Axioms of Probability

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Axioms of Probability

C.M. Liu

Feb. 25, 2007

www.csie.nctu.edu.tw/~cmliu/probability


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Contents

Introduction Sample Space and Events

Axioms of Probability

Basic Theorems. Continuity of Probability Function

Probabilities 0 and 1

Random Selection of Points from Intervals


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Introduction

Ancient Egypians, 3500 B.C. Use a four-sided die-shaped bone.

Hounds and Jackals.

Studies of chances of events in 15th century

Luca Paccioli (1445 1514)

Niccolo Tartaglia (1499 1557)

Girolamo Cardano (1501 1576)

Galileo Galilei (1564 1642)


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Introduction

Real Progress from 1654 Blaise Pascal (1623 1662).

Pierre de Fermat (1601 1665).

Christian Huygens (1629 1695). On Calculations in Games of

Chance Major Breakthrough

James Bernoulli (1654 1705).

Abraham de Moivre (1667 1754).

Pascal

Fermat

Moivre

Bernoulli

Christiaan Huygens


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Introduction

18th century Laplace, Poisson, Gauss expanded the growth of probability

and its application.

19th Century

Advanced the works to put it on firm mathematical grounds.

Pafnuty Chebyshev Andrei Markov.

Aleksandre Lyapunov.

20th Century

David Hilbert (1862 1943)

23 problems whose solutions were crucial to the advancementof mathematics.

Andrei Kolmogorov (1903-1987)

Combined the notion ofsample space, introduced by Richard vonMises, and measure theory and presented his axiom system for

probability theory in 1933.

Laplace

Kolmogorov


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2. Sample Space and Events

Basic definitions:Sample Space, S: The set of all possible

outcomes.

Sample Points: The outcomes.

Event: A subset of the sample space.


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Experiment (eg. Tossing a die)Outcome(sample point)

Sample space={all outcomes}

Event: subset of sample space

Ex1.1 tossing a coin once

sample space S = {H, T}

Ex1.2 flipping a coin and tossing a die if T

or flipping a coin again if H

S={T1,T2,T3,T4,T5,T6,HT,HH


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Ex1.3 measuring the lifetime of a light bulb

S={x: x 0}

E={x: x 100} is the event that the light

bulb lasts at least 100 hours

Ex1.4 all families with 1, 2, or 3 children

(genders specified)

S={b,g,bg,gb,bb,gg,bbb,bgb,bbg,bgg,ggg,gbg,ggb,gbb}


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2. Sample Space and Events--Subset Relationship

Subset EFxExF

Equal E=FEF and FE

Intersection The intersection of E and F, written E F, is the set of

elements that belong to both E and F. EF=EF={x: xE and xF}

Union The union of E and F, written E F, is the set of elements

that belong to either E or F. EF={x: xE or xF}.


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ComplementThe complement of E, written Ec, is the set of all

elements that are not in E.

E

c

={x: xE}.Difference of Two Events.

The set of elements belong to E but not in F.

E-F={x: xE and xF}.

Mutually Exclusive.

The joint occurrence of any two event isimpossible.

EF=


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Venn Diagrams

EFEF

Ec (Ec G)F

E FE F

E

G

E F

Intuitive justification, create counter-examples, and shows invalidity.


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For any three events A, B, and C definedon a sample space S,

Commutativity: EF=FE, EF=FE.

Associativity: E(FG)=(EF)G, E(FG)=(EF)GDistributative Laws: (EF)H =(EH)(FH)

(EF)H=(EH)(FH),

DeMorgans Laws:(EF)c=EcFc,

(EF)c=EcFcIUn

i

c

i

cn

i

i EE00 ==

=

IU

=

=

=

11 i

c

i

c

i

i EE

UIn

i

c

i

cn

i

i EE

00 ==

=

UI

=

=

=

11 i

c

i

c

i

i EE


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Proof of DeMorgans Laws

ccc FEFE )( ccc FEFE )(

cFExLet )(

)( FExThen

FxandExSo ,

)(,cc

FExHence

ccFExLet

cc

FxandExThen FxandExSo ,

FExTherefore ,

cFExThus )(,


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3. Axioms of Probability

Axioms of Probability: LetSbe the sample space of a random phenomenon.

Suppose that to each eventAofS. a number denoted byP(A),is associated withA. IfPsatisfies the following axioms,

then it is called a probability and the numberP(A)is said tobe the probability of A.

P(A) 0 for any event A.

P(S) = 1 where S is the sample space.

If {Ai}, i=1,2,, is a sequence of mutually exclusiveevents (that is, AiAj= for all ij), then

=

=

=1

1

)()(i

iii

APAP U


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Theorem 1.1The probability of the empty set P()=0.


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Theorem 1.2If {Ai}, i=1,2,n, are mutually exclusive (that

is, AiAj= for all ij), then ==

=n

i

ii

n

i

APAP

11

)()(U


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Examples

Flipping a fair or unbiased Coin Events

Sample space, S

Probability on sample space and events.

Probability on unbiased coin.

Probability on biased coin.


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Theorem 1.3Let Sbe the sample space of an experiment. If

Shas Npoints that are all equally likely occur,then for any eventAofS,

where N(A)is the number of points ofA.

ExampleFlipping a fair coin three times and A be the

event at least two heads.

N

ANAP

)()( =


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Proof on Theorem 1.3


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Examples


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Solutions


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4. Basic Theorems

Theorem 1.4

Theorem 1.5

Corollary

thenBAIf ,

)()()()( APBPBAPABP c ==

).(1)(, APAPAeventanyForc =

)()(, BPAPthenBAIf


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Proof


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4. Basic Theorems

Theorem 1.6

Inclusion-Exclusion Principle

Theorem 1.7

).()()()( ABPBPAPBAP +=

).()1(...)(

)()()(

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1

1

1 1

1

1 111

n

nn

i

n

ij

n

jk

kji

n

i

n

ij

ji

n

i

i

n

i

i

AAAPAAAP

AAPAPAP

++

=

=

+= +=

= +===

U

)()()( cABPABPAP +=


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4. Basic Theorems S

A B


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4. Basic Theorems

Ex 1.15 In a community of 400 adults, 300 bike orswim or do both, 160 swim, and 120 swim and bike.What is the probability that an adult, selected atrandom from this community, bike?

Sol: A: event that the person swims

B: event that the person bikes

P(AUB)=300/400, P(A)=160/400,

P(AB)=120/400P(B)=P(AUB)+P(AB)-P(A)

= 300/400+120/400-160/400=260/400= 0.65


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4. Basic Theorems Ex 1.16A number is chosen at random from the

set of numbers {1, 2, 3, , 1000}. What is theprobability that it is divisible by 3 or 5(I.e. either3 or 5 or both)?

Sol: A: event that the outcome is divisible by 3

B: event that the outcome is divisible by 5

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

=333/1000+200/1000-66/1000=467/1000


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4. Basic TheoremsInclusion-Exclusion Principle


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4. Basic Theorems

Inclusion-Exclusion Principle

)...()1(...)(

)()()...(

211

21

nn

kji

jiin

AAAPAAAP

AAPAPAAAP

++

=

UUU


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4. Basic Theorems-- Examples


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4. Basic Theorems-- Examples(c.1)


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4. Basic Theorems

Theorem 1.7 P(A) = P(AB) + P(ABc)

Proof:

)()()()(

)(

cc

c

cc

ABPABPABABPAPexclusivemutuallyareABandABSince

ABABBBAASA

+==

===


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Examples (c.2)


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Continuous Functions Let Rdenote the set of all real numbers.

is called continuous at a point if

It is called continuous on Rif it is continuous at all points.

Sequential Criterion

f(x)is continuous on Rif and only if, for every convergentsequence in R.

5. Continuity of Probability Functions

.: RRf Rc

)()(lim cfxfcx

=

)lim()(lim nn

nn

xfxf

=

Rc

=1}{ nnx


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Increasing Sequence of Events of Sample Space

Decreasing Sequence of Events of Sample Space

For increasing events

For increasing sequence of events, means the event thatat least one Ei occurs

For decreasing sequence of events, means the event thatevery Ei occurs

+121 nn EEEE

+121 nn EEEE

nn

nn EE

= = 1lim

nn

nn

EE

==

1lim

Applicable to the probabilitydensity function ?

nn

E

lim

nn

E

lim


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Theorem 1.8 Continuity of Probability Function

For any increasing or decreasing sequence of events,

.lim)(lim nn

nn

EPEP

=


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Example


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6. Probabilities 0 and 1

Not correct speculation IfEand F are events with probabilities 1 and 0, respectively, it

is not correct to say that E is the sample space and F is theempty space.

Ex. P(1/3) in (0, 1).


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7. Random Selection of Pointsfrom Intervals

Randomly selected from an IntervalA point is said to be randomly selected from an interval (a, b)

if any two subintervals of (a, b) that have the same length areequally likely to include the point. The probability associatedwith the event that the subinterval (, ) contains the point isdefined to be (-)/(b-a).


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dealine

Section 1.1-1.2 (page 10): 11, 12, 14, 16, 19.

Section 1.4 (page 23): 14, 22, 28, 31

Section 1.7 (page 34): 3, 10.

Review (page 36): 10, 12, 14.

Axioms of Probability

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