Chap 1 Axioms of probability Ghahramani 3rd edition

p2. Outline 1.1 Introduction 1.2 Sample space and events 1.3 Axioms of probability 1.4 Basic Theorems 1.5 Continuity of probability function 1.6 Probabilities 0 and Random selection of points from intervals

p Introduction Advent of Probability as a math discipline 1. Ancient Egypt 4-sided die 3500 B.C. 6-sided die 1600 B.C. 2. China playing card 7th-10th centuries

p4. Introduction Studies of Chances of Events Italy Luca Paccioli( ) Niccolo Tartaglia( ) Girolamo Cardano( ) Galileo Galielei( ) France Blaise Pascal( ) Pierre de Fermat( ) Christian Huygens( ) Dutch 1657 first prob. book “ On Calculations in Games of Chance"

p5. Introduction James Bernoulli( ) Abraham de Moivre( ) Pierre-Simon Laplace( ) Simeon Denis Poisson( ) Karl Friedrich Gauss( ) Russia Pafnuty Chebyshev( ) Andrei Markov( ) Aleksandr Lyapunov( )

p6. 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

p 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}

p8. 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}

p9. 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 can be defined straightforward. Can also define if {E 1, E 2, … } is a set of events

p10. Sample space and events 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 E = ES = E(FUF c ) = EF U EF c

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

p Basic Theorem Theorem 1.4 P(A c ) = 1 – P(A) Theorem 1.5 If A B, then P(B-A)=P(BA c )=P(B)-P(A) Corollary If A B, then P(A) <= P(B) Theorem 1.6 P(AUB) = P(A)+P(B)-P(AB)

p13. Basic Theorem 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=260/400= 0.65

p14. 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 P(AUB)=P(A)+P(B)-P(AB) =333/ / /1000 =467/1000

p15. Basic Theorem Inclusion-Exclusion Principle

p Continuity of probability function Recall the continuity of a function f: R  R 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 it is called decreasing if

p17. Continuity of probability function Thm 1.8(continuity of probability function) For any increasing or decreasing sequence of events, {E n, n>=1}: lim P(E n )=P(lim E n )

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}, P(A)=0 2. B=(0,1)-A, P(B)=1

p 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).