Conditional Probability. Given an event M with non zero probability and the condition P( M ) > 0, Independent Events P A P (AB) B P (B)

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1 Coditioal robability Give a evet M with o zero probability ad the coditio ( M ) > 0, ( / M ) ( M ) ( M ) Idepedet Evets () ( ) ( ) () ( ) ( ) ( )

2 Examples ) Let items be chose at radom from a lot cotaiig items of which 4 are defective. If {both items are defective} ad {both items are o-defective}, fid () ad ) (). S ca occur i 66 ways, ca occur i ways, ca occur i 8 ways, The 8 4 ( ). 66 The probability that at least oe item is defective(c) {at least oe item is defective} is the complemet of ; that is, C c. Thus by Theorem., C p( C) ( ) ( ) 4 9 Note: The odds that a evet with probability p occurs is defied to be the ratio p: (-p). Thus 4 the odds that at least oe item is defective is or 9:4 which is read 9 to 4. : ) Three horses, ad C are i a race; is twice as likely to wi as ad is twice as likely to wi as C. What are their respective probabilities of wiig, i.e. ( ), ( ) ad ( C )? ccordigly, () 4p 4, 7 () p, 7 (C) p 7 p + p + 4p or 7p or p 7

3 robabilities o the Real Lie ssume ζ is the set of positive real umber t i 0 t t, for ay 0 t t, z t 0 α (t) dt where α (t) 0 prob. of a evet { } Note: t t 0,. evet is { } t t t 0 t t - 0 t < t zt α (t) dt t

4 Examples oits a ad b are selected at radom such that b 0 ad 0 a as show below. Fid the probability p that the distace d betwee a ad b is greater tha. d a - b - b 0 a The sample space S X p area of ( ) area of S 6

5 Total robability Give mutually exclusive evets,,...,, ad S Theorem of Total robability: ( ) ( ) ι ι ι ( ) ( ) ( ) ( ) ( ) ( ) N L i i

6 ayes Theorum i posterieor i β i i j probability, ( ) i ( ) i ( ) prior probability i I geeral we ca state the followig state sample { sample state} { state} { sample } { state} state

7 Example: Samplig problem 80 g items ad 0 d items. O the secod draw, we have, 0 80 defect o st draw ( ), ( ) ( ) 9 defect o d draw, 99 ( ) ( ) ( ) + ( )

8 Examples Three Maufactures Supplier #---50% Supplier #---0% Supplier #---0% 95% compoets pass 90% compoets pass 80% compoets pass Give that a compoet passes ispectio, fid the probability that it has bee supplied by each supplier. Solutio: Let E i evet that a particular compoet is supplied by the ith supplier. Let {ay oe compoet passes ispectio tests.} E i [ ] E E EL ( E ) E ( 0.50)( 0.95) ( 0.50)( 0.95) + ( 0.0)( 0.90) + ( 0.0)( 0.80) 0.55 E E 0.97, 0.78

9 Example: O the spot check of the breakdow of a machie idicated that it is caused by overheatig of the motor. ast records show (correct)0.75 (correct/overheat)0.9 (overheat)0.8 a) Fid the probability that a breakdow is caused by the overheatig of the motor, give that the o the spot check is correct, idicatig that such is the case. b) Fid the probability of makig a correct aalysis from a o the spot check, give that a breakdow is ot caused by the overheatig of the motor. Let (correct aalysis of the cause of the breakdow is made through o the spot checks) E(breakdow is caused by overheatig) {}robability of correct aalysis through o the spot checks. ( E) a) [ ] [ ] ( 4 )( 9 ) E E [ ] ( 0 ) 4 Let E evet that breakdow is ot caused by overheatig. ( E' ) ( E) ; 5 [ D ] robability of makig correct o - the - spot aalysis give b) E' failure ot by overheatig ( ) ( ) ( ) + ( ') ( E E E E' ) ' ' 0. 5 E E robability of makig the correct aalysis from a o spot check, give that a breakdow is ot caused by overheatig. ( )( ) ( ) ( ) 5

10 Example Oe coi i a roll of cois has two heads. The others are fair cois. coi is selected at radom ad tossed m times. What is the probability that the coi tossed is two-headed, give that all m tosses are head. How large must m be for this probability to be greater tha ½? Solutio: Let evet two-headed coi selected. Let evet all m throws are head. a) / N-/ ( ( ) ) m ( ) (.5) m m (-(.5) ) + ( ) 0. 5 b ) For > l( ) l

11 Example The Simpso test is used by Compay to test 000 employees. Maagemet estimates that % of employees use drugs. The test is 98% accurate whe someoe is actually doig drugs, ad is % false positive. a) perso has a positive test. What is the probability that the perso truly uses drugs? b) If a perso tests egative, what is the probability he uses drugs? Solutios: ( D + ) + ( ) T ( D) (.0)(.9ε) a) D.48 + T ( T ) (. )(.9ε) + (.99)(.0) ( ).99.0 ND D ( ) T - T + T - T T D(0).0*.0 b) d D T ( T ).0*.0*.99*.97 +

12 Example game show cotestat selects oe of three curtais that he thiks hides a car. The other two curtais hide goats. fter he chooses, host opes oe of the remaiig curtais at radom to reveal a goat. Should the cotestat stay with his choice, or should he switch to the remaiig curtai? Geeralize to curtais, oe car ad - goats, where oe curtai is selected by the cotestat ad - of the - emaiig curtais are opeed by the host to reveal goats. Solutio: Let - (car)/(choosig car) (sg/hc)(shows goat/havig car) (goat)/(choosig goat) (hg/sg)(have goat/shows goat) (hc/sg)(havig a car/shows goat) (sh/hg)(shows goat/have goat) Usig aye s Rule sg ( hc) sg ( hg) hc hc ; hg hg sg ( sg) sg ( sg) ssumig the host selects a curtai at radom, the. (sg)(sg/car)(car)+(sg/goat)(goat) * + * ( if at radom, if ot, the prob. ) The: ( hc sg) * * hg sg We see it makes o differece if he chages or ot.

13 If the host kows where the goats are, host deliberately turs over curtai with goat sg hc hc sg ( hc) + ( hg ) * + * hg sg sg hg * ( )( ) + ( )( ) * * Hece, if the game is rigged, it pays to chage the selectio. You ca double your chace.

14 Whe there are curtais, sg ( hc) hc hc sg * sg We have oe car ad - goats. The,, sg sg or if game is rigged, hc hg - (hc) /, (hg) ( sg) ( sg / hc) ( hc) + ( sg / hg) ( hg) x/ + ( - )/( -)x( -)/ ( -)/ { hc / sg} Or, if the game is rigged, ( ) x hc sg { / }. { sg / hc} { hc} { sg} ( this is equal to ).

15 Cotiuig ( hg / sg ) ( sg / hg) ( hg) ( sg) - x or, if fixed * We see agai, there is a advatage to switchig curtais if you assume that the game host kows which curtai is hidig the car.

16 Idepedet Evets If ad are idepedet, the _ ad _ ad ad roof of (a) ( ) ( ) ad ( ) ( ) So Thus ( ) ( ) ( ) ( ) + ( ) ( ) ( ) + ( ) ( ) ( ) ( ) ( ) ( ) [ ( ) ] ( ) ( ) ( ) ( ) ( ) QED

17 roof of (b) ( ) ( ) ( ) ( ) + ( ) ( ) ( ) + ( ) ( ) ()[ - ()] ()() QED roof of (c) Usig DeMorga s Theorem, ( ) ( ) ( ) ( ) + ( ) ( ) ( ) ( ) [ ( ) ] ( ( ) )( ( ) ) ( ) ( ) QED

18 Defiitio ( ) sequece of evets,,,..., is said to be mutuallyidepede t if for each set for distict idicies, i,i,...i that are elemets of {,,..., } we have : ( i i... i ) ( i )... ( i ) κ Κ Κ NOTE : Ca have ( C) ( ) ( ) ( C) but ( ) ( ) ( ) Example: Experimet of tossig two dice. S {( i, j) : i, 6} j Let: First die results i,, or First die results i,4,5 C The sum of the two faces is 9 ( ), ( ), ( C ) 9 C 6 6 ut, ( ) ( ) ( ) 6 6 ot mutually idepedet. ( C) ( ) ( ) ( C) 4 {(, ),(,),(, ), (,4),(,5),(,6) } {(,6) } C {(,6),( 4,5),( 5,4) } C {(,6) }

19 Example bag cotais four marbles umbered,,, ad 4. Defie: - Draw marbles ad - Draw marbles ad C - Draw marbles ad 4 re,, C mutually idepedet? Solutio: robability of drawig ay two marbles is /4 +/4/. robability of drawig ay two first, ad the ay other two (assumig replacemet) is always (.5)x(.5) ()()()(C) etc, etc Hece, the evets are mutually idepedet.

20 Example Experimet of tossig two dice. S {( i, j) : i, 6} j Let: First die results i,, or First die results i,4,5 C The sum of the two faces is 9 C C {(, ), (,),(,),(,4), (,5 ), (,6) } {(,6) } {(,6), ( 4,5 ), ( 5,4) } C {(,6) } Thus,, ad C are ot mutually idepedet. 6 ut, ( ) ( ) ( ) 6 6 ( ), ( ), ( C) 9 4

Chapter 6 Conditional Probability

Chapter 6 Conditional Probability Lecture Notes o robability Coditioal robability 6. Suppose RE a radom experimet S sample space C subset of S φ (i.e. (C > 0 A ay evet Give that C must occur, the the probability that A happe is the coditioal