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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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2. Sample Space and Events
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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2. Sample Space and Events
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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2. Sample Space and Events--Subset Relationship
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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2. Sample Space and Events--Subset Relationship
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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2. Sample Space and Events--Subset Relationship
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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3. Axioms of Probability
Theorem 1.1The probability of the empty set P()=0.
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3. Axioms of Probability
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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3. Axioms of Probability
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(...)(
)()()(
21
12
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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5. Continuity of Probability Functions
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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5. Continuity of Probability Functions
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.
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