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ADDITIONAL MATHEMATICS

PROJECT WORK 2/2010

TITILE : THEORY OF PROBABILITY

NAME : KYRIOS JOYCE ERDAYA

RAJOO

IC NO : 930603-10-5700CLASS : 5 MULIA

TEACHER : MRS.MALLIKA


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a) History of Probability

The scientific study of probability is a moderndevelopment. Gambling shows that there has been aninterest in quantifying the ideas of probability for millennia,

but exact mathematical descriptions of use in thoseproblems only arose much later.

According to Richard Jeffrey, "Before the middle of theseventeenth century, the term 'probable' (Latinprobabilis)meant approvable, and was applied in that sense,univocally, to opinion and to action. A probable action oropinion was one such as sensible people would undertakeor hold, in the circumstances. However, in legal contexts

especially, 'probable' could also apply to propositions forwhich there was good evidence.

Aside from some elementary considerations made byGirolamo Cardano in the 16th century, the doctrine ofprobabilities dates to the correspondence of Pierre de
http://en.wikipedia.org/wiki/Gamblinghttp://en.wikipedia.org/wiki/Girolamo_Cardanohttp://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Girolamo_Cardanohttp://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Gambling


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Fermat and Blaise Pascal (1654). Christiaan Huygens(1657) gave the earliest known scientific treatment of thesubject. Jakob Bernoulli's Ars Conjectandi (posthumous,1713) and Abraham de Moivre's Doctrine of Chances

(1718) treated the subject as a branch of mathematics.See Ian Hacking'sThe Emergence of ProbabilityandJamesFranklin's The Science of Conjecture for histories of theearly development of the very concept of mathematicalprobability.

The theory of errors may be traced back to Roger Cotes'sOpera Miscellanea (posthumous, 1722), but a memoir

prepared byThomas Simpson in 1755 (printed 1756) firstapplied the theory to the discussion of errors ofobservation. The reprint (1757) of this memoir lays downthe axioms that positive and negative errors are equallyprobable, and that there are certain assignable limitswithin which all errors may be supposed to fall; continuouserrors are discussed and a probability curve is given.

Pierre-Simon Laplace (1774) made the first attempt to

deduce a rule for the combination of observations from theprinciples of the theory of probabilities. He represented thelaw of probability of errors by a curvey= (x),xbeing anyerror andy its probability, and laid down three propertiesof this curve:

1. it is symmetric as to they-axis;2. thex-axis is an asymptote, the probability of the error

being 0;3. the area enclosed is 1, it being certain that an errorexists.

He also gave (1781) a formula for the law of facility of error(a term due to Lagrange, 1774), but one which led to
http://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/Christiaan_Huygenshttp://en.wikipedia.org/wiki/Jakob_Bernoullihttp://en.wikipedia.org/wiki/Ars_Conjectandihttp://en.wikipedia.org/wiki/Abraham_de_Moivrehttp://en.wikipedia.org/wiki/Doctrine_of_Chanceshttp://en.wikipedia.org/wiki/Ian_Hackinghttp://en.wikipedia.org/wiki/James_Franklin_(philosopher)http://en.wikipedia.org/wiki/James_Franklin_(philosopher)http://en.wikipedia.org/wiki/Roger_Coteshttp://en.wikipedia.org/wiki/Thomas_Simpsonhttp://en.wikipedia.org/wiki/Pierre-Simon_Laplacehttp://en.wikipedia.org/wiki/Asymptotehttp://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/Christiaan_Huygenshttp://en.wikipedia.org/wiki/Jakob_Bernoullihttp://en.wikipedia.org/wiki/Ars_Conjectandihttp://en.wikipedia.org/wiki/Abraham_de_Moivrehttp://en.wikipedia.org/wiki/Doctrine_of_Chanceshttp://en.wikipedia.org/wiki/Ian_Hackinghttp://en.wikipedia.org/wiki/James_Franklin_(philosopher)http://en.wikipedia.org/wiki/James_Franklin_(philosopher)http://en.wikipedia.org/wiki/Roger_Coteshttp://en.wikipedia.org/wiki/Thomas_Simpsonhttp://en.wikipedia.org/wiki/Pierre-Simon_Laplacehttp://en.wikipedia.org/wiki/Asymptote


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unmanageable equations. Daniel Bernoulli (1778)introduced the principle of the maximum product of theprobabilities of a system of concurrent errors.

b) Application in life and importance

i) Weather forcasting

Suppose you want to go on a picnic this afternoon, and theweather report says that the chance of rain is 70%? Do youever wonder where that 70% came from?

Forecasts like these can be calculated by the people whowork for the National Weather Service when they look at all
http://en.wikipedia.org/wiki/Daniel_Bernoullihttp://en.wikipedia.org/wiki/Daniel_Bernoulli


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other days in their historical database that have the sameweather characteristics (temperature, pressure, humidity,etc.) and determine that on 70% of similar days in thepast, it rained.

As we've seen, to find basic probability we divide thenumber of favorable outcomes by the total number ofpossible outcomes in our sample space. If we're looking forthe chance it will rain, this will be the number of days inour database that it rained divided by the total number ofsimilar days in our database. If our meteorologist has datafor 100 days with similar weather conditions (the sample

space and therefore the denominator of our fraction), andon 70 of these days it rained (a favorable outcome), theprobability of rain on the next similar day is 70/100 or 70%.

Since a 50% probability means that an event is as likely tooccur as not, 70%, which is greater than 50%, means thatit is more likely to rain than not. But what is the probabilitythat it won'train? Remember that because the favorableoutcomes represent all the possible ways that an event can

occur, the sum of the various probabilities must equal 1 or100%, so 100% - 70% = 30%, and the probability that itwon't rain is 30%.

Batting averages

Let's say your favorite baseball player is batting 300. Whatdoes this mean?

A batting average involves calculating the probability of aplayer's getting a hit. The sample space is the total numberof at-bats a player has had, not including walks. A hit is afavorable outcome. Thus if in 10 at-bats a player gets 3hits, his or her batting average is 3/10 or 30%. For baseball
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stats we multiply all the percentages by 10, so a 30%probability translates to a 300 batting average.This meansthat when a Major Leaguer with a batting average of 300steps up to the plate, he has only a 30% chance of getting

a hit - and since most batters hit below 300, you can seehow hard it is to get a hit in the Major Leagues!

c) Theorical Probabilities and Empirical

Probabilities (differences)


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The term empirical means "based on observation or

experiment." An empirical probability is generally, but

not always, given with a number indicating the possiblepercent error (e.g. 80+/-3%). A theoretical probability,

however, is one that is calculated based on theory, i.e.,

without running any experiments.

Empirical Probability of an event is an "estimate" thatthe event will happen based on how often the event occursafter collecting data or running an experiment (in a largenumber of trials). It is based specifically on directobservations or experiences.

Theoretical Probability of an event is the number ofways that the event can occur, divided by the total numberof outcomes. It is finding the probability of events thatcome from a sample space of known equally likelyoutcomes.


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Comparing Empirical and TheoreticalProbabilities

Sharon and Sandra roll two dice 50 times and record their

results in the accompanying chart.1.) What is their empirical probability of rolling a 7?2.) What is the theoretical probability of rolling a 7?3.) How do the empirical and theoretical probabilitiescompare?

Solution:1.) Empirical probability(experimental probability orobserved probability) is 13/50= 26%.

2.) Theoretical probability (based upon what is possiblewhen working with two dice) = 6/36 = 1/6 = 16.7% (checkout the table at the right of possible sums when rolling two

dice).3.) Sharon and Sandra rolled more 7's than would beexpected theoretically.

Sum Of the rollsof two dice3, 5, 5, 4, 6, 7, 7, 5, 9, 10,

12, 9, 6, 5, 7, 8, 7, 4, 11, 6,8, 8, 10, 6, 7, 4, 4, 5, 7, 9,

9, 7, 8, 11, 6, 5, 4, 7, 7, 4,

3, 6, 7, 7, 7, 8, 6, 7, 8, 9


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a ) Suppose you are palying the Monopoly game with two

of your friends. To start the game, each player will have to

toss the die once. The player who obtains the highest

number will start the game. List all the possible outcomes

when the die is tossed once.

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


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b ) instead of one die, two dice can also be tossedsimultaneousla by each player. The player will move the

token according to the sum of all dots on both turned-up

faces. For example, if the two dice are tossed

simultaneously and 2 appears on one die and 3

appears on other the other, the outcome of the toss is

(2,3). Hence, the player shall move the token 5 places.

Note: the events (2,3) and (3,2) should be treated as twodifferent 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 tree diagram.


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Answer :


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a )Table 1 show the sum of all dots on both turned-upfaces when two dices are tossed simultaneously.

Sum of the dots on

both turned-up faces

(x)

Possible outcomes Probability, P(x)

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),(3,2),(4,1) 4/36

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


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

Table 1(i)

b ) Table of possible outcomes of the following events and

their corresponding probabilities.

Possible outcomes Probability P(X)

A

(the two numbers are

not the same)

( 36-6)

= 3030/36

B

(the product of the two

numbers is greater than

36)

0 0

C

(both numbers are

prime or the

difference between

two numbers is odd)

P = Both number are prime

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

(3,5), (5,3), (5,5)}

Q = Difference of 2 number is

odd

Q = { (1,2), (1,4), (1,6), (2,1),22/36


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(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) }

D

(the sum of the two

numbers are even

and both numbers are

prime)

P = Both number are primeP = {(2,2), (2,3), (2,5), (3,3),

(3,5), (5,3), (5,5)}

R = The sum of two 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)}

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Table 1(ii)


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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 (

)

Frequency

( )

2

2 2 4 8

3 4 12 36

4 4 16 645 9 45 225

6 4 24 144

7 11 77 539

8 4 32 256

9 6 54 486

10 3 30 300

11 1 11 121

12 2 24 288

= 50 = 329 = 2467

Table 2

a)Based on table 2 that you have completed, determine

the value of :


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i) Mean

ii)Varience

iii) Standard deviation of the data

b) Predict the value of the mean if the number of tosses is

increased to 100 times.

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

iii)Standard deviation of the new data

Was your prediction proven ?

Solution :

a ]


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i) Mean =

=

= 6.58

ii) Variance =

= -

= (6.58)2

= 6.044

iii) Standard deviation =

=

= 2.458


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b ]

Sum of the two

numbers ( )

Frequency (

)

2

2 4 8 16

3 5 15 45

4 6 24 96

5 16 80 400

6 12 72 432

7 21 147 1029

8 10 80 640

9 8 72 648

10 9 90 900

11 5 55 605

12 4 48 576

= 100 = 691 = 5387

Prediction of mean = 6.91


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c ]

i. Mean

= 6.91

ii. Variance = -

= 2

= 6.122

iii. Standard deviation =

= 2.474

Prediction is proven.


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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 = x P(x)

Variance = x 2P(x) (mean) 2

a ) Based on Table 1, determine the actual mean, the

variance and the standard deviation or the sum of all dots

on the turned-up faces by using the formulae given.

b ) Compare the mean, variance and standard deviationobtained in Part 4 and Part 5. What can you say about the

values ? explain in your own words your interpretation and

your understanding of the values that you have obtainrd

and relate your answers to the Teoretical and Empirical

Probabilities.

c ) If n is the number of times two dices are tossed

simultaneously, what is the range of mean of the sum of all

dots on the turned-up faces as n changes? Make your

conjecture and support your conjecture.


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Solutions :

a)

Mean = x P(x)

=

= 7

Variance = x2 P(x) (mean) 2

=

- (7)2

= 54.83 49


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= 5.83

Standard deviation =

= 2.415

b)

Part 4 Part 5

n = 50 n = 100

Mean 6.58 6.91 7.00

Variance 6.044 6.122 5.83

Standard

deviation

2.458 2.474 2.415

We can see that, the mean, variance and standard

deviation that we obtained through experiment in part 4are different but close to the theoretical value in part 5.

For mean, when the number of trial increased from n=50

to n=100, its value get closer (from 6.58 to 6.91) to the

theoretical value. This is in accordance to the Law of Large

Number. We will discuss Law of Large Number in next

section.

Nevertheless, the empirical variance and empirical

standard deviation that we obtained i part 4 get further


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from the theoretical value in part 5. This violates the Law

of Large Number. This is probably due to

a. The sample (n=100) is not large enough to see the

change of value of mean, variance and standard

deviation.

b. Law of Large Number is not an absolute law. Violation

of this law is still possible though the probability is

relative low.

In conclusion, the empirical mean, variance and standarddeviation can be different from the theoretical value. When

the number of trial (number of sample) getting bigger, the

empirical value should get closer to the theoretical value.

However, violation of this rule is still possible, especially

when the number of trial (or sample) is not large enough.

C )

The range of the mean

Conjecture: As the number of toss, n, increases, the meanwill get closer to 7. 7 is the theoretical mean.
http://void%280%29/


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