, where A is the compliment of A - the addition rule Mutually exclusive: Pr( A B)

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2 Proility. Discrete Rdom Vriles: Coditiol Proility Biomil Distriutio. Cotiuous Rdom Vriles: Proility Desity Fuctios Norml Distriutio. Smplig & Estimtio Itroductio to Proility & Smple Spce Proility ssigs umeric vlue to the likelihood of evet occurrig. Proility is cocered with outcomes or results of trils i rdom eperimets. A rdom eperimet is oe where: The possile umer of outcomes is fiite. All outcomes re eqully likely. The results re ucerti. The proility tht evet occurs is: umer of outcomes for tht evet totl umer of ll possile outcomes If evet is impossile, the proility tht this evet occurs =. If evet is certi, the proility tht this evet occurs =. So, the proility tht y evet occurs is etwee d iclusive. i.e. Pr evet ) Pr A) Pr A), where A is the complimet of A Pr A B) Pr A) Pr B) Pr A B) - the dditio rule Mutully eclusive: Pr A B) Idepedet: Pr A B) Pr A) Pr B) or Pr A B) Pr A) A smple spce shows ll possile outcomes Commo smple spces re Ve digrms, Tree digrms d tles.

3 F X D r w Emple : A fmily hs three childre. Wht is the proility tht ) they re ll oys? ) The st is oy d the d d the rd re girls? c) There is oe oy d two girls? d) They re ot ll oys? Solutio: Smple Spce: Boy Boy Boy Girl Girl Boy Girl ) Pr B d B d B) = ) PrB d G d G) = 8 8 Girl specific order) c) Pr oe oy oly) order ot specific = PrBGG)+PrGBG)+PrGGB) = Boy Girl Girl Boy Boy Girl 8 d) PrTht they re ot ll oys) = Prll oys) = So o d mes X multiply) o or mes + dd)

4 F X D r w Emple. A mthemtics studet clcultes his chces of pssig the et test ccordig to the results o erlier tests. If he pssed the lst test he thiks his chces re.7 of pssig the et test. If he filed the lst test he estimtes tht the proility of pssig the et test is.. Drw proility tree digrm to illustrte the possile results otied o the et two tests, give tht he filed the previous test. Fid the proility tht o the et two tests the studet will: ) pss oth; ) pss the first ut ot the secod; c) fil the first d pss the secod; d) fil oth. Previous st Test d Test Solutio: ) Pr P d P) =..7 =. ) Pr P d F) =.. =. c) Pr F d P) =.. =. d) Pr F d F) =.. =. Note : the sum of these four swers. F.. P F.7.. P F P. F Emple. From ur cotiig 7 lue d red lls, lls re tke t rdom i) with replcemet d ii) without replcemet Fid the proility tht: ) oth lls re lue; ) the first ll is red d the secod is lue; c) oe is red d the other is lue. Solutio: 7 Blue, Red i) With Replcemet ) Pr B d B) = 7 ) Pr R d B) = 7 c) Pr R d B or B d R) = 7 i) Without Replcemet ) Pr B d B) = ) Pr R d B) = c) Pr R d B or B d R) =

5 Emple : Simo visits the detist every 6 moths for checkup. The proility tht he will eed his teeth cleed is., the proility tht he will eed fillig is. d the proility tht he will eed oth is.. Wht is the proility tht he will ot eed his teeth cleed o visit, ut will eed fillig? Wht is the proility tht she will ot eed either of these tretmets? Solutio: Let F = he will eed fillig. Let C = he will eed cle. F F Totl C... C..6.6 Totl..9 Pr C F). Pr C F). 6

6 F X D r w Emple : A mrksm ever misses the trget, ut he hs lousy im d his rrows ld ywhere o the trget. The trget comprises three cocetric circles with rdii cm, cm, d cm with score vlues of poits, poits d poit respectively. i) For y sigle shot wht is the proility tht the resultig score is: ) four? ) two? c) oe? d) zero? ii) He fires three rrows oe fter the other). Wht is the proility tht the resultig score is: ) two? ) three? c) four? d) five? e)? Solutio: Need the res of ech prt of the trget. Are of Trget = ) Are of = ) Are of = ) ) 8 Are of = ) ) 6 Let X = the score from sigle shot i) ) PrX = )= 8 8 ) PrX = )= 6 6 c) PrX = )= d) PrX = )= he ever misses) cm cm c m Let Y = the score from three shots ii) ) Pr Y =) = 6 ) PrY = ) = Pr,, ) = c) PrY = ) = Pr,, or,, or,, ) = d) Pr Y = ) = Pr,, or,, or,, ) = 96 9 e) Pr Y >) = PrY ) = 6 6 Sheet A, Sheet B Smple Spce E A,,,, 6, 8,,,,,,

7 Sheet A. From pulic opiio poll it ws foud tht out of people wted Sudy shoppig. Out of three rdomly selected people wht is the proility tht: ) ll three wted Sudy shoppig; ) oe wted Sudy shoppig; c) oly the first d third wted Sudy shoppig; d) the first d secod wted Sudy shoppig.. Perso A is treted with drug X for his prticulr ilmet. Perso B with differet complit is treted with drug Y d perso C, with differet complit gi, is treted with drug Z. Drugs X, Y, Z hve success rtes of 7 i, 8 i 9 d i respectively. Fid the proility tht: ) ll three re cured; ) oe re cured; c) oly B is cured; d) oth B d C re cured.. Bo X cotis lue lls d red lls. Bo Y cotis lue lls d 6 red lls. Oe ll is rdomly selected from ech of the oes X d Y. Fid the proility tht, of the lls tke ) oth re red; ) oth re lue; c) either re lue; d) oly oe is red hvig come from Bo X; e) oly oe is red hvig come from Bo Y; f) oly oe ll is red.. Two dice, oe white d oe lue, re tossed. The white oe is fir die d the lue oe is weighted such tht the Pr) = Pr) =., Pr) =., Pr) = Pr) = Pr6) =.. Fid the proility tht: ) 6 tured up o oth dice; ) eve umer tured up o oth dice; c) the white die showed eve umer d the lue die odd umer; d) the white die showed odd umer d the lue die eve umer; e) oe die showed odd umer d the other die eve umer. Aswers 8. ) 7. ). ) 8 7 ) ) c) c) d) d) 9 ) c) 8 d) e) 9 9 f). ) 6 ) c) d) e)

8 Sheet B. From group of studets d techers, two people re selected t rdom. Wht is the proility tht: ) o studet is selected; ) o techers re selected; c) the first perso selected is studet d the secod is techer; d) oe perso selected is studet d the other techer.. The proility tht it ris tomorrow is. If it ris tomorrow the the proility tht it is fie the et dy is. Fid the proility tht: ) it ris tomorrow d is fie the et dy; ) it ris oth dys.. Two ideticl oes X d Y coti lls. Bo X cotis red d 7 lck lls while Bo Y cotis red d 9 lck lls. A o is chose t rdom d from tht o ll is selected t rdom. Wht is the proility tht: ) Bo X ws chose d the ll ws red; ) Bo Y ws chose d the ll ws red; c) the ll ws lck d it cme from Bo X. Aswers. ) 8 ) c) 6 d) 8. ) ). ) ) c) 7

9 F X D r w. Coditiol Proility Whe you re give etr iformtio out the outcome KEYWORD: GIVEN) For coditiol proility PrA B) the proility tht A occurs give tht B hs occurred. Pr A B) Rule: Pr A B) Pr B) Two evets A d B re idepedet if the occurrece of oe evet hs o effect o the proility of the occurrece of the other. o Pr A B) Pr A) o Pr A B) Pr A) Pr B) Two types of prolems: Type Emple : A die is throw. Fid the proility tht three turs up give tht the umer is odd. Solutio: Let A = three is throw Let B = odd is throw Pr A B) Pr A B) Pr B) Pr d odd) Pr A B) Pr odd) 6 Emple : From pck of plyig crds, oe is drw. If it is hert, wht is the proility tht it is the ce of herts? Let A = ce of herts drw Let B = hert is drw Pr A B) Pr A B) Pr B) Pr ce of herts d hert) Pr A B) Pr hert) TYPE Emple : I certi VCE mthemtics emitio, % of the cdidtes were girls, d 9% of these girls pssed i mthemtics. The rest of the cdidtes were oys d 8% of these pssed i mthemtics. ) Wht ws the overll percetge of cdidtes who pssed i mthemtics? ) A rdomly selected mthemtics pper ws foud to e the pper of studet who hs pssed i mthemtics. Wht is the proility tht this studet ws girl? Solutio: Girl odd 6. odd odd' Pss Fil. 8.8 Pss Boy

10 F X D r w ) % pssed = =.87 = 87. % ) Pr Girl give tht the studet pssed) Pr Girl studet pssed ) Pr studet pssed ) = Altertive pproch: Krugh or Two-wy) Tle. Girl Boy Totl Pss Fil Totl..8. Emple : There is oly oe us service pssig m s house ech morig. If the us is o time the he rrives t work o time o verge of 9 out occsios. If the us is lte the he rrives t work o time o oly verge of out times. The us is lte % of the time. Fid the proility tht the us ws lte o dy he ws lte to work. Solutio: Bus Arrive t work.6 Lte. Lte. O time. 8. Lte O Time.9 O time Pr Bus lte kowig M ws lte for work) Pr Bus lte M lte for work )..6 Pr M lte for work ) or Bus O time Lte Totl O time Work Lte.8.. Totl

11 Emple : The proility tht Moic rememers to do her homework is.7, while the proility tht Ptrick rememers to do his homework is.. If these evets re idepedet, the wht is the proility tht: oth will do their homework Moic will do her homework ut Ptrick forgets? Solutio: Let M = Moic does her homework Let P = Ptrick does his homework Pr M P) Pr M ) Pr P).7..8 Pr M P) Pr M ) Pr P).7.6. Idepedet Idepedet E B,,,,, 7, 8, 9,,,, 7, 9

12 Discrete Proility Discrete Rdom Vriles DRV) d Discrete Proility Distriutios DPD) A DRV is vrile. The vlue of DRV is usully iteger d coutle. Some emples d o-emples of DRV s o If X = the umer of oys i fmily, X is DRV o If Y = d iigs score of cricket mtch, Y is DRV o If A = the height of Yer studet t RSC, A is ot DRV o If T = the time tke to get home, T is ot DRV. A DPD is tle with two colums or rows) tht shows ll the possile vlues of DRV with ech of its respective possiilities. Emple : Cosider fmily of three childre. i) Use proility tree to list ll the possile fmilies. ii) If the proility of girl is, fid the proility of ech of the possile fmilies iii) iv) occurrig. Fid the proility distriutio of the discrete rdom vrile, X, where X is the umer of oys i the fmily. Usig your swer to iii) fid:. PrX = ). PrX < ) c. PrX = X ) 6 d. : Pr X ) 8 e. : Pr X ) Solutio: B G B G B G B G B G B G B G BBB BBG BGB BGG GBB GBG GGB GGG

13 8 Pr BBB) Pr BBG) Pr BGB) 8 Pr BGG) ii) Pr GBB) 8 Pr GBG) 8 Pr GGB) 7 Pr GGG) iii) PrX = ) iv) ) Pr X ) ) Pr X ) Pr X or or ) or PrX =)) Pr X X ) c) Pr X X ) Pr X ) 6 6 Pr X ) Pr X ) d) Pr X ) 8 e) Pr X ) E C,,,6, 7, 9 E C,,,,, 7,

14 Epected Vlue, EX) The epected vlue, EX), is the sme s the verge, me or. The geerl rule: E X ).Pr X ) the epected vlue is equl to the sum of ech vlue of X multiplied y its proility Also: E f )) f ) Pr X ) Note: Mode: Most Commo Properties of EX) Medi: Middle vlue i) EX)= EX) Me: Averge ii) EX+)= EX)+ iii) E)= iv) EX+Y) = EX) + EY) Emple: For the followig proility distriutio, fid: PrX = ). ) EX) d) Mode. ) EX + ) e) Medi. c) EX ). Solutio: ) E X ).Pr X ) PrX = ). Pr X ) Totl.7 = EX) ) E X ) ).Pr X ) PrX = ) ).Pr X ) Totl.7 = EX+) c) E X ) ).Pr X ) PrX = ).Pr X ) Totl 8. = EX ) d) Mode: = e) Medi: = ED,,,,, 7, 8, 9, i)ii),,,, 6, 7

15 The Vrice, VrX) of DRVX) The vrice mesures the spred of distriutio. Vr X ) E X )... E X ) Pr X ) ) E X )) or Pr X ) The stdrd devitio of X, SD X ) Vr X ). Commo Nottio Me EX) Vrice VrX) Stdrd Devitio SDX) Emple: Cosider these two distriutios: PrX = ) y PrY = y) PrX = ). Pr X ).Pr X ) E X ) =.9 E X ) = 8.9 y PrY = y) y. Pr Y y) y y.pr Y y) E Y ) =.9 E Y ) = 9.8 Both distriutios hve the sme me,, ut the y-vlues hve more spred. Vr X ) E X SD X ) ) X Y Vr Y ) E Y.9 SD Y ) ).9.8

16 Emple: A o cotis three white d two red lls. The lls re tke out oe t time d ot replced) util red ll is otied. ) fid the proility distriutio for the umer of lls chose. ) How my drws do you epect util you get red? c) Fid the vrice d stdrd vrice. Solutio: Let X = the umer of lls chose. Order of lls R WR WWR WWWR PrX = ). Pr X ).Pr X ) ) EX) = c) VrX) = ) = SDX)= 9 6 E X ) = E X ) = Useful property: VAR X ) Vr X ) ED 9c, iii) c,, c, c, 6, 7

17 The reltioship etwee the me d the stdrd devitio. ) For my proility distriutios ut ot ll), out 9% of the distriutio lies withi two stdrd devitios of the me. i.e. Pr X ). 9 Emple: For the fmily of three childre, used efore, where the chce of the irth of girl ws : PrX = ). Pr X ).Pr X ) Vr X ) E X.6..7 SD X ) EX)=. EX )= 7. 6 ) ) ).897 Pr X ) Pr.97 X.897) the vlues of X the umer of oys) tht lie etwee these two umers re, &. 7 Pr X ) Pr X ).96 or 9.6% ED d, d, 6c, 7c, 8 Review questios chpter

18 The Biomil Distriutio emple of Discrete Proility The Biomil Theorem Epsios of the form Cosider: ) ) ) ) Now: ) ) ) There is ptter: the idetity, There is idetity for Ad there is for. Iterestig is tht the co-efficiets of the epsios elog to Pscl s Trigle: PASCAL S TRIANGLE Etc.. I geerl the Biomil Epsio is: r r r Here we hve the: st term, d term, rd term, the geerl term r r r d the lst term..

19 Properties of comitios: d d i geerl r r Emple: Usig the iomil epsio epd 7. Solutio: 7,, **There is lwys oe more term th the power. ** For ech term the sum of the idices dd up to ) 7) 7) 7) 7) 7) 7) 7) 7) 7) 7) 7

20 The Biomil Proility Distriutio CHARACTERISTICS I iomil eperimet,. there re two possile outcomes for ech tril. success this is the desired outcome filure this outcome is ot desired.. the proility of success is the sme for ech tril. i.e. the trils re idepedet. # Note: Trils of this type re clled BERNOULLI trils. proouced Burooey) The FORMULA for the PROBABILITY of BINOMIAL r.v. Pr X ) p p where Pr X ) = Pr gettig successes i trils) d p = PrSuccess) NB. This formul tkes ito ccout ll possile orders. Whe do you use the Biomil Distriutio? Whe the situtio hs BOTH of the chrcteristics: & This usully ivolves: * smplig with replcemet OR * smplig without replcemet from lrge popultio OR * o smplig t ll: just oservig Nottio: The rdom vrile X hs iomil distriutio with idepedet trils d p = proility of success, is writte s: e.g. X ~ Bi,.) X ~ Bi, p) or X d Bi, p) Emple: A mchie mufcturig clcultors is kow to hve defective rte of i. Fid the proility tht i smple of 6 clcultors tke t rdom: ) ectly two re defective; ) o more th re defective. possile outcomes defective or ot defective. Let X = the umer of defective clcultors. X ~ Bi6, 6 9 ) Pr X ). 98

) Pr X ) Pr X ) Pr X ) Pr X 6 ) 9 6 6 9 6 9.

I the et times t-t, ) Wht is the proility of gettig

21 ) Pr X ) Pr X ) Pr X ) Pr X 6 ) E A,,, 6, 7,,,,,, 6, 8, 9,, Usig The Grphics Clcultor Emple: A hitter hs proility of of gettig hit ech time t t, with ech t-t idepedet of other t-t. I the et times t-t, ) Wht is the proility of gettig ectly three hits? ) Wht is the proility of gettig t lest two hits?

22 The grph of the iomil proility distriutio Emple: Fid the proility distriutio of the umer of girls i fmily of three childre. Assume tht the proility of girl eig or is.. Hece grph Pr X ) versus, where X = the umer of girls i the fmily. Solutio: X ~ Bi,.) Pr X ) Pr X ) Repet for: ll comitios of =,, 8 d p =.,.,.6 = = = 8 p =. p =. p =.6 The effect of the prmeters vriles) d p o the shpe of the grph: o As the vlue of icreses, the pek of the grph shifts to the right. i.e. the epected vlue the me) icreses) o Whe p =., the curve is perfectly symmetricl. o Whe p <. the distriutio is skewed to the right or positively skewed) NOTE: Skewess refers to the til. o Whe p >. the distriutio is skewed to the left egtively skewed). E B Q,,

23 Epecttio d Vrice of the Biomil Distriutio The me of iomil distriutio DRV c e foud y: E X ).Pr X ). p The vrice of iomil distriutio DRV c e foud y: E X ) p p) The stdrd devitio : SD X ) p p). Emple: A iomil rdom vrile hs me of d vrice of. fid the prmeters d p. Solutio: X ~ Bi, p) p p p p) p p) p) p p 9 Emple: Give the 9% cofidece limits for the umer of girls i fmily of eight childre. Assume Prgirl)=.. Solutio: Let X = the umer of girls i the fmily X ~ Bi8,.) We kow: Pr X ) ).88..7) 7. Pr.88 7.) Pr 7) E B,, 6, 7, 8, 9,

Biomil Distriutio: Solvig for Emple: A group of people meet for fcy dress prty.

Assume tht the proility of perso t the prty hvig prticulr zodic sig is.

there will e t lest oe Scorpio is greter th.8?

24 Biomil Distriutio: Solvig for Emple: A group of people meet for fcy dress prty. Ech perso comes dressed i somethig relted to his or her zodic sig. Assume tht the proility of perso t the prty hvig prticulr zodic sig is. ) Wht is the lest umer of people who eed to tted the prty so tht the proility tht there will e t lest oe Scorpio is greter th.8? Solutio: Let X = the umer of Scorpios t the prty Pr X ).8 Pr X ). Pr X ).. solve ^., t lest 9 people ) Wht is the lest umer of people who eed to e t the prty so tht the proility tht there will e ectly Scorpios is greter th %? Solutio: = 6 CTRL T) C t use ivbiomn ) commd s it is ot Cumultive Proility.

c) Wht is the lest umer of people who eed to

8 swer people Go to tleset put i umer, the go

25 c) Wht is the lest umer of people who eed to tted the prty so tht the proility of fewer th two Scorpios is closest to.8? Solutio: Pr X ).8 Pr X ).8 swer people Go to tleset put i umer, the go to tle d scroll dow to the vlue tht is closest to.8. OR E C Q,,,,, 6, 7 Chpter Review

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Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Lecture 17

Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Lecture 17 CS 70 Discrete Mthemtics d Proility Theory Sprig 206 Ro d Wlrd Lecture 7 Vrice We hve see i the previous ote tht if we toss coi times with is p, the the expected umer of heds is p. Wht this mes is tht