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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




                      2
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)




                             3
                                                   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.
                                            5
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



                                 7
   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}




                              8
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}.


                            9
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
                                 14
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.




                            17
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
 event at least two heads.

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Proof on Theorem 1.3




            19
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                                        )

                                                   24
                        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 )




                                  29
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 )




                                      32
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→∞

                                    34
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

                                   35
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).




                              38
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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