Chapter 3 = 1/3. 6. In both cases the one black ball is equally likely to be in either of the 4 positions. Hence the answer is 1/2.

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1 Chapter roblems {6 dfferet} {6, dfferet}/{dfferet} { st 6,d 6} {s t 6,d 6} / 6 / 6 / 6 / 6 ould also have bee solved by usg redued sample spae for gve that outomes dffer t s the same as askg for the probablty that 6 s hose whe of the umbers,,,,, 6 are radomly hose {6 sum of } {(6, } / 6 /6 {6 sum of } {(6, } / 6 / {6 sum of } {(6, } / 6 / {6 sum of 0} {(6, } / 6 / {6 sum of } {(6, } / 6 / {6 sum of } {E has N S has } { E has, N S has } { N S has } {at least oe 6 sum of } Otherwse twe the probablty gve roblem 6 6 I both ases the oe blak ball s equally lkely to be ether of the postos Hee the aswer s / g ad b at least oe b} / / / 0 Chapter

2 / {A w w} { A w, w} {w} { A w, B w, C w} { A w, B w, C w} {w} 0 /0 BA ( s (a B A s A ( s Whh ould have bee see by otg that, gve the ae of spades s hose, the other ard s equally lkely to be ay of the remag ards, of whh are aes (b B A B ( A ( (a ((( 0 (b Let F deote the evet that she faled the th exam F F (( F F F F E, E E 6 E E E, E E E E Hee, p Chapter

3 Let E be the evet that a radomly hose pregat wome has a etop pregay ad S the evet that the hose perso s a smoker The the problem states that Hee, E S E S, S S E SE/E E S S E S S E S S S S /66 S 6 Wth S beg survval ad C beg C seto of a radomly hose delvery, we have that S S C S C 6( S C C Hee S C D 6, C 0, C D (a DC D C D 0 (b D C DC/C 0/ 6 voted Id Id (a Id voted voted type type (6 (6 6( ( (b {Lb voted} ( {Co voted} 6(0 (6 6( ( ( (6 6( ( 6 (d {voted} (6 6( ( 6 That s, 6 peret of the voters voted Chapter

4 Choose a radom member of the lass Let A be the evet that ths perso atteds the party ad let W be the evet that ths perso s a woma (a W A AW W AW W A M M where M W ( ( (6 Therefore, peret of the attedees were wome (b A ( (6 Therefore, peret of the lass atteded 0 (a F C FC C 0/0 0 (b C F FC/F 0/ /6 0 (a {husbad uder } ( 6/00 6 (b {wfe over husbad over} {both over}/{husbad over} ( /00 (/00 / ( {wfe over husbad uder} 6/ a b ! 6 6 w w trasferred}{w tr} w R tr}{r tr} {w trasferred w} { w w tr} { w tr} { w} / Chapter

5 (a {g g at least oe g } / / / (b Se we have o formato about the ball the ur, the aswer s / 6 Let M be the evet that the perso s male, ad let C be the evet that he or she s olor bld Also, let p deote the proporto of the populato that s male M C C M M C M M C M M (0 p (0 p (00( p Method (b s orret as t wll eable oe to estmate the average umber of workers per ar Method (a gves too muh weght to ars arryg a lot of workers For stae, suppose there are 0 ars, trasportg a sgle worker ad the other arryg workers The of the workers were a ar arryg workers ad so f you radomly hoose a worker the wth probablty / the worker would have bee a ar arryg workers ad wth probablty / the worker would have bee a ar arryg worker Let A deote the evet that the ext ard s the ae of spades ad let B be the evet that t s the two of lubs (a {A} {ext ard s a ae}{a ext ard s a ae} (b Let C be the evet that the two of lubs appeared amog the frst 0 ards B B CC B C C 0 6 Let A be the evet that oe of the fal balls were ever used ad let B deote the evet that of the frst balls hose had prevously bee used The, A A B 0 B 0 A B B A B B A B B Let B ad W be the evets that the marble s blak ad whte, respetvely, ad let B be the evet that box s hose The, B B B B B B B (/(/ (/(/ / W B B (/ (/ B W / W / Chapter

6 Let C be the evet that the tumor s aerous, ad let N be the evet that the dotor does ot all The β C N NC ( N ( NCC ( ( NCC ( ( NC ( C ( α α ( α α α α wth strt equalty uless α Let E be the evet the hld seleted s the eldest, ad let F be the evet that the famly has hldre The, F E EF ( E ( F ( EF ( F ( EF ( p (/ (/ (/ (/ Thus, F E, F E Let V be the evet that the letter s a vowel The E V V E E V E E V A A (/ (/ (/ (/ (/(/ / G C C G G C G G C G G /6 {A superor A far, B poor} { A far, B poor A superor A superor} { A far, B poor} Chapter

7 6 {C woma} {wome {wome A} { A} {wome C} { C} B} { B} {wome C} { C} (a {far h} (b {far hh} ( {tals w} {a o a} {o a, a {o a} ((6 (( 0 0 (6 ( (a (b ( (d Chapter

8 {ae} {ae terhaged seleted} 6 {ae terhaged ot seleted} 6 {A falure} (0( (0( (0( (0( 0 { headed heads} ( ( {th heads} {heads 0 { h th th } { } th th } { } 6 Let M ad F deote, respetvely, the evets that the polyholder s male ad that the polyholder s female Codtog o whh s the ase gves the followg A A A A A A A A m p α p p α p m M α A A M α A f f ( α ( α F( α F( α Hee, we eed to show that p α p [ α > (p m α p f ( α m f or equvaletly, that m f p ( α α p [ α ( a ] > α( αp f p m Chapter

9 Fatorg out α( α gves the equvalet odto m f p p > pf m or (p m p f > 0 whh follows beause p m p f Itutvely, the equalty follows beause gve the formato that the polyholder had a lam year makes t more lkely that t was a type polyholder havg a larger lam probablty That s, the polyholder s more lkely to me male f p m > p f (or more lkely to be female f the equalty s reversed tha wthout ths formato, thus rasg the probablty of a lam the followg year {all whte} 6 { all whte} 6 {all whte} (a {slver other slver foud} { S other, S foud} { S foud} To ompute these probabltes, odto o the abet seleted / { S foud A}/ { S foud B}/ / Let C be the evet that the patet has aer, ad let E be the evet that the test dates a elevated SA level The, wth p C, C E ECC ( ( ECC ( ( EC ( C ( Smlarly, E ( CC ( C E E ( CC ( E ( C C ( p p 6( p Chapter

10 0 Choose a perso at radom {they have adet} {a good}{g} {a ave}{ave} {a bad b} (0( (( (0( ( {A s good o adet} (( {A s average o adet} Let R be the evet that she reeves a ob offer (a R R strogstrog R moderatemoderate R weakweak (( (( (( 6 (b strog R Smlarly, R strog strog R (( moderate R, weak R 6 6 ( strog R Smlarly, R strog strog R (( moderate R, weak R Let M, T, W, Th, F be the evets that the mal s reeved o that day Also, let A be the evet that she s aepted ad R that she s reeted (a M M AA M RR ((6 (0( Chapter

11 (b T M T M T A A T R R M ( (6 (( 6 ( A M T W M T W M T W A A ( 0 (6 ((6 (( (d A Th Th A A Th ((6 ((6 (( (e A o mal o mal A A o mal ((6 ((6 (( Let W ad F be the evets that ompoet works ad that the system futos WF W / W F F F (/ {Boy, F} 6 x {Boy 0 6 x {F} 0 6 x so depedee x x 6 or x A dret hek ow shows that sophomore grls (whh the above shows s eessary s also suffet for depedee of sex ad lass 6 {ew} { ew type } p ( p p 0 Chapter

12 (a p( p (b p ( p ( {up o frst up after } {up frst, up after }/[p ( p] pp( p/[p ( p] / (a All we kow whe the proedure eds s that the two most flps were ether H, T, or T, H Thus, heads H, T H, T or T, H H, T H, T T, H p( p p( p ( p p (b No, wth ths ew proedure the result wll be heads (tals wheever the frst flp s tals (heads Hee, t wll be heads wth probablty p (a /6 (b /6 ( The oly way whh the patter H, H, H, H a our frst s for the frst flps to all be heads, for oe a tal appears t follows that a tal wll preede the frst ru of heads (ad so T, H, H, H wll appear frst Hee, the probablty that T, H, H, H ours frst s /6 60 From the formato of the problem we a olude that both of Smth s parets have oe blue ad oe brow eyed gee Note that at brth, Smth was equally lkely to reeve ether a blue gee or a brow gee from eah paret Let X deote the umber of blue gees that Smth reeved (a {Smth blue gee} {X X } / / / (b Codto o whether Smth has a blue-eyed gee {hld blue} {blue blue gee}(/ {blue o blue}(/ (/(/ / ( Frst ompute {Smth blue hld brow} {hld brow Smth blue}/ / Now odto o whether Smth has a blue gee gve that frst hld has brow eyes {seod hld brow} {brow Smth blue}/ {brow Smth o blue}/ / / / / Chapter

13 6 Beause the o-albo hld has a albo sblg we kow that both ts parets are arrers Hee, the probablty that the o-albo hld s ot a arrer s A, A A, a or a, A or A, A Where the frst gee member eah gee par s from the mother ad the seod from the father Hee, wth probablty / the o-albo hld s a arrer (a Codto o whether the o-albo hld s a arrer Wth C deotg ths evet, ad O the evet that the th offsprg s albo, we have: O O CC O C C (/(/ 0(/ /6 ( (b O O O O O ( O O C C O O C (/ (/ (/ / 6 / 6 0(/ 0 C 6 (a {both ht at least oe ht} {both ht} {at least oe ht} p p /( q q (b {Barb ht at least oe ht} p /( q q Q p, ad we have assumed that the outomes of the shots are depedet 6 Cosder the fal roud of the duel Let q x p x (a {A ot ht} {A ot ht at least oe s ht} {A ot ht, B ht}/{at least oe s ht} q B p A /( q A q B (b {both ht} {both ht at least oe s ht} {both ht}/{at least oe ht} p A p B /( q A q B ( (q A q B ( q A q B (d { rouds A uht} { rouds, A uht}/{a uht} ( qaqb paqb q p /( q q B A (q A q B ( q A q B A B Chapter

14 (e rouds both ht} { rouds both ht}/{both ht} ( qaqb pa pb p p /( q q B A (q A q B ( q A q B Note that (, (d, ad (e all have the same aswer 6 If use (a wll w wth probablty p If use strategy (b the {w} {w both orret}p {w exatly orret}p( p {w ether orret}( p p p( p 0 p Thus, both strateges gve the same probablty of wg 6 (a {orret agree} {orret, agree}/{agree} p /[p ( p ] 6/ / whe p 6 (b / A B 66 (a [I ( ( ] ( (b Let E { ad lose}, E {,, all lose} E {, lose}, E {,, lose} The desred probablty s 6 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E (a ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( p ( p k Chapter

15 6 Let C deote the evet that relay s losed, ad let F be the evet that urret flows from A to B C C F CC F F p ( p p F C C p p C C p p p p p p ( p p p p p p p p p p 6 (a (b (a (b ( 0 (d ( 6 (d 6 0 (a {arrer wthout} / / / / / / (b / {Braves w} {B B ws of } / {B B ws of } / {B B ws of } / {B B ws 0 of } / 6 where {B B ws of } s obtaed by odtog o the outome of the other seres For stae {B B w of } {B D or G w of, B w of } / {B D or G w of, B w of } / By symmetry {D w} {G w} ad as the probabltes must sum to we have {D w} {G w} 6 Chapter

16 Let f deote for ad a agast a erta plae of legslature The stuatos whh a gve steerg ommttees vote s desve are as follows: gve member other members of SC other oul members for both for or agast for oe for, oe agast at least for agast oe for, oe agast at least for agast both for of agast {desve} p p( p p p( p(6p ( p p ( p p ( pp( p(6p ( p p ( p p ( pp p( p (a /6, (b /, ( 0/, (d /, (e / Let A be the probablty that A ws whe A rolls frst, ad let B be the probablty that B ws whe B rolls frst Usg that the sum of the de s wth probablty /, we obta upo odtog o whether A rolls a that Smlarly, A ( B B ( A 6 6 Solvg these equatos gves that A / (ad that B /6 (a The probablty that a famly has sos s /; the probablty that a famly has exatly so s / Therefore, o average, every four famles wll have oe famly wth sos ad two famles wth so Therefore, three out of every four sos wll be eldest sos Aother argumet s to hoose a hld at radom Lettg E be the evet that the hld s a eldest so, lettg S be the evet that t s a so, ad lettg A be the evet that the hld s famly has at least oe so, ES E S S E ( E A E A 0 / Chapter

17 (b Usg the preedg otato ES E S S E ( E A E A / 6 Codto o outome of tal tral Hee, E before F E b F EE E b F FF E b F ether E or F[ E F] E E b F( E F] E b F E E F (a Ths s equal to the odtoal probablty that the frst tral results outome (F gve that t results ether or, gvg the result / More formally, wth L beg the evet that outome s the last to our F L L ( F F ( (/ (/ / L ( / (b Wth S beg the evet that the seod tral results outome, we have F S L L ( FS FS ( (/ (/ /6 L ( / (a Beause there wll be games f eah player ws oe of the frst two games ad the oe of them ws the ext two, games p( p[p ( p ] (b Let A be the evet that A ws Codtog o the outome of the frst two games gves A A a, ap A a, bp( p A b, a( pp A b, b( p p Ap( p where the otato a, b meas, for stae, that A ws the frst ad B ws the seod game The fal equato used that A a, b A b, a A Solvg, gves A p p( p 6 Chapter

18 Eah roll that s ether a or a eve umber wll be a wth probablty p / 6 eve / 6 / / Hee, from Example we see that the desred probablty s (/ (/ (/ (/ 6 (/ 0 (a A (/, f < (/, f (/ (/ (b ( Codto o whether they tally play eah other Ths gves where s the probablty they both w gve they do ot play eah other (d There wll be losers, ad thus that umber of games (e Se the players game are equally lkely to be ay of the B pars t follows that (f Se the evets B are mutually exlusve (/ 0 (/ B ( B ( (/ (a A ( ( [ ] ( smlar to (a wth or A A replag Chapter

19 Chapter (b ad (d Let ( deote the probablty that A ws whe A eeds more ad B eeds more ad A(B s to flp The, ( (, These equatos a be reursvely solved startg wth 0,,0 0 (a Codto o the o flp {throw s red} 6 6 (b {r rr} } { } { rr rrr ( {A rr} } { ( } { rr A A rr / (b A ws B ws C ws art (a remas the same The possbltes for part (b beome more umerous 6 Usg the ht {A B} / / ( 0 0 (/ where the fal equalty uses 0 (

20 (b AB φ A B (/, by part (a, se B s also equally lkely to be ay of the subsets { th all heads} k ( / k 0 ( / k No they are odtoally depedet gve the o seleted (a J votes gulty J ad J vote gulty} {J, J, J all vote gulty}/{j ad J vote gulty} ( 0 ( 0 ( 0 ( 0 (b J gulty oe of J, J votes gulty} ((( ((( 0 0 (( (( ( {J gulty J, J vote oet} (( (( 0 0 ( ( E are odtoally depedet gve the gult or oee of the defedat 0 Let N deote the evet that oe of the trals result outome,, The N N N N N N ( p ( p ( p p Hee, the probablty that both outomes our at least oe s ( p ( p (p 0 Chapter

21 Theoretal Exerses AB A AB AB AB A B A A B If A B A B A, A B 0, B A, B A B BA A Let F be the evet that a frst bor s hose Also, let S be the evet that the famly hose method a s of sze a (F F S S m b (F Thus, we must show that or, equvaletly, or, / m / m Cosderg the oeffets of the term, shows that t s suffet to establsh that or equvaletly whh follows se ( 0 0 Chapter

22 Let N deote the evet that the ball s ot foud a searh of box, ad let B deote the evet that t s box B N N N B B B B N ( α ( α ( α f f B B Noe are true 6 E E [ E ] (a They wll all be whte f the last ball wthdraw from the ur (whe all balls are wthdraw s whte As t s equally lkely to by ay of the m balls the result follows g g b (b RBG RBG G last r b g r b g r b bg b g Hee, the aswer s ( r b( r b g r b g r g (a A A CC A C C > B CC B C C B (b For the evets gve the ht CAA ( ( (/ 6(/ 6 A C / /6 /6 Beause /6 A s a weghted average of A C ad A C, t follows from the result A C > A that A C < A Smlarly, / B C > B > B C However, AB C 0 < AB C A B C /, AB AC BC / But, ABC / 0 A, /6 For k, A, A,k 6/(6 /(6 Also, for k r, A, A k,r /(6 ( p log( /, or, log( p Chapter

23 a ( a s the probablty that the frst head appears o the th flp ad probablty that all flps lad o tals ( a s the Codto o the tal flp If t lads o heads the A wll w wth probablty,m whereas f t lads tals the B wll w wth probablty m, (ad so A wll w wth probablty m, Let N go to fty Example {r suesses before m falures} {r th suess ours before tral m r} m r r r p ( p r r 6 If the frst tral s a suess, the the remag must result a odd umber of suesses, whereas f t s a falure, the the remag must result a eve umber of suesses / (/(/ (/(/ / (/(/(6/ (/(/(/ (/(/(/ / / (b ( Codto o the result of tral to obta ( (d Must show that or equvaletly, that But the rght had sde s equal to ( ( ( Chapter

24 Codto o whe the frst tal ours, p, ( p, 0 α α p ( α ( p α p ( α p ( (b, {A reeves frst votes} (, {A reeves frst ad at least of the ext } ( (,m m, m m (d,m {A always ahead} {A always A reeves last vote} m {A always B reeves last vote} m,m, m m m m m (e The oeture of ( s true whe m (, m 0 Assume t whe m k Now suppose that m k By (d ad the duto hypothess we have that,m m m m m m m m m m whh ompletes the proof p ( ( p (p ( p ( p ( p p by the duto hypothess p ( p p ( p Chapter

25 , / Assume that a,b / whe k a b ad ow suppose a b k Now a,b {last s whte frst a are whte} a b a {last s whte frst b are blak} b a b {last s whte ether frst a are whte or frst b are blak} a! b! a! b! a! b! a b b a ( a b! ( a b! ( a b! a b where the duto hypothess was used to obta the fal odtoal probablty above The probablty that a gve otestat does ot beat all the members of some gve subset of k other otestats s, by depedee, (/ k Therefore B, the probablty that oe of the other k otestats beats all the members of a gve subset of k otestats, s [ (/ k ] k Hee, Boole s equalty we have that B [ (/ ] k k k k k Hee, f [ (/ ] < the there s a postve probablty that oe of the k k evets B our, whh meas that there s a postve probablty that for every set of k otestats there s a otestat who beats eah member of ths set E F EF/F E FGG F EFG FG FG F EFG F E FG G F EFG F The result ow follows se EF EFG EFG E, E,, E are odtoally depedet gve F f for all subsets,, r of,,, r ( E E F E r F Chapter

26 Not true Let F E {ext m heads frst heads} {frst m are heads}/frst heads} 0 p m dp 0 p dp m Chapter

Section 2:00 ~ 2:50 pm Thursday in Maryland 202 Sep. 29, 2005

Section 2:00 ~ 2:50 pm Thursday in Maryland 202 Sep. 29, 2005 Seto 2:00 ~ 2:50 pm Thursday Marylad 202 Sep. 29, 2005. Homework assgmets set ad 2 revews: Set : P. A box otas 3 marbles, red, gree, ad blue. Cosder a expermet that ossts of takg marble from the box, the