e fruit E Events E re E PREE i PREE The complement of event 1 - PDF document

Conditional probability July 20g 2020 The Monty Hall 1 problem 2 Conditional probability 3 Bayes Rude Probability Rule 4 Total time Last Sf Sample space Sample Point Outcome Wj w we was 1 1 g n WA of Prc wid f l wn n E Pr 1 wi is I PREE e fruit E Events E re E PREE i PREE The complement of event

1 The Monty Hall Problem I z z three doors There are a as is behind one of A car the doors two doors Goats behind the other but does not open it a door 1 The contestant picks one of the 2 Then Hall's assistant opens other two doors revealing a goat is then given the 3 The contestant option of or switching to sticking with their current choice the other unopened door 4 He if andonly if their the car she win chosen door is the correct one contestant have a better Question Does the chance of winning he she switches door if is equally likely to Prize Assume the behind any of the doors three be

sample space I 2 3 Pr L g c's a cg g wz r.efeggggeggg.ge g g c ay W LO G assume happens we w I 2 3 1 142,3 e g g Prez sprE5f PRET E i winning by switching A a 3 2 I I 2 3 Prez B Pr 2 Pr pV ED I 3 D O 0 Pr 5 t 15 23 23 13 Pr EDs Pru 0 0 l J

33 0 Pr 0 I 13 1 13 D prez 13 Pr i 3 is x EZE not switching by winning Stick to 2 3 I 2 3 I Pr Pr Pr 23 PREZ l D 3 3 Ez Iz Iz e Ot D ED Pr b Xl Ig P8 B l 2 2Jstgx0 pr O pr 0 SO 3 3 13 a better chance of winning if have we switching strategy use we

2 Conditional probability coin flips First flip is heads Examples Two Probability of two heads g uniform probability space AH HT TH TT R A f H Hy AT first flip heads is EVA r New sample space TH Htt two heads f HH TT HT Event B The probability of two heads if the first flip is heads B given A The Probability of is PREBIA 42 Two coin flips At least one of the flips is Example heads of two heads probability Event A Ef HI HH Tht Event Bsf htt given A The Probability of B Pr BIA Ig

non uniform enample A r Prew Red 340 Bene q 4110 Green 410 orange Expermint f Red Green orange Blue r pr retired or green t 7 B A A if 4 greens 3 reds B f3reds Ff Another example B A p N considers.sfb2 g Pz with PrEnJspn a Pa BzBg Http Asfb2g3 et e joint ooo p µ fMt probability Bsf 3.4 p PREBIAT Tipp pm hB or

sample Point Assume A WE then pros Pr WAA Pr wlA r pretty Torres Then EBP.rcwnt PREB.AT PREBIA s Prca PRCA The conditional Probability of 13 Definition is given A R AAB Pr BRA pr BIA B Aeg µ A given PRETTY Pratt PRE one more example 3 balls into 3 bins suppose I toss at the time one 1st bin is empty 2ndbin is empty A B is Pr AIB what 000 se z Mls 33 27 f gzg3 r 3 i 2 A f 2,333 333 1333 AAB Bsf 23 B IAN BI 8 I IA Is 23 s 8 1131

1 PKA.ms At B Pr f s 2qfI pVTBJprEB3sfBI s z8ygPrCAh B s ftp.B zz Bayesian Inference 8 A way update knowledge after making an observation an estimate of probability of may have we a event A knowledge prior given can update this After B occures event we to PRCA 1B estimate Prior probability PEAT interpretation In this Pr AIB Posterior Probability Example There a certain disease for test is a new affect Person 1 when the test applied to an comes up positive of test 90 l the cases and negative in False negative 101 2 when applied to a healthy Person the test a

of cases and positive comes up negative 80 False Positive 215 that only of Population the 5 Suppose disease A this has prior random Person is tested positive Q when a the Person is the probability what that disease the has At B Pr Let's define events is Positive Bo Test Ae affected 0.9 PREBTA K PRCA 0.05 0.2 PrEAAB PREBIAJPREAT Pr AIB PrEB PVCB t PRE AAB BI AT PREAABF.PK Pr puffy fPrCAJprEB7 prCAABJ PREFAB

Pr AhB3sPr BID PRCA I PRAT Pr PREFAB Prof Pr B F BIT PREBIAJP.CA Pr AIB PrEBIA3PrCAJpprCBIATf PrcAJ 3 Bayes Rule PLA 1B PCBIAJPCAJPREAIB.IS P PVE AAB B Tj Prc Anoypret pr ABIA 4 Total Probability Rule bins containing Imagine two number of some red and green Bini Bin 2 One bin is chosen with equal Bad BBB B Bo 507 Probability a ball is drawn uniformly at rqndom Then b ko Yo Yo what is the probability that Ya BOB We Picked Bin i given that G I oo a green baltwas drain B Bz

prosit green PrE9rj green ABD ePr green MBB PrEgreenrigs Pr PrEgreen Big PrCBDtPr green IBD pr Bz 12 12 Ip 25 12 B is Partitioned into Definition Event n Aig if events An AA TTn 1 B B for all Air Aj 0 2 Az A An i An sn An are A i.e 4 mutually exclusive total the probability is Then PEBA Ari pr B PRIBLADPREAD s

for the bays rule So PREBIAITPREAD PREBIAITPREAD Pr Ail B PREB PREBIAJJPREAD g