Introduc)ontoProbability

AkshatShankar,MansiGoel

There is a 99.99% probability that we’ll do…

…in today’s class !

Outlineo  Introduc)ono  SampleSpaceandEventso  ProbabilityDefinedonEventso  Condi)onalProbabili)eso  Independenceo  Bayes’Formula

Introduc)ono  Probabilityisameasureofhowlikelyitisforan

eventtohappeno  Wenameaprobabilitywithanumberfrom0to1o  Ifaneventiscertaintohappen,thentheprobability

oftheeventis1o  Ifaneventiscertainnottohappen,thenthe

probabilityoftheeventis0o  Ifitisuncertainwhetherornotaneventwill

happen,thenitsprobabilityissomefrac)onbetween0and1

Outlineo  Introduc)ono  SampleSpaceandEventso  ProbabilityDefinedonEventso  Condi)onalProbabili)eso  Independenceo  Bayes’Formula

SampleSpaceandEvents

{HH}

(HT}

{TH}

{TT}

HH

HT

TH

TT

{}

{HH,HT,TH,TT}

o  Experiment:Repeatableprocedurewithwell-definedpossibleoutcomesn  Tossacointwice

o  Samplespace:possibleoutcomesofanexperimentn  S={HH,HT,TH,TT}

o  Event:asubsetofpossibleoutcomesn  A={HT,TH}

{HH,HT,TH}

(HH,HT,TT}

{HH,TH,TT}

{HT,TH,TT}

{HH,HT}

{HH,TH}

(HH,TT}

{HT,TH}

{HT,TT}

{TH,TT}

SampleSpace Events

Outlineo  Introduc)ono  SampleSpaceandEventso  ProbabilityDefinedonEventso  Condi)onalProbabili)eso  Independenceo  Bayes’Formula

Defini)onofProbabilityo  Probabilityofanevent:anumberassignedtoaneventPr(A)

n  Axiom1:Pr(A)≥0n  Axiom2:Pr(S)=1n  Axiom3:Foreverysequenceofdisjointevents Pr( ) Pr( )i iii

A A=∑U

Event

SampleSpace

1

0

Probability

Outlineo  Introduc)ono  SampleSpaceandEventso  ProbabilityDefinedonEventso  Condi)onalProbabili)eso  Independenceo  Bayes’Formula

o  Considertheprobabilitythatonewouldn’tbeabletoreachtoofficeo  Giventhatthereisasnowstormtodayo  Nowconsidertheprobabilitythatonewouldn’tbeabletoreachtoofficeo  Ofcoursethesecondprobabilityincreasesgiventhenewinforma)on

Condi)oning

o  Ais“it’srainingnow”.o  P(A)indryCaliforniais.01o  Bis“itwasrainingtenminutesago”o  P(A|B)means“whatistheprobabilityofitrainingnowifitwasraining10minutes

ago”o  P(A|B)isprobablywayhigherthanP(A)o  PerhapsP(A|B)is.10o  Intui)on:TheknowledgeaboutBshouldchangeoures)mateoftheprobabilityofA

Condi)oningo  Example:Thereisaboxwhichcontains4darkchocolatesofLindt,2milkchocolates

ofLindt,1darkchocolateofAmuland5milkchocolatesofAmul.o  SupposeIcloseyoureyesandthenyoupickachocolatefromthebox.o  Whatistheprobabilitythatadarkchocolatewouldbepicked?o  SupposeItellyouthatthechocolateyouhavepickedisAmul,whatisthe

probabilitythatyouhavepickedadarkchocolate?

Lindt Amul

Dark 4 1

Milk 2 5

Probabilitythatdarkchocolateispicked=5/11

Probabilitythatdarkchocolateispicked=1/6

GiventhatAmulchocolatehasbeenpicked,thesamplespacechanges.

Sonowonly6Amulchocolates(andnot11)havetobeconsidered.

Andfromthese6,thereisonlyonewhichisdark.

Conditioning o  If A and G are events with Pr(G) > 0, the conditional

probability of A given G is

)Pr()Pr()|Pr(

GGAGA ∩

=

GA∩

GG

AA

Probability is the ratio of two circles

Additionally it is given that G has occurred.

Conditional Probability is the proportion of A in G.

Occurrence of G increases the chance of A..

GA∩

Outlineo  Introduc)ono  SampleSpaceandEventso  ProbabilityDefinedonEventso  Condi)onalProbabili)eso  Independenceo  Bayes’Formula

JointProbabilityo  ForeventsAandB,jointprobability

standsfortheprobabilitythatbotheventshappen.

o  Example:A={HH,TT},B={HH,HT,TH},whatisthejointprobability?

o  A=boththecoinsshowthesamefaceo  B=Atleastoneheadsisthere.o  {HH}

)Pr( BA∩

)Pr( BA∩

=∩ BA 4/1)Pr( =∩BA

Independenceo  TwoeventsAandBareindependentincase

o  OccurrenceofBdoesn’tchangetheprobabilityofA

o  Asetofevents{Ai}isindependentincase

Pr( ) Pr( )i iiiA A=∏I

)Pr()Pr()Pr( BABA =∩

)Pr()Pr()Pr().Pr(

)Pr()Pr()|Pr( A

BBA

BBABA ==

∩=

Outlineo  Introduc)ono  SampleSpaceandEventso  ProbabilityDefinedonEventso  Condi)onalProbabili)eso  Independenceo  Bayes’Formula

ThefollowingtextisOp)onal,asalltheproblemsofProbabilitycanbedoneusingthe

conceptsexplainedpreviously.

Bayes’Rule…o Bayes’LawisnamedforThomasBayes,aneighteenthcenturymathema)cian.o Initsmostbasicform,ifweknowP(B|A),o WecanapplyBayes’LawtodetermineP(A|B)

P(B|A)P(A|B)

o  GiventwoeventsAandBandsupposethatPr(A)>0.Theno  Example:

Bayes’Rule

Pr(W|R) R ¬RW 0.7 0.4¬W 0.3 0.6

R:Itisarainyday

W:Thegrassiswet

Pr(R|W)=?

Pr(R)=0.8

)Pr()Pr()|Pr(

)Pr()Pr()|Pr(

ABBA

ABAAB =

∩=

Bayes’Rule

Pr(W|R) R ¬R

W 0.7 0.4

Thegrassiswetgiventhatitrains

R W

Informa=on

Pr(W|R)

Inference

Pr(R|W)

Thegrassiswetgiventhatitdoesn’train

Bayes’Rule

)Pr()Pr()|Pr()|Pr(

WRRWWR =

Pr(W)=Probabilitythatthegrassiswet

=Probabilitythatthegrassiswetgivenitrained+Probabilitythatthe

grassiswetgiventhatitdidn’train

=Pr(W|R)*Pr(R)+Pr(W|¬R)*Pr(¬R)

=0.7*0.8+0.4*0.2

=0.56+0.08=0.64

875.064.08.07.0)|Pr( =

×=WR