# 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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Fermat
Introduction
Real Progress from 1654                Pascal
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 –
Moivre
1754).

Bernoulli

Christiaan Huygens   4
Laplace

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.                                                      Kolmogorov
Aleksandre Lyapunov.
20th Century
David Hilbert (1862 – 1943)
23 problems whose solutions were crucial to the advancement
of mathematics.
Andrei Kolmogorov (1903-1987)
• Combined the notion of sample space, introduced by Richard von
Mises, and measure theory and presented his axiom system for
probability theory in 1933.
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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
E⊂F⇔x∈E⇒x∈F
Equal
E=F⇔E⊂F and F⊂E
Intersection
The intersection of E and F, written E ∩ F, is the set of
elements that belong to both E and F.
E∩F=EF={x: x∈E and x∈F}
Union
The union of E and F, written E ∪ F, is the set of elements
that belong to either E or F.
E∪F={x: x∈E or x∈F}.

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2. Sample Space and Events--
Subset Relationship
Complement
The complement of E, written Ec, is the set of all
elements that are not in E.
Ec={x: x∉E}.
Difference of Two Events.
The set of elements belong to E but not in F.
E-F={x: x∈E and x∉F}.
Mutually Exclusive.
The joint occurrence of any two event is
impossible.
E∩F= φ                 10
2. Sample Space and Events--
Subset Relationship
Venn Diagrams       Intuitive justification, create counter-
examples, and shows invalidity.

∩
E∩F                          ∪
E∪F
E           F             E                  F

Ec                        (Ec G)∪F
∪

E         F
E
G
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2. Sample Space and Events--
Subset Relationship
For any three events A, B, and C defined
on a sample space S,
Commutativity: E∪F=F∪E, EF=FE.
Associativity: E∪(F∪G)=(E∪F)∪G, E(FG)=(EF)G
Distributative Laws: (EF)∪H =(E∪H)(F∪H)
(E∪F)H=(EH)∪(FH),
DeMorgan’s Laws:           c
c=EcFc,     n       n
∞ 
c  ∞
(E∪F)              U Ei  = I Ei
      
c
 U Ei  = I Eic
      
 i =0  i =0     i =1    i =1
(EF)c=Ec∪Fc                c                c
 n       n
∞         ∞
 I Ei  = U Ei
      
c
 I Ei  = U Eic
      
 i =0       i =0    i =1      i =1

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Proof of DeMorgan’s Laws
( E ∪ F )c ⊆ E c F c         ( E ∪ F )c ⊇ E c F c

Let x ∈ ( E ∪ F )   c           Let x ∈ E c F c

Then    x ∉ (E ∪ F )      Then      x ∈ E c and x ∈ F c
So, x ∉ E and x ∉ F            So, x ∉ E and x ∉ F

Hence, x ∈ ( E c F c )        Therefore, x ∉ E ∪ F

Thus, x ∈ ( E ∪ F ) c

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3. Axioms of Probability
Axioms of Probability:
Let S be the sample space of a random phenomenon.
Suppose that to each event A of S. a number denoted by
P(A), is associated with A. If P satisfies the following axioms,
then it is called a probability and the number P(A) is said to
be 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 exclusive
events (that is, AiAj=φ for all i≠j), then
∞         ∞
P ( U Ai ) = ∑ P ( Ai )
i =1       i =1
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3. Axioms of Probability
Theorem 1.1
The probability of the empty set P(φ)=0.

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3. Axioms of Probability
Theorem 1.2
If {Ai}, i=1,2,…n, are mutually exclusive (that
n
is, Ai∩Aj=φ for all i≠j), then P(U A ) = P( A )
n

i =1
i   ∑
i =1
i

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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 1.3
Let S be the sample space of an experiment. If
S has N points that are all equally likely occur,
then for any event A of S,
N ( A)
P ( A) =
N
where N(A) is the number of points of A.
Example
Flipping a fair coin three times and A be the

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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
For any event A, P ( Ac ) = 1 − P ( A).
Theorem 1.5
If A ⊆ B , then
P ( B − A) = P ( BAc ) = P ( B ) − P ( A)
Corollary
If A ⊆ B, then P( A) ≤ P( B)

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Proof

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4. Basic Theorems
Theorem 1.6
P ( A ∪ B ) = P ( A ) + P ( B ) − P ( AB ).
Inclusion-Exclusion Principle
n                      n                  n −1      n
P (U Ai ) =
i =1
∑
i =1
P ( Ai ) −   ∑ ∑
i =1    j = i +1
P ( Ai A j )

n−2      n −1       n
+   ∑ ∑ ∑
i =1
P ( A i A j A k ) − ... + ( − 1 ) n − 1 P ( A1 A 2 ⋅ ⋅ ⋅ A n ).
j = i + 1k = j + 1

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

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A   B

4. Basic Theorems   S

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4. Basic Theorems
Ex 1.15 In a community of 400 adults, 300 bike or
swim or do both, 160 swim, and 120 swim and bike.
What is the probability that an adult, selected at
random 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)=120/400
P(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 1.16 A number is chosen at random from the
set of numbers {1, 2, 3, …, 1000}. What is the
probability that it is divisible by 3 or 5(I.e. either
3 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 Theorems
Inclusion-Exclusion Principle

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4. Basic Theorems
Inclusion-Exclusion Principle

P ( A1 U A2 U ... U An ) =      ∑ P( Ai ) − ∑ P( Ai A j )
+   ∑   P ( Ai A j Ak ) − ... + ( −1)n −1 P ( A1 A2 ... An )

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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:

A = AS = A( B ∪ B c ) = AB ∪ AB c
Since AB and AB c are mutually exclusive
P( A) = P( AB ∪ AB c ) = P( AB ) + P( AB c )

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Examples (c.2)

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5. Continuity of Probability Functions
Continuous Functions
Let R denote the set of all real numbers.
f : R → R. is called continuous at a point c ∈ R if
lim f ( x ) = f (c )
x→c

It is called continuous on R if it is continuous at all points.
c∈R
Sequential Criterion
f(x) is continuous on R if and only if, for every convergent
sequence { xn }∞=1 in R.
n

lim f ( xn ) = f ( lim xn )
n→∞                 n→∞

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5. Continuity of Probability Functions
Increasing Sequence of Events of Sample Space
E1 ⊆ E 2 ⊆ ⋅ ⋅ ⋅ ⊆ E n ⊆ E n +1 ⋅ ⋅ ⋅
Decreasing Sequence of Events of Sample Space

E1 ⊇ E 2 ⊇ ⋅ ⋅ ⋅ ⊇ E n ⊇ E n +1 ⋅ ⋅ ⋅     Applicable to the probability
density function ?
For increasing events
For increasing sequence of events, lim E n
n→∞
means the event that
at least one Ei occurs            ∞
lim E n = ∪ E n
n→∞           n =1

For decreasing sequence of events, lim E n      means the event that
n→∞
every Ei occurs             ∞
lim E n = ∩ E n
n→∞         n =1

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5. Continuity of Probability Functions
Theorem 1.8 Continuity of Probability Function
For any increasing or decreasing sequence of events,

n→ ∞
{
lim P ( E n ) = P lim E n .
n→ ∞
}

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Example

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6. Probabilities 0 and 1
Not correct speculation
If E and F are events with probabilities 1 and 0, respectively, it
is not correct to say that E is the sample space and F is the
empty space.

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

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7. Random Selection of Points
from Intervals
Randomly selected from an Interval
A point is said to be randomly selected from an interval (a, b)
if any two subintervals of (a, b) that have the same length are
equally likely to include the point. The probability associated
with the event that the subinterval (α, β) contains the point is
defined to be (β-α)/(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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