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Axioms of Probability C.M. Liu Feb. 25, 2007 www.csie.nctu.edu.tw/~cmliu/probability 1 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. 6 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 11 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 12 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 13 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. 15 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 16 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. 18 Proof on Theorem 1.3 19 Examples 20 Solutions 21 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) 22 Proof 23 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 25 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 26 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 27 4. Basic Theorems Inclusion-Exclusion Principle 28 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 30 4. Basic Theorems-- Examples (c.1) 31 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) 33 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→ ∞ } 36 Example 37 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). 39 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. 40

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