Chapter 1: Introduction to Probability

Transcription

1 Hery Star ad Joh W. Woods, obablty, Statstcs, ad Radom Varables for Egeers, th ed., Pearso Educato Ic.,. ISBN: Chater : Itroducto to obablty Sectos. Itroducto: Why Study obablty?. The Dfferet Kds of obablty obablty as Ituto obablty as the Rato of Favorable to Total Outcomes (Classcal Theory) obablty as a Measure of Frequecy of Occurrece obablty Based o a Axomatc Theory 5. Msuses, Mscalculatos, ad Paradoxes obablty. Sets, Felds, ad Evets 8 Examles of Samle Saces 8.5 Axomatc Defto of obablty 5. Jot, Codtoal, ad Total obabltes; Ideedece Comoud Exermets. Bayes Theorem ad Alcatos 5.8 Combatorcs 8 Occuacy oblems Extesos ad Alcatos.9 Beroull Trals Bomal ad Multomal obablty Laws 8 Multomal obablty Law 5. Asymtotc Behavor of the Bomal Law: The Posso Law 5. Normal Aroxmato to the Bomal Law Summary 5 oblems Refereces Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

2 A uderstadg of obablty ad Statstcs s ecessary most f ot all wor related to scece ad egeerg. Statstcs: the study of ad the dealg wth data. obablty: the study of the leless of result, acto or evet occurrg. Ofte based o ror owledge or the statstcs of smlar or ast evets! Terms: Radom Varables, Radom ocesses or Stochastc ocesses For ay measured heomeo there wll be Ucertaty, Exected Varatos, Radomess, or eve Exected Errors cluded. whe a outcome s o-determstc where a exact value s subject to errors e.g. ose, measuremet Easy examles of such heomeo clude all games of chace Flg cos, rollg dce, dealg cards, etc. Egeerg Alcatos clude Realstc sgals wth ose or characterstc uow arts Sgal-to-ose Ratos, Nose-Power Measuremets, Bacgroud Nose Exected Values, Varaces, Dstrbutos Thermal Moto, Electro Movemet Relablty, Qualty, Falure Rates, etc. Thermal Moto, Electro Movemet obablty theory s ecessary for egeerg system modelg ad smulatos. uow tal codtos (radom) osy measuremets, exected accuraces, etc. durg oerato Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

3 Textboo Itro Quote By Euge Merzbacher: The robablty doctre of quatum mechacs asserts that the determato... s a roerty heret ature ad ot merely a rofesso of our temorary gorace from whch we exect to be releved by a future better ad more comlete theory. The covetoal terretato thus dees the ossblty of a deal theory that would ecomass the umerable exermetally verfed redctos of quatum mechacs but would be free of ts suosed defects, the most otorous "merfecto" of quatum mechacs beg the abadomet of strct classcal determsm. Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

4 Dfferet ds of obablty Suggested that there are essetally tyes obablty by Ituto o Lucy Numbers obablty as the Rato of Favorable to Total Outcomes (Classcal Theory) o Measured Statstcal Exectatos obablty as a Measure of the Frequecy of Outcomes obablty Based o Axomatc Theory Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

5 Deftos of used obablty Exermet A exermet s some acto that results a outcome. A radom exermet s oe whch the outcome s ucerta before the exermet s erformed. Possble Outcomes A descrto of all ossble exermetal outcomes. The set of ossble outcomes may be dscrete or form a cotuum. Trals Evet The sgle erformace of a well-defed exermet. A elemetary evet s oe for whch there s oly oe outcome. A comoste evet s oe for whch the desred result ca be acheved multle ways. Multle outcomes result the evet descrbed. Equally Lely Evets/Outcomes Whe the set of evets or each of the ossble outcomes s equally lely to occur. A term that s used syoymously to equally lely outcomes s radom. Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 5 of 5 ECE 8

6 obablty as the Rato of Favorable to Total Outcomes (Classcal Theory) Dce examle There are ossble outcomes A elemetal evet ca be defed as the total of the two de The total umber of outcome resultg each uque evet s ow. The robablty of each evet ca be comuted ad descrbed f the de are far. So the true odds ca be comuted ad a gamblg game wth sewed odds the houses favor ca be created from: htts://e.weda.org/w/cras Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

7 obablty as the Rato of Favorable to Total Outcomes (Classcal Theory) flg two cos examle Fl two cos: What are the ossble outcomes {HH, HT, TH, TT} Defe a evet as the gettg of gettg at least oe Tal. obablty s the favorable outcomes/total outcomes, = / Possble Outcomes wth obabltes: HH robablty / HT robablty / TH robablty / TT robablty ¼ Possble evets: oe head, oe tal, at least oe head, at least oe tal, at most oe head, at most oe tal, two heads, two tals o heads, or o tals. Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

8 obablty as a Measure of the Frequecy of Outcomes Exermet: Selectg a sequece of radom umbers. The radom umbers are betwee ad. Determg the relatve frequecy of a sgle umber as from to, umbers are selected. The statstcs of observed evets s aroachg /. (fte trals?) Fgure.- Evet = {occurrece of umber 5} (Numbers derved from webste RANDOM.ORG). Fgure.- Evet = {occurrece of umber } Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 8 of 5 ECE 8

9 obablty Based o a Axomatc Theory Develo the coheret mathematcal theory: Statstcs collected data o radom exermets o Possble outcomes, samle sace, evets, etc. From the statstcs, robablty structure ca be observed ad defed o Radom rocesses follow defed robablstc models of erformace. Mathematcal roertes aled to robablty derves derve ew/alterate exectatos o obablstc exectatos ca be verfed by statstcal measuremet. Ths ca be cosdered as modelg a system ror to or stead of erformg a exermet. Note that the results are oly as good as the model or theory match the actual exermet. Msuses, Mscalculatos, ad Paradoxes obablty Old tme quotato There are three ds of les: les, damed les, ad statstcs! htts://e.weda.org/w/les,_damed_les,_ad_statstcs From the CNN headles Math s racst: How data s drvg equalty, by Amee Rawls, Setember, htt://moey.c.com//9//techology/weaos-of-math-destructo/dex.html Aother examle As a erso, you are a uque dvdual ad ot a statstcal robablty but future chaces may be based o others le you that have come before. The class as a whole may exhbt statstcal exectatos although t s made u of uque dvduals. For Sc-F readers. Issac Asmov s Foudato Trlogy sychohstory used to redct the future. Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 9 of 5 ECE 8

10 Sets, Felds ad Evets Cocetually Defg a oblem Relatve Frequecy Aroach (statstcs) Set Theory Aroach (formal math) Ve Dagrams (ctures based o set theory) If you le ctures try to use Ve Dagrams to hel uderstad the cocets. Fgure.- Ve dagrams for set oeratos. Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

11 Set Theory Deftos (A revew?!) Set Subset Sace A collecto of objects ow as elemets A a, a,, a The set whose elemets are all members of aother set (usually larger but ossble the same sze). B a, a,, a therefore B A The set cotag the largest umber of elemets or all elemets from all the subsets of terest. For robablty, the set cotag the evet descrto of all ossble exermetal outcomes. A S, for all subsets Null Set or Emty Set The set cotag o elemets A Ve Dagrams ca hel whe cosderg set theory A grahcal (geometrc) reresetato of sets that ca rovde a way to vsualze set theory ad robablty cocets ad ca lead to a uderstadg of the related mathematcal cocets. from: Robert M. Gray ad Lee D. Davsso, A Itroducto to Statstcal Sgal ocessg, Cambrdge Uversty ess,. A df fle verso ca be foud at htt://www-ee.staford.edu/~gray/s.html Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

12 Equalty More Set Theory Deftos Set A equals set B f ad oly f (ff) every elemet of A s a elemet of B AND every elemet of B s a elemet of A. A B ff A B ad B A Sum or Uo (logc OR fucto) The sum or uo of sets results a set that cotas all of the elemets that are elemets of every set beg summed. Laws for Uos S A A A A B B A A A A A A A S S A B A, f B A oducts or Itersecto (logc AND fucto) The roduct or tersecto of sets results a set that cotas all of the elemets that are reset every oe of the sets. Laws for Itersectos S A B B A A A A A A S A A B B, f B A Mutually Exclusve or Dsjot Sets Mutually exclusve or dsjot sets of o elemets commo. A B NOTE: The tersecto of two dsjot sets s a set the ull set! A N Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

13 Comlemet The comlemet of a set s the set cotag all elemets the sace that are ot elemets of the set. Laws for Comlemet A A ad A A S S S A A A B, f B A A B, f B A DeMorga s Law A B A B A B A B Dffereces The dfferece of two sets, A-B, s the set cotag the elemets of A that are ot elemets of B. Laws for Dffereces A B A B A A B B B A A A A A A A A A A S S A A A B Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

14 Ve Dagram set theory cocets. (D ctures that ca hel you uderstad set theory) from: Robert M. Gray ad Lee D. Davsso, A Itroducto to Statstcal Sgal ocessg, Cambrdge Uversty ess,. Pdf fle verso foud at htt://www-ee.staford.edu/~gray/s.html (a) The sace (b) Subset G (c) Subset F (d) The Comlemet of F (e) Itersecto of F ad G (f) Uo of F ad G Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

15 More Ve Dagrams from: Robert M. Gray ad Lee D. Davsso, A Itroducto to Statstcal Sgal ocessg, Cambrdge Uversty ess,. Pdf fle verso foud at htt://www-ee.staford.edu/~gray/s.html (a) Dfferece F-G (b) Dfferece F-G Uo wth Dfferece G-F F G G F If evets ca be descrbe set theory or Ve Dagrams, the robablty ca drectly use the cocets ad results of set theory! What ca be sad about F G? [ read as the robablty of evet F uo evet G ] F G F G G F G F F G F G Therefore, F G F G F G Set algebra s ofte used to hel defe robabltes Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 5 of 5 ECE 8

16 Equaltes Set Algebra from: Robert M. Gray ad Lee D. Davsso, A Itroducto to Statstcal Sgal ocessg, Cambrdge Uversty ess,. Aedx A, Set Theory. Pdf fle verso foud at htt://www-ee.staford.edu/~gray/s.html Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

17 Axomatc Deftos Usg Sets For evet A Dsjot Sets If A S A B A B A B, the Comlemet (comlemetary sets) (defg the comlemet may be easer sometmes) If A A A A S A A, the A A Not a Dsjot Sets (soluto) If A B, the A B??? Maulato () B A A B, the uo of dsjot sets A B A A B A A B A Maulato () Maulato () Substtuto for () B A B A B, the uo of dsjot sets B A B A B A B A B A B B A B, rearragg from () A B A A B A B A B Note that we ca geerally defe a boud where A B A B A B A B equalty holds for A ad B beg dsjot sets! Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

18 Examle: -sded de A: The robablty of rollg a or a, evet A, A B: The robablty of rollg a or 5, evet B,5 5 B 5 C: The robablty of evet A or evet B, evet C A B C A B A B A B Note: C A B,,5 C C 5 Whe doubt, wrte t out to double chec your results! A Ve dagram may also hel. Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 8 of 5 ECE 8

19 obablty of A uo of evets (from secto.5) A exteso of the set theory for uos. Fgure.5- Parttog to seve dsjot regos Δ,...,Δ.) E j j If A B A B A B, what about A B C??? A B C A B C A B A C B C A B C Ca you recogze a atter E E E E j j E j E j E E j E + sgles - doubles + trles - quads etc What about A B C D E F??? Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 9 of 5 ECE 8

20 Secto.: More Deftos obablty, the relatve frequecy method: The umber of trals ad the umber of tmes a evet occurs ca be descrbed as N N A N B N C the relatve frequecy s the r A N N A ote that N N N A N B N N C r A rb rc Whe exermetal results aear wth statstcal regularty, the relatve frequecy teds to aroach the robablty of the evet. ad Where A lm r N A A B C A s defed as the robablty of evet A. Mathematcal defto of robablty:. A. A B C, for mutually exclusve evets. A mossble evet, A, ca be rereseted as. A certa evet, A, ca be rereseted as A. A. Odds or robabltes ca be assged to every ossble outcome of a future tral, exermet, cotests, game that has some ror hstorcal bass of evets or outcomes. Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

21 Jot obablty Defg robablty based o multle evets two classes for cosderatos. Ideedet exermets: The outcome of oe exermet s ot affected by ast or future exermets. o flg cos o reeatg a exermet after tal codtos have bee restored o Note: these roblems are tycally easer to solve Deedet exermets: The result of each subsequet exermet s affected by the results of revous exermets. o drawg cards from a dec of cards o drawg straws o selectg ames from a hat o for each subsequet exermet, the revous results chage the ossble outcomes for the ext evet. o Note: these roblems ca be very dffcult to solve (the ext exermet chages based o revous outcomes!) Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

22 Codtoal obablty Defg the codtoal robablty of evet A gve that evet B has occurred. Usg a Ve dagram, we ow that B has occurred the the robablty that A has occurred gve B must relate to the area of the tersecto of A ad B A B A B B, for B or A B A B, for B B For elemetary evets, A B A B A, B, for B B B Secal cases for A B, B A, ad B A. If A s a subset of B, the the codtoal robablty must be A B B A B A B, for A B Therefore, t ca be sad that A B B A B A B A, for A B If B s a subset of A, the the codtoal robablty becomes If A ad B are mutually exclusve, B B B A B A B, for B A B A B A B, for B A B Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

23 Ideedece Two evets, A ad B, are deedet f ad oly f A B A B Ideedece s tycally assumed whe there s o aaret hyscal mechasm by whch the two evets could deed o each other. For evets derved from deedet elemetal evets, ther deedece may ot be obvous but may be able to be derved. Ideedece ca be exteded to more tha two evets, for examle three, A, B, ad C. The codtos for deedece of three evets s A B A B B C B C A C A C A B C A B C Note that t s ot suffcet to establsh ar-wse deedece; the etre set of equatos s requred. For multle evets, every set of evets from dow must be verfed. Ths mles that equatos must be verfed for deedet evets. Imortat oertes of Ideedece Uos hel smlfyg the tersecto term f evets are deedet! Ideedet tersecto wth a Uo A B A B A B A B A B A B A B C A B C There wll be some examle roblems where you must determe f evets are deedet order to solve the roblem. swtch roblems homewor ad slls Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

24 Total obablty For a sace, S, that cossts of multle mutually exclusve evets, the robablty of a radom evet, B, occurrg sace S, ca be descrbed based o the codtoal robabltes assocated wth each of the ossble evets. oof: S A A A A ad A A, for j j B B S B A A A A B A B A B A B A B B A B A B A B A But B A B A A, for A Therefore B B A A B A A B A A Remember your math roertes: dstrbutve, assocatve, commutatve etc. aled to set theory. Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

25 Exermet : A bag of marbles, draw A bag of marbles: -blue, -red, oe-yellow Objects: Marbles Attrbutes: Color (Blue, Red, Yellow) Exermet: Draw oe marble, wth relacemet Samle Sace: {B, R, Y} obablty (relatve frequecy method) The robablty for each ossble evet the samle sace s. Evet obablty Blue / Red / Yellow / Total / Ths exermet would be easy to ru ad verfy after lots of trals. see Matlab Sec_Marble.m trals = vs. vs. (reeat executo a few tmes) (Aother roblem: f we ra trals, what s the robablty that we get evets that exactly match the robablty? -Blue, -Red, Yellow - a much harder roblem) Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 5 of 5 ECE 8

26 Exermet : A bag of marbles, draw Exermet: Draw oe marble, relace, draw a secod marble. wth relacemet Samle Sace: {BB, BR, BY, RR, RB, RY, YB, YR, YY} Defe the robablty of each evet the samle sace. Jot obablty Whe a desred outcome cossts of multle evets. (Read the jot robablty of evets A ad B). A, B Statstcally Ideedet Evets Whe the robablty of a evet does ot deed uo ay other ror evets. If trals are erformed wth relacemet ad/or the tal codtos are restored, you exect tral outcomes to be deedet. A, B B, A A B Therefore The margal robablty of each evet s ot affected by ror/other evets. The robablty of evet A gve evet B occurred s the same as the robablty of evet A ad vce versa. A B A B A B ad Alcable for multle objects wth sgle attrbutes ad wth relacemet. st-rows\ d -col Blue Red Yellow Blue 9 Red Yellow Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

27 Next Cocet Codtoal obablty Whe the robablty of a evet deeds uo ror evets. If trals are erformed wthout relacemet ad/or the tal codtos are ot restored, you exect tral outcomes to be deedet o ror results or codtos. A B A whe A follows B The jot robablty s. A, B B, A A B B B A A Alcable for objects that have multle attrbutes ad/or for trals erformed wthout relacemet. Exermet : A bag of marbles, draw wthout relacemet Exermet: Draw two marbles, wthout relacemet Samle Sace: {BB, BR, BY, RR, RB, RY, YB, YR} Therefore st-rows \ d -col Blue Red Yellow d Marble Blue Red Yellow st Marble Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

28 Matlab Marble Smulato Examles: Sec_Marble.m examle to show small versus large umber of samle statstcs vs. robablty Sec_Marble.m examle to valdate robablty ad/or small versus large umber of trals Sec_Marble.m examle to valdate robablty ad/or small versus large umber of trals Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 8 of 5 ECE 8

29 Resstor Examle: Jot ad Codtoal obablty Assume we have a buch of resstors (5) of varous medaces ad owers Smlar to old textboo roblems (more realstc resstor values) 5 ohms ohms ohms Subtotal ¼ watt ½ watt 5 55 watt 5 5 Subtotal Each object has two attrbutes: medace (ohms) ad ower ratg (watts) Margal obabltes: (uses subtotals) (¼ watt) = /5 (½ watt) = 55/5 ( watt) = 5/5 (5 ohms) = 8/5 ( ohms) = 5/5 ( ohms) = /5 These are called the margal robabltes whe fewer tha all the attrbutes are cosdered (or do t matter). Jot obabltes: dvded each member of the table by 5! 5 ohms ohms ohms Subtotal ¼ watt /5=. /5=. /5=. /5=. ½ watt /5=. /5=. 5/5=. 55/5=. watt /5=. /5=. 5/5=. 5/5=. Subtotal 8/5=.5 5/5=. /5=. 5/5=. These are called the jot robabltes whe all uque attrbutes must be cosdered. (Cocet of total robablty thgs that sum to.) Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 9 of 5 ECE 8

30 Codtoal obabltes: Whe oe attrbutes robablty s determed based o the exstece (or o-exstece) of aother attrbute. Therefore, The robablty of a ¼ watt resstor gve that the medace s 5 ohm. (¼ watt gve that the medace s 5 ohms) = (¼ watt 5 ohms) = /8 =.5 5 ohms ¼ watt /8=.5 ½ watt /8=.5 watt /8=.5 Total 8/8=. Smle math that does ot wor to fd the soluto: (they are ot deedet) (¼ watt) = /5 ad (5 ohms) = 8/5 (¼ watt) x (5 ohms) = /5 x 8/5 = 5/5 =.9 NO!!! Not deedet!! Math that does wor A B A B B A, B B What about (5 ohms gve the ower s ¼ watt) 5 ohms ohms ohms Total ¼ watt /=.5 /=.8 /=. /=. (5 ohms ¼ watt) = (5 ¼) = / =.5 A B A B B A, B B Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

31 Ca you determe? (, ½) = () = (5, ½) = (½ 5) = (5 ½) = ( ) = Usg the table t s rather straght forward 5 ohms ohms ohms Subtotal ¼ watt ½ watt 5 55 watt 5 5 Subtotal Jot obabltes A B A, B (, ½) = (5, ½) = Codtoal obabltes A B A B B A, B B (½ ) = ( ½) = Margal obablty B B A A B ( ) = () = A A Are there multle ways to cocetually defe such roblems? Yes Relatve Frequecy Aroach (statstcs) Set Theory Aroach (formal math) Ve Dagrams (ctures based o set theory) All ways to derve equatos that form desred robabltes. The Relatve Frequecy Aroach s the slowest ad requres the most wor! Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

32 A or ad A Posteror obablty (Sec.. Bayes Theorem) The robabltes defed for the exected outcomes, A, are referred to as a ror robabltes (before the evet). They descrbe the robablty before the actual exermet or exermetal results are ow. After a evet has occurred, the outcome B s ow. The robablty of the evet belogg to oe of the exected outcomes ca be defed as A B or from before A B B A A A B B A B B A A, for B B Usg the cocet of total robablty B B A A B A A B We also have the followg forms A B A A B A A B A A B A A B A A or A j B B A A j B A A j B A A j B j Ths robablty s referred to as the a osteror robablty (after the evet). It s also referred to as Bayes Theorem. Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

33 Examle More Resstors B B B B B 5 B Subtotal ohm 5 8 ohm 8 ohm Subtotal 8 What s the robablty of selectg a ohm resstor from a radom b? 5 Gve B margal robablty B # B B B 8 B B 5 B B B A A B A A B B A A B. 8 Assumg a ohm resstor s selected, what s the robablty t came from b? A B B 5 B A A B A A B A A B A A B B B B B B.8 B. 89 Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

34 Dgtal Trasmssos A dgtal commucato system seds a sequece of ad, each of whch are receved at the other ed of a l. Assume that the robablty that s receved correctly s.9 ad that a s receved correctly s.9. Alterately, the robablty that a or s ot receved correctly s. (the cross-over robablty, ). Wth the sequece, the robablty that a s set s % ad that a oe s set s %. [S s Sed ad R s Receve} S. S. R.9 R.9 S S R. R. S S Fgure.- a) What s the robablty that a zero s receved? Total obablty: B B A A B A A B R R S S R S S R.9... R b) What s the robablty that a oe s receved? R R S S R S S R...9. R... A A Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

35 Dgtal Commucatos (cotued) S. S. R S. 9 R S. 9 R S. R S. c) What s the robablty that a receved zero was trasmtted as a? Bayes Theorem A B B A A B A A B A A B A A S R R S S R R S S R S S R S S R S S.9..5 S R d) What s the robablty that a receved oe was trasmtted as a? Bayes Theorem A B B A A B A A B A A B A A S R R S S R R S S R S S R S S S R R S S Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 5 of 5 ECE 8

36 Dgtal Commucatos (cotued) S. S. R S. 9 R S. 9 R S. R S. e) What s the robablty that a receved zero was trasmtted as a? Bayes Theorem A B B A A B A A B A A B A A S R R S S R R S S R S S R S S S R R S S Note: S R S R f) What s the robablty that a receved oe was trasmtted as a? Bayes Theorem A B B A A B A A B A A B A A S R R S S R R S S R S S R S S S R R S S Note: S R S R.85.. Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

37 Dgtal Commucatos (cotued) S. S. R S. 9 R S. 9 R S. R S. e) What s the robablty that a symbol s receved error? Error R S S R S S Error Alterately, Error S R R S R R Whch way s easer? Error Notce that you were told orgally that there was a. chace of recevg a symbol error! Summary: A-ror obabltes S. S. R S. 9 R S. 9 R S. R S. Comuted Total obablty R. 58 R. Bayes Theorem (A-osteror obabltes) R. 9 S R S. 85 S R. 9 S R. Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

38 Examle. : Amylod test: s t a good test for Alzhemer s? A amylod test for Alzhemer s dsease had reorted results/formato for eole 5 ad older. Alzhemer s atets wth dsease = 9% had amylod rote Alzhemer s free atets = % had amylod rote Geeral oulato facts for Alzhemer s Total Alzhemer s robablty = % Total o-alzhemer s robablty = -% = 9% The setu a-ror robabltes (gve) am Alz. 9 ad am oalz. Alz. ad oalz. 9 What we wat to ow f someoe had the amylod rote, what s the robablty they have Alzhemer s? Usg Bayes Theorem Alz am??? Alz am am Alz Alz am But we eed to ow am determe the total robablty am am Alz Alz am oalz oalz am Therefore.9.. Alz am. The dagoss s better tha %, but for comleteess what about the o- Alzhemer s oulato too may for a good test..9 oalz am. 8. too hgh a robablty for a good test Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 8 of 5 ECE 8

39 .8 Combatorcs Some mortat math before dog more robablty From Merram Webster s Dctoary. htt:// Combatoral - of or relatg to the arragemet of, oerato o, ad selecto of dscrete mathematcal elemets belogg to fte sets or mag u geometrc cofguratos. For a oulato of sze the set cotas elemets (a dec of 5 layg cards) A suboulato of sze r ca be defed (draw 5 cards at radom from the dec) How may uque suboulatos of r ca we exects (otce that the same r elemets ca be selected umerous ways). () Samlg wth relacemet s the easy way Possble combatos r () Samlg wthout relacemet Possble combatos r! r! Next cosderatos how may ways ca the r thgs be selected Possble selectos r r r r! Now we ca cosder the uque combatos Uque combatos PossbleCombato PossbleSelecto! r! r! We have ow defed a oerator to determe uque values for choose r C r! also sometmes show as r r C r C, r! r!! r! r! Ths s also called a bomal coeffcet. There s also some mortat deftos! C r ad!! C r!!! Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 9 of 5 ECE 8

40 Why are they called bomal coeffcets? Bomal Powers x or x y - the coeffcets for the varous ower resultg from summed elemets to the th ower. x y x y The coeffcets ca also be selected usg Pascal s Tragle also called Bomal Exaso Each row starts (ad eds) wth ad the sums the adjacet coeffcets from the ext hgher row. So, by secto a b a a b a b a b b. Now, f a fl a co tmes, what are the ossble combatos ad how may tmes do they occur? What f a sad that: a=heads ad b=tals H^ H^xT^ H^xT^ H^xT^ T^ Also of ote Lettg x=y= x y x y Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

41 Multomal Coeffcets Theorem.8-: Let cossts of multle subsets, each of r elemets such that K r The umber of ways whch the oulato of elemets ca be arttoed to K suboulatos of whch each cotas r elemet s! r! r! r! r K! 5 Card Draw Combatoral How may ways ca 5 cards be draw from a dec of 5 layg cards? C r r! r! r! 5 5 5!,598,9 5 5!5! If you are oer layer see htts://e.weda.org/w/poer_robablty Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

42 .9 Beroull Trals A reeated tral ca tae the form of:. Reeated exermets where the relatve frequecy of occurrece s of terest. The creato of a ew exermet that cossts of a defed umber of elemetary evets Beroull Trals: Determg the robablty that a evet occurs tmes deedet trals of a exermet. For some exermet let: A ad A q where q The for a exermet where we get evet A s followed by ot A (.e., B AAAA ) B A A A A q But what about the other ways to have evet A s trals? Note that for each stace, the robablty of occurrg wll be the same as just defed so how may of them are there? AA AA, AAAA, AAAA, AAAA, AAAA, AAAA The umber of occurreces ca be defed usg bomal coeffcets ad the Bomal Theorem. The umber of staces s defed by the bomal coeffcet, C or. the umber of ways to select elemets out of a set of elemets... Where!!! Therefore, to descrbe the desred outcome of A s trals, the robablty s Therefore A occurg tmes trals q!! q Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

43 Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8 Beroull Trals The robablty that a evet occurs tmes deedet trals of a exermet ca be defed as q Aoccurg tmes trals Examle Flg Cos The robablty for each outcome of flg a co tmes, where (H)= ad (T)=q wth q H : q HHHH H & T: q HHHT H & T: q HHTT H & T: q HTTT T: q TTTT What f =. ad q=.? A ufar co!!

44 Examle Bary Commucatos Examle : For a bt-error-rate (BER) of a bary data stream, what s the robablty of error a -bt word?.995. Examle : For a bt-error-rate (BER) of a bary data stream, what s the robablty of errors a -bt word?. 985 Examle : What s that robablty of havg oe or more errors bts? or Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

45 Power Ball Lottery The lottery s a 9 choose 5 game combed wth a choose game. see MATLAB code Power Ball total combatos = 98. ob of + balls s.59 m ayout s $. ob of + balls s 8.9 m ayout s $. ob of + balls s.9 m ayout s $. ob of + balls s 9.95 m ayout s $. ob of + balls s 8.5 m ayout s $. ob of + balls s.8 m ayout s $. ob of + balls s 59.5 m ayout s $. ob of + balls s 9. m ayout s $. ob of + balls s 55. m ayout s $. ob of + balls s 99 m ayout s $5. ob of 5+ balls s.88e+ m ayout s $. ob of 5+ balls s.9e+8 m ayout s $. ob of wg somethg s.8. Exected Wgs er $ wthout Jacot = $. Exected Wgs er $ m $M Jacot = $. Sgle Wer Brea Eve Jacot = $9,9,8. Total comuted wthout cosderg taxes. Max US tax rate 9.%, MI tax rate.5%)!! Sgle Wer Brea Eve Jacot wth taxes= $88,88,. Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 5 of 5 ECE 8

46 Examle Baseball/Softball Statstcs Examle : A batter has a.5 battg average. What s the robablty that the batter gets ht at bats? Aoccurg tmes trals q.5. 5!.5.5.!! Examle : A batter has a.5 battg average. What s the robablty that the batter gets ht at bats?.5. 5!.5.5.!! 8 Examle : A batter has a.5 battg average. What s the robablty that the batter gets at least ht at bats?! !! 5 5 Examle : A batter has a.5 battg average. What s the robablty that the batter gets at most ht at bats?!!!.5.5!!! Defg a layer havg a httg slum how may at bats utl t s a slum? How may at bats would the batter eed to tae to have a 9% (or 99%) robablty of gettg at least oe ht. Cabrera s average.was.? (see Excel Sread Sheet m. 9 m Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8

47 Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall of 5 ECE 8 Examle -. from Cooer-McGllem I layg a ooet of equal ablty, whch s more robable: q Aoccurg tmes trals a) To w games out of, or to w 5 games out of 9? 5..5!!! q !! 9! q Therefore, wg out of s more robable. b) To w at least games out of, or to w at least 5 games out of The robabltes are the same! (You should have a 5-5 chace of wg or losg)!

48 Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 8 of 5 ECE 8. Asymtotc Behavor of the Bomal Law For Beroull Trals or Bomal Law exactly successes trals q b Aoccurg tmes trals, ; For Summato Bomal Law or fewer successes trals q b B, ;, ; Whe gets large, there are some aroxmatos that ca be used Why aroxmate, the combatoral fucto ca cause calculators ad comuters to loose umercal recso ad roduce correct results f they roduce results at all. Posso robablty mass fucto (mf) ad aroxmato from future chaters Codtos:,,, but s a costat term, the q b!, ; The, f we cosder a fte umber of trals that s ex!!, ; b

49 Examle.-. Comuter Comoet Falure = umber of comoets =, = comoet falure rate er year = - Assumg the comuter fals f or more comoet fals. usg the aroxmato What s the robablty the comuter wll stll be worg oe year from ow? The robablty of falures s b b!! ;, ex ;, ex ex b! ;, ex. 8 Examle.-. Radom ots tme Suose deedet ots (evets) are laced radomly tme from to T. We wat to observe the terval for a short erod. t t T. What s the robablty of observg exactly ots (evets) the terval? Gut reactos to the exercse: For t t you would exect the umber of ots =!! T Setu bomal b ;, ex Lettg ad T b! T! ;, ex ex T Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 9 of 5 ECE 8

50 Alcato of Posso Physcs: radoactve decay Telecommucatos: lag the sze of a telehoe call ceter or server farm for the teret Bology: water olluto or orgasm motorg Otcs: desgg otmal recevers based o hotos receved er secod Examle.-. Web Server O the average, assume there are access request er mute. If the server ca hadle at most accesses er mute, what s the robablty that ay oe mute terval that the web ste would be saturated? b ;,! ex ad = to B osaturato B saturato 5 ex! ex! 5!! ex ex See Matlab soluto t does ot agree wth the textboo result?! The rob of satuarato s.55. The rob of o satuarato s.985. Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 5 of 5 ECE 8

51 . Normal Aroxmato to the Bomal Law (DeMovre-Lalace Theorem) Ths s a aroxmato of the bomal dstrbuto whe the umber of trals ( of choose ) s large ad other assumtos are met. Assumtos: q ad q q q ex q [Asde: we wll be dscussg the law of large umbers ad that sums of larger umbers of evets aear as a Gaussa dstrbuto. Ths s the frst examle ad you have t bee told what a Gaussa dstrbuto s yet.] Text examle: co tosses, equally lely head or tal, robablty of heads..5 ex Note: assumtos vald for <= <=. The robablty for exactly 5 heads would be ex Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 5 of 5 ECE 8

52 ECE Alcatos of Beroull Trals () Bt errors bary trasmssos: Degree of error detecto ad correcto eeded. The theoretcal valdato of erformace of the system after extra bts for error correcto have bee added. bt-error-rate may also crease f a greater badwdth s eeded because of the extra bts () Radar (or smlar) sgal detecto: After settg a sgal detecto threshold, the exected sgal should be above the threshold whe beg receved for a fxed umber of samle tmes. If the sgal s above the threshold for m (or more) of samle erods, oe may also say the sgal has bee detected. Detecto, m s m s s Oe ca also defe a ose threshold where the ose should ot be above a artcularly level more tha m (or more) of tme samles. False _ Alarm, m a m () System relablty mrovemet usg redudacy. a a If a ut has a ow falure rate, by cororatg redudat uts, the system wll have a loger exected lfetme. Imortat whe dealg wth systems that caot be servced, systems that may be very exesve to servce, systems that requre very hgh relablty, system wth comoets wth hgh falure rates, etc.. (e.g. satelltes, comuter hard-ds farms, teret order etry servers). Defg the robablty that oe of the redudat elemets s stll worg Fuctoal ( All _ Faled Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 5 of 5 ECE 8

53 Hyergeometrc Dstrbuto related formato From: htt://e.weda.org/w/hyergeometrc_dstrbuto I robablty theory ad statstcs, the hyergeometrc dstrbuto s a dscrete robablty dstrbuto (robablty mass fucto) that descrbes the umber of successes a sequece of draws from a fte oulato wthout relacemet. A tycal examle s the followg: There s a shmet of N objects whch D are defectve. The hyergeometrc dstrbuto descrbes the robablty that a samle of dstctve objects draw from the shmet exactly x objects are defectve. x X, N, D, D x N N D x for max, D N x m, D The equato s derved based o a o-relacemet Beroull Trals Where the deomator term defes the umber of tral ossbltes, the st umerator term defes the umber of ways to acheve the desred x, ad the d umerator term defes the fllg of the remader of the set. Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 5 of 5 ECE 8

54 Qualty Cotrol Examle A batch of 5 tems cotas defectve tems. Suose tems are selected at radom ad tested. What s the robablty that exactly 5 of the tems tested are defectve? The umber of ways of selectg tems out of a batch of 5 s the umber of combatos of sze from a set of 5 objects: C 5 5 5!!! The umber of ways of selectg 5 defectve ad 5 odefectve tems from the batch of 5 s the roduct N x N where N s the umber of ways of selectg the 5 tems from the set of defectve tems, ad N s the umber of ways of selectg 5 tems from the odefectve tems. C 5 C5!! 5!5! 5!5! Thus the robablty that exactly 5 tested tems are defectve s the desred ways the selecto ca be made dvded by the total umber of ways selecto ca be made, or C C C! 5!5!! 5!5! 5!!! Aother Use: From The Mesota State Lottery a better descrto tha Mchga There are N objects whch D are of terest. The hyergeometrc dstrbuto descrbes the robablty that a samle of dstctve objects draw from the total set exactly x objects are of terest. Lotteres N= umber of balls to be selected at radom D = the balls that you wat selected = the umber of balls draw x = the umber of desred balls the set that s draw htts:// Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 5 of 5 ECE 8

55 Examle: Mchga s Classc Lotto ze Structure For Classc Lotto (web ste data) Match ze Odds of Wg of Jacot,,5 5 of $,5 (guarateed),9 of $ (guarateed) 8 of $5 (guarateed) 5 Overall Odds: Matlab Odds Match Odds of Wg Percet obablty of 5 <x -5 % 5 of 8..% of 8.9.% of % of..% of.9.85% of.85% Chace of wg.% ROI er dollar wthout jacot ~ $. see Matlab smulato MI_Lotto.m Matlab Note: bomal coeffcet = choose(,) Keo ayoe? Aother Mchga gamblg game 8 balls, draw, you eed to match of selected for = to. Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 55 of 5 ECE 8

56 MI Keo A Keo tcet wth the ayouts s show! Aother hyergeometrc desty fucto N= umber of balls to be selected at radom (8) D = the balls that you wat selected (D) = the umber of balls draw () x = the umber of desred balls the set that s draw (:D) MI Keo ROI ad [w] ROI er $ [w] I geeral, you get $.5 bac for every $ layed. I dd ot clude a cer bet. The overall odds of a Kcer (,,,, 5, ) umber beg or hgher are :.. see MI_Keo.m o the web ste for more formato ad results. Examles for Chater more otes o le. Notes ad fgures are based o or tae from materals the course textboo: obablty, Statstcs ad Radom ocesses for Egeers, th ed., Hery Star ad Joh W. Woods, Pearso Educato, Ic.,. B.J. Bazu, Fall 5 of 5 ECE 8