1. Axioms of probability

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1 1. Axioms of probability 1.1 Introduction Dice were believed to be invented in China during 7~10th AD. Luca Paccioli( ) Italian (studies of chances of events) Niccolo Tartaglia( ) Girolamo Cardano( ) Galileo Galielei( ) Blaise Pascal( ) French Pierre de Fermat( ) Christian Huygens( ) Dutch first book On Calculations in Games of Chance)(1655) p2. 1

2 Introduction James Bernoulli( ) Abraham de Moivre( ) Pierre-Simon Laplace( ) Simeon Denis Poisson( ) Karl Friedrich Gauss( ) Pafnuty Chebyshev( ) Russian Andrei Markov( ) Aleksandr Lyapunov( ) p3. Introduction 1900 David Hilbert( ) pointed out the problem of the axiomatic treatment of the theory of probability Emile Borel( ) Serge Bernstein( ) Richard von Mises( ) *1933 Andrei Kolmogorov( ) Russian successfully axiomatized the theory of probability p4. 2

3 1.2 Sample space and events Experiment (eg. Tossing a die) Outcome(sample point; element) Sample space={all outcomes} Event: subset of sample space Ex 1.1 tossing a coin once sample space S = {H, T} Ex 1.2 flipping a coin and tossing a die if T(ail) or flipping a coin again if H(ead) S={T1,T2,T3,T4,T5,T6,HT,HH} p5. Sample space and events Ex 1.3 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 Ex 1.4 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} p6. 3

4 Sample space and events Event E has occurred in an experiment: If the outcome of an experiment belongs to E. Take events E, F as sets and sample space S then E F, EF = E I F, E U F, E F, E c = S E can be defined straightforward. Can also define U n n i Ei i E i i E = 1, I = 1, U = 1 i, Ii = 1Ei if {E 1, E 2, } is a set of events p7. Sample space and events Commutative laws EUF=FUE, EF=FE Associative laws EU(FUG)=(EUF)UG Distributive laws (EF)UH=(EUH)(FUH) (EUF)H=(EH)U(FH) De Morgan s 1 st law: (E U F) c = E c F c De Morgan s 2 nd law: (EF) c = E c U F c Definition E and F are mutually exclusive if and only if EF = φ E = ES = E(FUF c ) = EF U EF c p8. 4

Venn Diagram p9. 1.3 Axioms of probability Definition (Probability Axioms) S: sample space A: event, A S Pr: a function for each event A, i.e. Pr: 2 S R A) is said to

5 Venn Diagram p Axioms of probability Definition (Probability Axioms) S: sample space A: event, A S Pr: a function for each event A, i.e. Pr: 2 S R A) is said to be the probability of A if Axiom 1 A) >= 0 Axiom 2 S) = 1 Axiom 3 If {A 1, A 2, A 3, } is a sequence of mutually exclusive events, then 1 i i i=1 U = A ) = A ) i p10. 5

6 Classical definition of Probability Theorem 1.3 Let S be a sample space of an experiment. If S has N(S) sample points that are equally likely to occur, then for any event of A of S, A)=N(A)/N(S) where N(A) is the number of sample points of A. Ex 1.11 Let S be the sample space of flipping a coin three times and A be the event of at least two heads; then Ans: N(S)=8, N(A)=4, A)=N(A)/N(S)=1/2 p11. Classical definition of Probability Ex Two fair dice are thrown, what is the probability that the sum of these two dice is equal to 6? Ans: S={(1,1), (1,2),, (6,6)} contains 36 equally likely sample points. Therefore, N(S)=36. A={(1,5), (2,4),, (5,1)}. Therefore N(A)=5. A)=N(A)/N(S)=5/36 Ex 1.13 A number is selected randomly from the set of integers {1,2,..., 1000}. What is the probability that the number is divisible by 3? Ans: N(S)=1000, N(A)=333, A)=333/1000 p12. 6

7 1.4 Basic Theorem Theorem 1.4 A c ) = 1 A) Theorem 1.5 If A B, then B-A)=BA c )=B)-A) Corollary If A B, then A) <= B) p Basic Theorem Theorem 1.6 AUB) = A)+B)-AB) p14. 7

8 1.4 Basic Theorem Theorem 1.7 A) = AB)+AB c ) Proof: It is clear that A = AS = AB U AB c In addition, AB and AB c mutually exclusive. Therefore, A) = AB)+AB c ). S S S A B AB C AB p Basic Theorem (Cont d) 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 AUB)=300/400, A)=160/400, AB)=120/400 B)=AUB)+AB)-A) = 300/ / /400=260/400= 0.65 p16. 8

9 Basic Theorem 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 AUB)=A)+B)-AB) =333/ / /1000 =467/1000 p17. Basic Theorem Inclusion-Exclusion Principle A1 U A2 U... U An ) = + Ai ) Ai Aj ) A )... ( 1) n 1 i Aj Ak + A1 A2... An ) p18. 9

10 1.5 Continuity of probability function (Omission) Recall the continuity of a function f: R R lim f ( xn ) = f ( lim xn) n n fro every convergent seq {x n } in R. The continuity of probability function is similar. Def. A seq {E n, n>=1} of event of a sample space is called increasing if E1 E2 E3 L E n E n + 1 L; it is called decreasing if E1 E2 E3 L E n E n + 1 L. p Probabilities 0 and 1 If E and F are events with probabilities 1 and 0, then it is not correct to say that E is the sample space S and F is the empty set. Example: selecting a random point from (0,1) 1. A={1/3}, A)=0 2. B=(0,1)-A, B)=1 p20. 10

11 1.7 Random selection of points from intervals Def. A point is randomly selected from an interval (a, b). The probability the subinterval (c, d) contains the point is defined to be (d-c)/(b-a). Ex: Choose a number randomly from an interval of (1.5,5.8), what is probability that this number is locaded between (2.5,4.3)? Sol: P=( )/( )=1.8/4.3=18/43 Ex: Choose two numbers randomly from an interval of (1.2,5.2) and denote them by X and Y, respectively. what is probability that X+Y<4 Sol: Draw a rectangular graph, then. p21. 11

Chapter 1 Axioms of Probability. Wen-Guey Tzeng Computer Science Department National Chiao University

Chapter 1 Axioms of Probability Wen-Guey Tzeng Computer Science Department National Chiao University Introduction Luca Paccioli(1445-1514), Studies of chances of events Niccolo Tartaglia(1499-1557) Girolamo