( ) ( ) = for all Corollary: Rules of robablty The probablty of the unon of any two events and B s roof: ( Φ) = 0. F. ( B) = ( ) + ( B) ( B) If B then, ( ) ( B). roof: week 2
week 2 2 Incluson / Excluson formula: For any fnte collecton of events For any fnte collecton of events roof: By nducton n,...,, 2 ( ) ( ) ( ) ( ) ( ) > > > = = + = j n n k j k j j n n 2 U n,...,, 2 ( ) = = n n U
Example In a lottery there are 0 tckets numbered, 2, 3,, 0. Two numbers are drown for przes. You hold tckets and 2. What s the probablty that you wn at least one prze? week 2 3
Condtonal robablty Idea have performed a chance experment but don t know the outcome (ω), but have some partal nformaton (event ) about ω. Queston: gven ths partal nformaton what s the probablty that the outcome s n some event B? Example: Toss a con 3 tmes. We are nterested n event B that there are 2 or more heads. The sample space has 8 equally lkely outcomes. Ω = { HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} The probablty of the event B s Suppose we know that the frst con came up H. Let be the event the frst outcome s H. Then { HHH, HHT, HTH, HTT} and B = HHH, HHT, The condtonal probablty of B gven s 3 3 8 ( B) = = 4 4 8 = { HTH} ( ) week 2 4
Gven a probablty space (Ω, F, ) and events, B F wth () > 0 The condtonal probablty of B gven the nformaton that has occurred s ( ) ( B) B = Example: ( ) We toss a de. What s the probablty of observng the number 6 gven that the outcome s even? Does ths gve rse to a vald probablty measure? Theorem If F and () > 0 then (Ω, F, Q) s a probablty space where Q : s defned by Q(B) = (B ). roof: F R week 2 5
The fact that condtonal probablty s a vald probablty measure allows the followng: ( ) ( ) B = B,, B F, () >0 ( B B ) = ( B ) + ( B ) ( B B ) 2 2 2 for any, B, B 2 F, () >0. week 2 6
Multplcaton rule For any two events and B, For any 3 events, B and C, ( B) = ( B ) ( ) ( B C) = ( ) ( B ) ( C B) In general, n I = = n ( ) ( ) ( ) 2 3 2 n I = Example: n urn ntally contans 0 balls, 3 blue and 7 whte. We draw a ball and note ts colure; then we replace t and add one more of the same colure. We repeat ths process 3 tmes. What s the probablty that the frst 2 balls drawn are blue and the thrd one s whte? Soluton: week 2 7
Law of total probablty Defnton: For a probablty space (Ω, F, ), a partton of Ω s a countable collecton { B } of events such that B F, B B = Φ and B = Ω. Theorem: If { B, B,... 2 } s a partton of Ω such that B > 0 then roof: ( ) ( B ) ( ) j U for any. = B F ( ) week 2 8
( ) Examples. Calculaton of for the Urn example. B 2 2. In a certan populaton 5% of the females and 8% of the males are left-handed; 48% of the populaton are males. What proporton of the populaton s lefthanded? Suppose person from the populaton s chosen at random; what s the probablty that ths person s left-handed? week 2 9
Bayes Rule Example: test for a dsease correctly dagnoses a dseased person as havng the dsease wth probablty 0.85. The test ncorrectly dagnoses someone wthout the dsease as havng the dsease wth probablty 0. If % of the people n a populaton have the dsease, what s the probablty that a person from ths populaton who tests postve for the dsease actually has t? (a) 0.0085 (b) 0.079 (c) 0.075 (d) 0.500 (e) 0.9000 week 2 0
Independence Example: Roll a 6-sded de twce. Defne the followng events : 3 or less on frst roll B : Sum s odd. If occurrence of one event does not affect the probablty that the other occurs than, B are ndependent. week 2
Defnton Events and B are ndependent f Note: Independence dsjont. Two dsjont events are ndependent f and only f the probablty of one of them s zero. Generalzed to more than 2 events: collecton of events, 2 s (mutually) ndependent f for any subcollecton ( B) ( ) ( B) = {,... }, 2 {,... } m n ( ) = ( ) ( ) ( ) 2 m 2 m Note: parwze ndependence does not guarantee mutual ndependence. week 2 2
Defnton Events and B are ndependent f Note: Independence dsjont. Two dsjont events are ndependent f and only f the probablty of one of them s zero. Generalzed to more than 2 events: collecton of events, 2 s (mutually) ndependent f for any subcollecton ( B) ( ) ( B) = {,... }, 2 {,... } m n ( ) = ( ) ( ) ( ) 2 m 2 m Note: parwze ndependence does not guarantee mutual ndependence. week 2 3
Example Roll a de twce. Defne the followng events; : st de odd B: 2nd de odd C: sum s odd. week 2 4
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