introduc)on to probability · given that amul chocolate has been picked, the sample space changes....
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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
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