01-Axioms of Probability

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Chap 1 Axioms of Probability

Ghahramani 3rd edition


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p2.

Outline

1.1 I nt roduct ion1.2 Sample space and events

1.3 Axioms of probabilit y

1.4 Basic Theorems

1.5 Cont inuit y of probabil it y funct ion

1.6 Probabili t ies 0 and 11.7 Random select ion of point s f rom int ervals


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p3.

1.1 Introduction

Advent of Probabili t y as a math discipline

1. 4-sided die: Ancient Egypt ; 3500 B.C.

2. 6-sided die: 1600 B.C.

3. Dice: China; 7t h-10th centur ies

4. Playing cards: Much more recent


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p4.

Introduction

St udies of Chances of Event sI ta ly

Luca Paccioli (1445-1514)

Niccolo Tart aglia(1499-1557)

Girolamo Cardano(1501-1576)

Galileo Galielei(1564-1642)

France

Blaise Pascal(1623-1662)Pierre de Fermat(1601-1665)

Chr ist ian Huygens(1629-1695) Dutch

1657 fi rst prob. bookOn Calculat ions in Games of Chance"


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p5.

Introduction

James Bernoull i( 1654-1705)Abraham de Moivre(1667-1754)

Pierre-Simon Laplace(1749-1827)

Sim eon Denis Poisson( 1781-1840)Karl Fr iedr ich Gauss(1777-1855)

RussiaPafnut y Chebyshev(1821-1894)

Andrei Markov(1856-1922)

Aleksandr Lyapunov(1857-1918)


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p6.

Introduction

1900 David Hilbert (1862-1943) pointed out t he

problem of the axiomat ic t reat ment of the t heory

of probabil i t y

Emile Borel(1871-1956)

Serge Bernst ein( 1880-1968)

Richard von Mises(1883-1953)

* 1933 Andrei Kolmogorov( 1903-1087) Russian

successfully ax iomat ized t he t heory of probabili t y


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1.2 Sample space and events

Experiment (eg. Tossing a die)

Outcome(sample point )

Sample space= { al l outcomes}

Event : subset of sample space

Ex1.1 tossing a coin once

sample space S = { H, T}

Ex1.2 fl ipping a coin and t ossing a die if T

or f l ipping a coin again i f H

S= { T1,T2,T3,T4,T5,T6,HT,HH}


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p8.

Sample space and events

Ex1.3 measuring t he l i fet ime of a l ight bulb

S= { x : x 0}

E= { x: x 100} is t he event t hat t he l ightbulb last s at least 100 hours

Ex1.4 all families w it h 1, 2, or 3 children(genders specif ied)

S= { b,g,bg,gb,bb,gg,bbb,bgb,bbg,bgg,

ggg,gbg,ggb,gbb}


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p9.

Sample space and events Event E has occurred in an experiment :

I f t he out come of an experim ent belongs t o E.

Take event s E, F as sets and sample space S t hen

can be defined st raight forw ard.

Can also def ine

if { E1, E2, } is a set of event s

ESEFEFEFEEFFE c ,,,,

iiiiinii

ni EEEE

1111

,,,


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p10.

Sample space and events

Associat ive law s EU(FUG)= (EUF)UG Dist ribut ive law s (EF)UH= (EUH)(FUH)

(EUF)H= (EH)U(FH)

De Morgans 1 st law : (E U F) c = EcFc

De Morgans 2nd law : (EF) c = Ec U Fc

E = ES = E(FUFc) = EF U EFc


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p11.

1.3 Axioms of probabilityDefinit ion(Probabilit y Axioms)

S: sample space

A: event ,

Pr: a funct ion for each event A, i.e. Pr : 2S R

Pr(A) is said to be t he probabilit y of A if

Axiom 1 Pr (A) > = 0

Axiom 2 Pr(S) = 1

Axiom 3 I f { A1, A2, A3, } is a sequence of

mut ually exclusive events t hen

SA

)Pr()Pr(11

i iii

AA


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Basic Theorem

Ex 1.15 I n a comm unit y of 400 adult s, 300 bike orsw im or do bot h, 160 sw im, and 120 sw im and bike.What is t he probabilit y t hat an adult , select ed atrandom from t his comm unit y, bike?

Sol: A: event t hat t he person sw ims

B: event t hat t he 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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Basic Theorem Ex 1.16 A number is chosen at random from t he

set of numbers { 1, 2, 3, , 1000} . What is t heprobabil i t y t hat it is divisible by 3 or 5( I .e. eit her3 or 5 or both)?

Sol: A: event t hat t he out come is divisible by 3

B: event t hat t he out come is divisible by 5

P(AUB)= P(A)+ P(B) -P(AB)

= 333/ 1000+ 200/ 1000-66/ 1000= 467/ 1000


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p15.

Basic Theorem

I nclusion-Exclusion Pr inciple

)...()1(...)(

)()()...(

211

21

nn

kji

jiin

AAAPAAAP

AAPAPAAAP


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1.5 Continuity of probabilityfunction

Recall t he cont inuit y of a funct ion f : R R

fro every convergent seq { xn} in R.

The cont inuit y of probabilit y funct ion is sim ilar.

Def. A seq { En, n> = 1} of event of a sample space

is called increasing i f

it is called decreasing if

)lim()(lim n

n

n

n

xfxf

;1321 nn EEEEE

.1321 nn EEEEE


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Continuity of probability function Thm 1.8(cont inuit y of probabil i t y function)

For any increasing or decreasing sequence of

events, { En, n> = 1} :

l im P(En)= P( l im En)


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1.6 Probabilities 0 and 1 I f E and F are events w it h probabili t ies 1 and 0,

t hen it is not correct t o say t hat E is t he sample

space S and F is the empt y set .

Example: select ing a random point from ( 0,1)

1. A= { 1/ 3} , P( A) = 0

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


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1.7 Random selection of pointsfrom intervals Def. A point is randomly selected from an interval

(a, b) . The probabili t y t he subint erval ( c, d)

contains t he point is defined t o be (d-c) / (b-a).

01-Axioms of Probability

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