week1_Probability12

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MTE 3105: STATISTIK

SEMESTER

JAN-JUN 2012


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COURSE INTRODUCTION

SINOPSIS

HASIL PEMBELAJARAN

KANDUNGAN KURSUS PENILAIAN


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SINOPSIS

Mengimbas kembali konsep yangberkaitan dengan kebarangkalian

Menerokai statistik inferens

ujian-t, ujian khi kuasa dua, analisis varians(ANOVA) dalam pengujian hipotesis dan

regresi linear dalam menganalisis

perhubungan linear dalam dua

pembolehubah.


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SINOPSIS

Penggunaan teori pensampelan dan

penganggaran untuk menganggar min

populasi.

Kepentingan menggunakan kaedah

statistik yang sesuai dalam penyelesaian

masalah harian adalah dititikberatkan.


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Hasil Pembelajaran

Menerangkan aspek teori dan empirikal yangmendasari kebarangkalian

Mengaplikasikan teori persampelan dan

anggaran dalam menganggarkan min populasi Mengaplikasikan statistik inferens seperti ujian

khi kuasa dua, ANOVA dan regresi linear dalamujian hipotesis.

Menganalisis dan menyelesaikan masalahharian menggunakan kaedah statistik.


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THE TOPICS

Probability (3)

Sampling and estimation theory (9)

Hypothesis testing (12) Inferential statistics

The chi-square test (9)

The analysis of variance [ANOVA] (6) Linear regression (6)


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ASSESSMENT

50% COURSEWORK

50% EXAMINATION


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PROBABILITY

1. Introduction to

probability

Theoretical

Probability

Empiricalprobability

2. Compound eventsIndependent events

Mutually exclusiveevents

3. Addition Rule &

Multiplication Rule

4. Probability tree

diagrams

5. Conditional

Probability


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Experiment & outcomes

Event

any collection of outcomes froman experiment eg tossing a coin

Sample space

Set of all possible outcomes

IMPORTANT DEFINITIONS


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Estimating probability

Conduct (or observe) an experiment alarge number of times, and count thenumber of times event A actuallyoccurs, then an estimate of P(A) is

EXPTAL PROBABILITY

P(A) = Number of times A occurredTotal number of trials


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NOTATION: SOME REVISION

P :denotes a probability

A, B, : denote specific events

P(A) : denotes the probabilityof event A occurring


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The Classical approach(equally likely outcomes)

If a trial has S different possible outcomes,each with an equal chance of occurring, thenthe probability that event A occurs is

P(A)= number of ways A can occurtotal number of outcomesn(A)

n(S) =

THEORETICAL PROBABILITY


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EXPERIMENTAL VS THEORETICAL

Experimental probability uses therelative frequency approach and it isan approximation or estimation

Theoretical probability uses theclassical approach it is the actualexpected probability if the experimentis repeated over a large number oftimes.


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LAW OF LARGE NUMBERS

As a procedure is repeated again and

again, the relative frequency probability

(experimental probability) of an event

tends to approach the actual (theoretical)

probability.


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PROBABILITY LIMITS

The probability of an impossibleevent is 0.

The probability of a certain eventis 1.

0 P(A) 1

Impossibleto occur

Certainto occur


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PROBABILITY VALUES

Certain

Likely

50-50 Chance

Unlikely

Impossible

1

0.5

0


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The complement of event A,denoted by A, consists of all

outcomes in which event A doesnot occur.

COMPLEMENTARY EVENTS


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THE COMPLEMENT OF EVENT A

Total Area = 1

P (A)

P (A) = 1 - P (A)


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Any event combining two ormore simple events

COMPOUND EVENTS


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When finding the probability thatevent A occurs or event B occurs,

find the total number of ways A

can occur and the number of

ways B can occur, but ensure

that no outcome is counted morethan once.

PROB OF COMPOUND EVENTS


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P(A or B) = P(A) + P(B) - P(A and B)

where P(A and B) denotes the

probability that A and B bothoccur at the same time.

THE ADDITION RULE


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Kebarangkalian peristiwasaling eksklusif

P(A B) =P(A) +P(B) - P(A B)

Jika A dan B saling eksklusif,

makaP(A B) = P(A) + P (B)


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MUTUALLY EXCLUSIVE EVENTS

Events A and B are mutually

exclusiveif they cannot occur

simultaneously.


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Sets Presentations


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Sets Presentations


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NON-MUTUALLY EXCLUSIVE

Total Area = 1

P(A) P(B)

P(A and B)

Overlapping Events


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APPLYING THE ADDITION RULE

P(A or B)Addition Rule

Are

A and B

mutually

exclusive

?

P(A or B) = P(A)+ P(B) - P(A and B)

P(A or B) = P(A) + P(B)Yes

No


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Find the probability of randomly

selecting a man or a boy.

Contingency Table

Men Women Boys Girls Totals

Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 56 2223


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The events of randomly selecting a

man or a boy are mutually exclusive.

Contingency Table


Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 56 2223


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Find the probability of randomly

selecting a man or someone whosurvived.

Contingency Table


Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 56 2223


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The events of randomly selecting a

man or someone who survived arenot mutually exclusive.

Contingency Table


Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 56 2223


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P(man or survivor) = 1692 + 706 - 332 = 1756

2223 2223 2223 2223

Contingency Table


Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 56 2223

= 0.929


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The General Addition Rule

P ( A B) = P(A) + P(B) P(A B)

Non-mutual exclusive

The Special Addition Rule

P ( A B) = P(A ) + P(B )

Mutual exclusive


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Kebarangkalian peristiwa takbersandar

Kebarangkalian bagi peristiwa A dan Bberlaku ialah

dengan syarat A dan B tak bersandar

P(A B) =P(A) x P(B)

I d d d


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Independent andConditional Events

The conditional probability of anevent is the probability that the

event occurs under the assumptionthat another event has occurred.

Kebarangkalian bhw peristiwa B

berlaku diberi bahawa A telahberlaku ialah


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Conditional Probability

P( B l A) = The probability of B

given A= Kebarangkalian

peristiwa B berlaku diberi bhwperistiwa A telah berlaku


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Example

A fair die is thrown, if it is known that the

number obtained is an even number, what

is the probability that the number is 4?

d l P b b l /


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Conditional Probability/Kebarangkalian Bersyarat

)(

)()1(

)(

)()1(

AP

ABpABP

BP

BApBAP


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Let A = a 4 obtained

B = an even number is obtained

P ( A given B has occurred)


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so,)(

)()1(

)(

)()1(

APABpABP

BP

BApBAP

)()1()()1( APABPBPBAP


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If A and B are mutually exclusiveevents, then

and = 0, then

= 0

)( ABp BAP

(

)()1()()1( APABPBPBAP


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APPLYING THE MULTIPLICATION RULE


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APPLYING THE MULTIPLICATION RULE

P(A or B)Multiplication Rule

Are

A and B

independent

?

P(A and B) = P(A) P(B A)

P(A and B) = P(A) P(B)Yes

No


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FORMAL MULTIPLICATION RULE

P(A and B) = P(A) P(B|A)

If A and B are independentevents, P(B|A) is really the

same as P(B)


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INDEPENDENT EVENTS

P (A and B) = P (A B)

= P (event A occurs in a first trial andevent B occurs in a second trial)


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COMPUTING PROBABILITY

A student answers two questions, of which

the first is a true (T) or false (F) item

followed by a multiple choice item with 5

choices (A, B, C, D, and E)

Find the probability that the students

answers both questions correctly


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THE MULTIPLICATION RULE

TATB

TC

TD

TEFA

FB

FC

FD

FE

AB

C

D

E

A

B

C

D

E

T

F

P(T) = P(C) = P(T and C) =12

1

5

1

10


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PROBABILITY OF AT LEAST ONE

At least one is equivalent toone or more.

The complementof getting atleast one item of a particular type

is that you get no items of thattype.



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The complementof getting at leastone item of a particular type is thatyou get no items of that type

If P(A) = P(getting at least one),

then P(A) = 1 - P(A) where

P(A) is P (getting none)


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Find the probablility of a couplehave at least 1 girl among 3children.

If P(A) = P(getting at least 1 girl), then

P(A) = 1 - P(A)

where P(A) is P(getting no girls)


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If P(A) = P(getting at least 1 girl), then

P(A) = 1 - P(A)

where P(A) is P(getting no girls)

P(A) = (0.5)(0.5)(0.5) = 0.125

P(A) = 1 - 0.125 = 0.875


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Q1BA10

1(a) An experiment consists of threerepetitions of a Bernoulli trial with theprobability of success equal to 0.25 and

the probability of failure equal to 0.75.(i) Draw a tree diagram to illustrate

the situation.

(ii) What is the probability that nofailure is obtained in the three trials.

(iii) What is the probability that exactly


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(iii) What is the probability that exactlyone failure is obtained in the threetrials?

(b) There are 78 tennis players in a

competition. Table 1 gives detailinformation about them

56 school-level

players

29 male 7 left-handed

27 female 3 left-handed

22 state-level

players

12 male 6 left-handed

10 female 2 left-handed


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