiki/Maximum_likelihood_estimatehttp://en.wikipedia.org/wiki/Maximum_likelihood_estimatehttp://en.wikipedia.org/wiki/Bayesian_estimatehttp://en.wikipedia.org/wiki/Bayesian_estimatehttp://en.wikipedia.org/wiki/Prior_distributionhttp://en.wikipedia.org/wiki/Prior_distributionhttp://en.wikipedia.org/wiki/Prior_distributionhttp://en.wikipedia.org/wiki/Statistically_independenthttp://en.wikipedia.org/wiki/Statistically_independenthttp://en.wikipedia.org/wiki/Statistical_modelhttp://en.wikipedia.org/wiki/Statistical_modelhttp://en.wikipedia.org/wiki/Probability_distributionshttp://en.wikipedia.org/wiki/Frequency_(statistics)http://en.wikipedia.org/wiki/Experiencehttp://en.wikipedia.org/wiki/Observationhttp://en.wikipedia.org/wiki/Bayesian_inferencehttp://en.wikipedia.org/wiki/Posterior_probabilityhttp://en.wikipedia.org/wiki/Binomial_distributionhttp://en.wikipedia.org/wiki/Maximum_likelihood_estimatehttp://en.wikipedia.org/wiki/Bayesian_estimatehttp://en.wikipedia.org/wiki/Bayesian_estimatehttp://en.wikipedia.org/wiki/Prior_distributionhttp://en.wikipedia.org/wiki/Statistically_independenthttp://en.wikipedia.org/wiki/Statistical_modelhttp://en.wikipedia.org/wiki/Statistical_modelhttp://en.wikipedia.org/wiki/Probability_distributions


13/38

P a g e | 13

1mpirical and !heoreticalProaility

1mpirical Proaility of an event is an ;estimate; that the event will happenased on how often the event occurs after collecting data or running ane/periment 6in a large numer of trials7. It is ased specically on directoservations or e/periences.

Em(iri!l "ro#!#ility ormul!

P67 K proaility that an event, , willoccur.TopK numer of ways the specic eventoccurs.!ottom K numer of ways the e/perimentcould occur.

E'!m(le) A survey was conducteddetermine studentsC favorite reeds dogs. 1ach student chose only onereed.

Dog

Collie

$(!niel

!#

o'er

"it

#ull

Other

25 24 D4 E 4 2J

$hat is the proaility that a studenfavorite dog reed is 'aLAnswer) D4 out of the E4 students

chose 'a. !he proaility is

!heoretical Proaility of an event is the numer of ways that the event canoccur, divided y the total numer of outcomes. It is nding the proailityof events that come from a sample space of known e?ually likely outcomes.

Theoreti!l "ro#!#ility ormul!

P67 K proaility that an event, , willoccur.n"#K numer of e?ually likely outcomes

E'!m(le 1) Find the proaility ofrolling a si/ on a fair die.

Answer) !he sample space for rollindie is 3 e?ually likely results) M2, J, D>, 4, 3.!he proaility of rolling a 3 is one o

of 3 or


14/38

P a g e | 14

of .n"$# K numer of e?ually likely outcomesof sample space $.

E'!m(le 2- Find the proaility of tossing a fair die and getting an odd numer.Answer)event ) tossing an odd numeroutcomes in ) M2, D, 4sample space $) M2, J, D, >, 4, 3

@omparing 1mpirical and

!heoretical Proailities

Haren and Bason roll two dice 45 times andrecord their results in the accompanying chart.1/ $hat is their empirical proaility ofrolling a L2/ $hat is the theoretical proaility ofrolling a L3/ 8ow do the empirical and theoreticalproailities compareL

&um of the rolls oftwo dice

D, 4, 4, >, 3, , , 4,, 25,

2J, , 3, 4, , E, , >,22, 3,

E, E, 25, 3, , >, >, 4,

, ,, , E, 22, 3, 4, >, ,

, >,D, 3, , , , E, 3, ,

E,

$olution-2.7 1mpirical proaility 6e/perimental proaility oroserved proaility7 is 2DO45 K J3.

J.7 !heoretical proaility 6ased upon what ispossile when working with two dice7 K 3OD3 K 2O3K 23. 6check out the tale at the right of possilesums when rolling two dice7.D.7 Haren and Bason rolled more Cs than would ee/pected theoretically.


15/38

P a g e | 15


16/38

P a g e | 16

PA0! !$(


17/38

P a g e | 17

a7 "y friends and I are playing "onopoly. At the eginner, each

of us will toss a die once. !he player who otains the highest

numer will start the game. !he possile outcomes are)

M2, J, D, >, 4, 37 Instead of one dice, two dice can also e tossed

simultaneously y each player. !he player will move the

token according to the sum of all dots on oth turned*up

faces. For e/ample, if the two dice are tossed simultaneously

and ;J; appears on one die and ;D; appears on the other,

the outcome of the toss is 6J, D7. 8ence, the player shall

move the token 4 spaces. !he possile outcomes when two

dice are tossed simultaneously)

M62, 27, 62, J7, 62, D7, 62, >7, 62, 47, 62, 37, 6J, 27, 6J, J7, 6J, D7, 6J, >7, 6J, 47, 6J, 37, 6D, 27, 6D, J7, 6D, D7, 6D, >7, 6D, 47, 6D, 37, 6>, 27, 6>, J7, 6>, D7, 6>, >7, 6>, 47, 6>, 37, 64, 27, 64, J7, 64, D7, 64, >7, 64, 47, 64, 37, 63, 27, 63, J7, 63, D7, 63, >7, 63, 47, 63, 37


18/38

P a g e | 18

PA0! !8011


19/38

P a g e | 19

!ale 2 show the sum of all dots on oth turned*up faces when

two dice are tossed simultaneously.

&um of dots on

oth turned*upfaces "%#

Possile outcomes Proaility, P"%#

J 62,27 136

D 62,J7,6J,27 236

> 62,D7,6J,J7,6D,27 336

4 62,>7,6J,D7,6D,J7,6>,27 4

36

3 62,47,6J,>7,6D,D7,6>,J7,

64,27

5

36

62,37,6J,47,6D,>7,6>,D7,

64,J7,63,27

6

36

E 6J,37,6D,47,6>,>7,64,D7,

63,J7

5

36

6D,37,6>,47,63,D7,64,>7 4

36

25 6>,37,64,47,63,>7 336

22 64,37,63,47 236

2J 63,37 136

!A


20/38

P a g e | 20

A K Mthe two numers are not the same

K M62,J7,6J,27,62,D7,6D,27, 62,>7,6J,D7,6D,J7,6>,27,62,47,6J,>7,6>,J7,

64,27, 62,37,6J,47,6D,>7,6>,D7,64,J7,63,27, 6J,37,6D,47,64,D7,

63,J7, 6D,37,6>,47,63,D7,64,>7, 6>,37,63,>7, 64,37,63,47

P 6A7 Kn(A)n(S)

K30

36

< K Mthe product of the two numers is greater than D3

K { }

@ K M7,62,37, 6J,D7,6J,47,6D,J7,6D,>7,6D,37,6>,27,6>,D7,6>,47,

64,J7,64,>7,64,37,63,27,63,D7,63,47

P6@7 K9

36+17

36

K26

36


21/38

P a g e | 21

Q K M!he sum of the two numers are even and oth numers

are prime7

K M62,27,62,D7,62,47,6J,J7,6J,>7,6J,37,6D,27,6D,D7,6D,47,6>,J7,6>,>7,

6>,37,64,27,64,D7,

64,47,63,J7,63,>7,63,37 M6J,D7,6J,47,6D,J7,6D,D7,6D,47,64,J7,

64,D7,64,47

P 6Q7 K18

36 8

36

K1

9


22/38

P a g e | 22

PA0! F(R0


23/38

P a g e | 23

a7 An activity was conducted y tossing two dice

simultaneously 45 times. !he sums of all dots on oth

turned*up faces were oserved. !he fre?uency tale is

completed elow.

&um of the two numers "%# Fre?uency "f#J DD E> 34 3 DE J

>25 >22 D2J 2

!A


24/38

P a g e | 24

K 7.4596

KJ.D2JJ3ED2

7 New mean value if the numer of tosses is increased to 255times K

c7 !he prediction in 67 is tested y continuing Activity D6a7 until

the total numer of tosses is 255 times. !he value of

6i7 mean+

6ii7 variance+ and

6iii7 standard deviation

of the new data is estimated.

x f fx fx2

J 3 2J J>D J E2> 22 >> 234 2J 35 D55

3 2D E >3E 25 5 >5E 43 >>E 2J 25E J

25 5 5522 3 33 J32J E> 255E

f= 255 fx K34 f x2

3"4"


25/38

P a g e | 25

* mean K fx

N

K675

100 K 3.4

* SarianceK f x2

N x2

K5363

1006.752

KE.D34

* &tandard deviation K f x

2

N

x2

KJ.EJ33D2DD

!he prediction is proven.


26/38

P a g e | 26

PA0! FIS1


27/38

P a g e | 27

$hen two dice are tossed simultaneously, the actual mean and variance of

the sum of all dots on the turned*up faces can e determined y using the

formulae elow)

a7


28/38

P a g e | 28

"eanK

SarianceK1787

1872

K45.JE

&tandard deviationK 50.277777778

K.553EJ>3J

7

#able below shows the comparison of mean, variance an! stan!ar! !eviation of part $ an! part 3.

5&6# $ 5&6# 3

n37 n177

8ean +.72 +.93 9.77

:ariance 9.$34+ ;."+93 45.JE


29/38

P a g e | 29

theoretical value. >hen the number of trial )number of sample* ettin bier, the empirical

value shoul! et closer to the theoretical value. /owever, violation of this rule is still possible,

especially when the number of trial )or sample* is not lare enouh.

c7#he rane of the mean

6mean 7

on-ecture( &s the number of toss, n, increases, the mean will et closer to 9. 9 is the theoretical

mean.

Imae below support this con-ecture where we can see that, after 377 toss, the theoretical mean

become very close to the theoretical mean, which is ".3. )#ake note that this is experiment oftossin 1 !ie, but not 2 !ice as what we !o in our experiment*


30/38

P a g e | 30


31/38

P a g e | 31

FR0!8101TP'(0A!I(N


32/38

P a g e | 32

'aw of 'arge Numers

In proaility theory, the law of large numers 6''N7 is a theoremthat

descries the result of performing the same e/periment a large numer of

times. According to the law, the averageof the results otained from a large

numer of trials should e close to the e/pected value, and will tend to

ecome closer as more trials are performed.

For e/ample, a single roll of a si/*sided dieproduces one of the numers 2,

J, D, >, 4, 3, each with e?ual proaility. !herefore, the e/pected value of a

single die roll is

1+2+3+4+5+66

=3.5

According to the law of large numers, if a large numer of dice are rolled,

the average of their values 6sometimes called the sample mean7 is likely to

e close to D.4, with the accuracy increasing as more dice are rolled.

&imilarly, when a fair coinis Gipped once, the e/pected value of the numer

of heads is e?ual to one half. !herefore, according to the law of large

numers, the proportion of heads in a large numer of coin Gips should e

roughly one half. In particular, the proportion of heads after nGips will almost

surelyconvergeto one half as napproaches innity.

!hough the proportion of heads 6and tails7 approaches half, almost surelythe

asolute 6nominal7 dierence in the numer of heads and tails will ecome

large as the numer of Gips ecomes large. !hat is, the proaility that the

asolute dierence is a small numer approaches ero as numer of Gips

ecomes large. Also, almost surely the ratio of the asolute dierence to

numer of Gips will approach ero. Intuitively, e/pected asolute dierence

grows, ut at a slower rate than the numer of Gips, as the numer of Gips

grows.

!he ''N is important ecause it ;guarantees; stale long*term results for

random events. For e/ample, while a casino may lose money in a single spin

of the roulettewheel, its earnings will tend towards a predictale percentage

over a large numer of spins. Any winning streak y a player will eventually

e overcome y the parameters of the game. It is important to rememer
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/Sample_meanhttp://en.wikipedia.org/wiki/Fair_coinhttp://en.wikipedia.org/wiki/Almost_surelyhttp://en.wikipedia.org/wiki/Almost_surelyhttp://en.wikipedia.org/wiki/Limit_of_a_sequencehttp://en.wikipedia.org/wiki/Almost_surelyhttp://en.wikipedia.org/wiki/Roulettehttp://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/Sample_meanhttp://en.wikipedia.org/wiki/Fair_coinhttp://en.wikipedia.org/wiki/Almost_surelyhttp://en.wikipedia.org/wiki/Almost_surelyhttp://en.wikipedia.org/wiki/Limit_of_a_sequencehttp://en.wikipedia.org/wiki/Almost_surelyhttp://en.wikipedia.org/wiki/Roulette


33/38

P a g e | 33

that the ''N only applies 6as the name indicates7 when a large numberof

oservations are considered. !here is no principle that a small numer of

oservations will converge to the e/pected value or that a streak of one

value will immediately e ;alanced; y the others. &ee the :amlerCs

fallacy.
http://en.wikipedia.org/wiki/Gambler's_fallacyhttp://en.wikipedia.org/wiki/Gambler's_fallacyhttp://en.wikipedia.org/wiki/Gambler's_fallacyhttp://en.wikipedia.org/wiki/Gambler's_fallacy


34/38

P a g e | 34

01F'1@!I(N


35/38

P a g e | 35

!1A" $(0H I& I"P(0!AN!


36/38

P a g e | 36


37/38

P a g e | 37

@(N@'R&I(N


38/38

P a g e | 38

As a conclusion, now I know+

!he history of proaility from 2Ethcentury to J5th

century,

8ow to apply theory of proaility in daily life and itsimportance,

!wo categories of proaility) empirical proaility

and theoretical proaility and their dierences,8ow to conduct a dice*tossing activity to nd its

proaility,8ow to calculate the proaility,8ow to calculate mean, variance and standard

deviation using formula)

"ean, x fx

N , orx=x P(x)

Sariance, 2= f x2

N x2 , or

x2

P(x )x2

2=

&tandard deviation, =f x

2

N x , or

x2P(x )x2=

Aout 'aw of 'arge Numer 6''N7 and its relation inthis project work,"oral values otained from this project work.

Kerja Projcj

Documents

Transcript of Kerja Projcj