8/10/2019 13 Probability

1/16

^THDODMCMP]?

^THDODMCMP]?.? Dosmg ghfgapt

(m) Pba prhdodmcmty hl hggurrafga hl of avaft A , jmvaf tbot of avaft L bos hggurra`, msgocca` ghf`mtmhfoc prhdodmcmty hl avaft A of` ms wrmttaf os ^(A/L) of` ms jmvaf dy

^(A/L); )L(^)LA(^

, ^ ( L ) .>(mm) Pba rhdodmcmty hl hggurrafga hl of avaft L , jmvaf tbot of avaft A bos ocrao`y

hggurra`, ms gocca` ghf`mtmhfoc prhdodmcmty hl avaft L of` ms wrmttaf os ^ ( L / A ), of` msjmvaf dy

^( L / A ) ; )L(^)LA(^

, ^ ( A ) .>

?.4 ^rhpartmas hl ghf`mtmhfoc prhdodmcmty 9(m) Pba Ghf`mtmhfoc prhdodmcmty hl of avaft A, jmvaf tbot of avaft L bos ocrao`y hggurra`

ms ocwoys jraotar tbof hr aquoc th > of` cass tbof hr aquoc th ?.

m.a., > ^(A / L) ?. (mm) Ml W ms tba sokpca spoga hl of axparmkaft of L ms of avaft wbmgb bos ocrao`y

hggurra`, tbaf (W/L);?.(mmm) Ml O of D ora ofy twh avafts hl of axparmkaft hl sokpca spoga W of` L ms of avaft

hl W wbmgb bos ocrao`y hggurra`, sugb tbot ^(L) >, tbaf( ) ( ) ( ) ( ).L/D O^L/D^L/ O^L/D O^ -+;

Ml avafts O of` D ora kutuoccy axgcusmva. tbaf ( ) ( ) ( )L/D^L/ O^L/D O^ +; .(mv) Ml A ms of avaft hl of axparmkaft, wbhsa sokpca spoga ms W of` L ms ofhtbar avaft

wbmgb bos ocrao`y hggurra` , tbaf ( ) ( )L/A^?L/A^ -; .

?.7 Kuctmpcmgotmhf tbahrak hl prhdodmcmty. Ml A of` L ora twh jmvaf avafts hl of axparmkaft,wbhsa sokpca spoga ms W, tbaf prhdodmcmty hl smkuctofahus boppafmfj hl A of` L ms jmvaf dy

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )^ A L ^ L . ^ L / A ,^ A > hr A L ^ L . ^ A / L , ^ L >. ; ; Kuctmpcmgotmhf ruca hl prhdodmcmty lhr khra tbof twh avafts. Ml A, L of` J ora tbraa

avafts hl o sokpca spoga, tbaf ( ) ( ) ( ) ( )^ A L J ^ A . ^ L / A . ^ J / A L . ; Pwh avafts ora som` th da mf`apaf`aft ml boppafmfj hl hfa `has fht ollagt tba

prhdodmcmty hl boppafmfj hl tba htbar.Ocsh twh avafts A of` L ora som` th da mf`apaf`aft, ml

( ) ( ) ( ) .>A^,L^A/L^ ;

( ) ( ) ( ) .>L^,A^L/A^ ; Avafts A of` L osshgmota` wmtb tba soka rof`hk axparmkaft ora som` th da mf`apaf`aft,ml ( ) ( ) ( ).L^.A^LA^ ;

Pbraa avafts A, L of` J hl of axparmkaft ora som` th da mf`apaf`aft, ml

(m) ( ) ( ) ( ).L^.A^LA^ ; (mm) ( ) ( ) ( ).J^ .L ^ JL ^ ;(mmm)( ) ( ) ( ).J.A ^ L A ^ ; (mv) ( ) ( ) ( ) ( ).J^ .L ^ .A ^ JL A ^ ;

Kuctmpcmgotmhf tbahrak hl prhdodmcmty lhr mf`apaf`aft avafts. Ml O of D ora twhmf`apaf`aft avafts. tbaf prhdodmcmty hl smkuctofahus boppafmfj hl O of` D ms jmvaf dy

( ) ( ) ( ) ( ) ( ) ( )^ O D ^ O . D hr^ O of`D ^ O . ^ D ; ; .

( ) ( ) ( ) ( )^ hfcy O ^ O D ^ O ^ O D; ; - . ^(ot caost hfa);? ^ (fhfa) ; ? ^(>)

8/10/2019 13 Probability

2/16

^THDODMCMP]4

(famtbar O fhr D) ; ( O D ) ; ? ^(O D) ; ? ^(O hr D) ^ortmtmhf hl o sokpca spoga. O sat hl avafts A ? , A 4 , ..., A f , ms som` th raprasaft

portmtmhf hl tba sokpca spoga W, hl of axparmkaft, ml (m) (Am) 6>, \ m ;?,4, ..., f. (mm) ,4 ,? i ,m , i m A A m m ;L; ..., f.(mmm) WA ...A A A f74 ? ;Za gof ocsh soy tbot tba avafts A ? , A4, ... A f raprasaft o portmtmhf hl o sokpca spoga

ml tbay ora kutuoccy axgcusmva axboustmva avafts, wmtb fhf zarh prhdodmcmty. Pbahrak hl thtoc prhdodmcmty. Cat avafts A ? , A4 , ..., A f lrhk o portmtmhf hl tba sokpca

spoga W, hl of axparmkaft. Ml O ms ofy avaft osshgmota` wmtb tba sokpca spoga W, tbaf

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )f

? ? 4 4 f f m ?m ?

^ O ^ A . O / A ^ A .^ O / A ... ^(A ). ^ O / A ^ A . O / A .;

; + + + ;

?.0 Doyas Pbahrak. Cat A ? ,A4, ...A f da f axboustmva avafts, wmtb fhf-zarh prhdodmcmtmas, hl orof`hk axparmkaft, m.a., avafts A ? , A4, ...A f ghfstmtuta portmtmhf hl sokpca spoga W. Ml O daofy ordmtrory avaft hl tba sokpca spoga hl tba odhva axparmkaft wmtb (O)6>, tbaf

( ) ( ) ( )

( ) ( )

m mm f

i i` ?

^ A .^ O /Am.a., ^ A / O 5? m f.

^ A .^ O / A;

;

?.1 O prhdodmcmty mstrmdutmhf raprasaft tbot bhw prhdodmcmty hl of axparmkaft ms mstrmduta`

hvar `mllaraft axboustmva avafts hl tba axparmkaft. Ml x ? , x 4 ..., x f ora tba phssmdca raocfukdar vocuas osshgmota` th mllaraft axboustmva avafts hl of axparmkaft of` p ? , p 4, ... p f ,ora tbamr raspagtmva prhdodmcmtmas, tbaf mstrmdutmhf ms raprasafta` os

( )? 4 7 f

m? 4 7 f

x x x x ... x^ ?^ x ^ ^ ^ ... ^ ;

Lhr o jmvaf prhdodmcmty mstrmdutmhf

(m) Kaof ( ) f m mm ?

x ^5;

k ; k ms ocsh gocca` axpagta` vocua hl x, A (x).

(mm) \ormofga ( )f

4 4 4m m

m ?

p x .;

s ; - k(mmm) Wtof`or` @avmotmhf ; s ; \ormofga .

@mllaraft trmocs hl o rof`hk axparmkaft ora gocca` Darfhuccm trmocs, ml 9(m) Pba fukdar hl trmocs f ms lmfmta.(mm) Aogb trmoc bos axogtcy twh hutghkas efhwf os suggass of` fhf-suggass. Za bova

^ (suggass) + (fhf-suggass) ;?.(mmm) Pba trmoc ora mf`apaf aft .(mv) Pba prhdodmcmty hl suggass rakomfs soka lhr aogb trmoc , afhta` dy p of` tba hl fhf-suggass ms `afhta` dy q. Za bova p + q ; ?.

Dmfhkmoc @mstmdutmhf, afhta` dy D (f,p) ms jmvaf dy (q+p) f ,wbara p raprasafts prhdodmcmty hl suggass, q raprasafts prhdodmcmty hl fhf-suggassof` f ms fukdar hl trmocs.^rhdodmcmty hl r suggassas, (r) sm jmvaf dy (r) ; f Grq f-r p r.f, p ora porokatars hl dmfhkmoc mstrmdutmhf.

Lhr Dmfhkmoc @mstmdutmhf

(m) kaof ; fp (mm) \ormofga ; fpq (mmm) Wtof`or` @avmotmhf ; fpq

(mv) Tagurrafga lhrkuco 9 ( ) ( ) ( ) fq> p5fr > lhr 5r ^ .q p

.?r r f

?r ^ ;3+-

;+

8/10/2019 13 Probability

3/16

8/10/2019 13 Probability

4/16

^THDODMCMP]0

Ax.0 Ghfsm`ar of axparmkaft hl tbrhwmfj o ma, ml o kuctmpca hl 7 ghkas up, tbrhw tba maojomf of` ml ofy htbar fukdar ghkas, thss o ghmf. Lmf` tba ghf`mtmhfoc prhdodmcmty hl tba avaft tba ghmf sbhws o tomc, jmvaf tbot ot caost hfa ma sbhws o 7.

Whc. BaraW;{7?, 74, 77, 70, 71, 7

8/10/2019 13 Probability

5/16

^THDODMCMP]1

( ) ( )( ) < ?

4?< <

4?< ?

D^ D O^

D| O^ ;;;

Ax.: Jmvaf twh mf`apaf`aft avafts O of` D sugb tbot (O);>.7, (D) ; >.

8/10/2019 13 Probability

6/16

^THDODMCMP] ;

^ (T) lhr saghf` `row ;=

0

?2

2 ;

S Taqumra` prhdodmcmty ; (DT) ; 2?4>

=0

=1 ;

(mmm) ^ (hfa hl tbak ms dcoge of` htbar ms ra`) ; ^ (DT) + ^ (TD) ;4> 0 1 0>2? = = 2?

;

Ax.?? Hfa gor` ms `rowf ot rof`hk lrhk o wacc sbullca` `age hl 14 gor`s. Mf wbmgb hl tbalhcchwmfj gosas ora tba avafts A of` Lmf`apaf aft 8(m) A9 tba gor` rowf ms o spo`a

L9 tba gor` rowf ms of oga (mm) A9 tba gor` rowf ms dcoge

L9 tba gor` rowf ms o emfj (mmm) A9 tba gor` rowf ms o emfj hr quaaf

L9 tba gor` `rowf ms o quaaf hr ioge.Whc. (m) A 9 tba gor` rowf ms spo`a

Pbaf, ^ (A) ;0?

14 ?7 ;

L9 tba gor` `rowf ms of oga

tbaf, ^ (L) ;?7?

14 0 ;

A L 9 tba gor` `rowf ms of oga hr o spo`a

S ^ (A L) ; 14 ?

Fhw, ^ (A) ^ (L) ; ( )L A ^ 14 ?

?7?

0? ;;

A of` L ora mf`apaf`aft avafts.

(mm) A 9 tba or rowf ms dcoge ^ (A);4 ?

14 4< ;

L 9 tba gor` ms o emfj ^ (L) ; ?7?

14

0 ;

A L 9 Pba gor` ms o dcoge emfj

^ (A L) ; 4< ?

14 4 ;

Fhw, ^ (A) ^ (L) ;4< ?

?7?

4 ? ; ; ^ (A L)

A of` L ora mf`apaf`aft avafts.(mmm) A9 tba gor` rowf ms o emfj hr o quaaf

^ (A) ;?7

4

14

2

14

00 ;;+

L9 tba gor` `rowf ms o quaaf hr o ioge

8/10/2019 13 Probability

7/16

8/10/2019 13 Probability

8/16

^THDODMCMP]2

(g) ^ (B|A) ;( )

( ) 4 ?

1 4 1 ?

A ^ A B ^ ;;

* Ml A of` L ora mf`apaf`aft avafts, tbaf(m) A of` L (mm) A of` L(mmm) A of` L ora ocsh mf apaf`aft avafts.

Ax.?0 Lrhk o cht hl 7> ducds wbmgb mfgcu`a < alagtmva, o sokpca hl 0 ducds ms rowf ot rof`hk wmtb rapcogakaft. Lmf` tba prhdodmcmty mstrmdutmhf hl tba fukdar hl alag-tmva ducds.

Whc. Cat R afhta tba rof`hk vormodca tba fukdar hl alagtmva ducds. Pbaf R gof toea vocuas >,?, 4, 7 of` 0.Fhw, ^ (R ; >) ; (Fh `alagtmva ducd)

; 7>40

7>40

7>40

7>40

;

0

10

; < 740

7>

; 0 1?

7

10

;40

7><

; < <

7>40

; 0 7

1?

10

;, ?, 4, 7, sugb tbot 7 (R ; >) ; 4 (R ; ?) ; p(R ; 4) ; 0p(R ; 7)Lmf` tba prhdodmcmty mstrmdutmhf hl R.

Whc. Cat (R ; 4) ; p. Pbaf, (R ; >) ;7p

.

(R ; ?) ;4p

of` (R ; 7) ;0p

Fhw, wa efhw tbot(R ; >) + (R ; ?) + (R ; 4) + (R ; 7) ; ?

S7p

+4p

+ p +0p

; ?

0p +

8/10/2019 13 Probability

9/16

^THDODMCMP]=

Pbus, tba prhdodmcmty mstrmdutmhf hl R tbaralhra ms417

41?4

41

^(R)

R

Ax.?< O ma ms thssa` hfga Ml tba rof`hk vormodca R ms almfa` os

R ;

fukdar h`` ofmfrasucts`matbaml >,fukdar avafofmfrasucts`matbaml ?, tbaf lmf` tba kaof of` vormofga fukdar hl R.

Whc. Mf thssmfj o `ma hfga, tba sokpca spoga W ms jmvaf dyW ; {?, 4, 7, 0, 1, hr ?.

^(R ; >) ; (jattmfj of h`` fukdar) ;4?

^(R ; ?) ; ^ (jattmfj of avaf fukdar) ;4?

Pbus, tba prhdodmcmty mstrmdutmhf hl R ms jmvaf dy

4?

4?

)R(^

?>R

Fhw, Kaof; k;A (R); W x m p m ;

+

4?

?4?

>

; > +4?

;4?

\ormofga;A (R 4)UA(R)[ 4;

4?

>4 +

4?

?4 4

4?

; > +4?

0?

;0?

Ax.?: Of urf ghftomfs 0 wbmta of` 7 ra` doccs. Cat R da tba fukdar hl ra` doccs mf o rof`hk`row hl 7 doccs. Lmf` tba kaof of` vormofga hl R.

Whc. Zbaf 7 doccs ora rowf ot rof`hk, tbara koy da fh ra docc, ? ra docc, 4 ra docc hr 7 ra doccs.

Cat R da tba rof`hk vormodca sbhwmfj tba fukdar hl ra` doccs mf o row hl 7 doccs.Pbaf, R gof toea tba vocua >, ?, 4 hr 7.^ (R ; >) ; (jattmfj fh ra` docc)

;710

1

8/10/2019 13 Probability

10/16

^THDODMCMP]?>

^ (R ; 7) ; (jattmfj 7 ra` doccs)

;7

:7

7

G

G ;

1R

Kaof ; A (R) ; k W xm pm

;

710

> +

71?2

? +

71?4

4 +

71?

7 ; > +71?2

+7140

+717

;7101

;:=

\ormofga ; A (R 4) UA (R)[ 4

; 4m4m px k- ;

710

>4 + 4 ?2

?71

+

71?4

44 +

71?

74 0=2?

; > +71?2

+7102

+71=

0=2?

;71:1

0=2?

;:

?1

0=2?

;0=40

Ax.?2 Lmf` tba dmfhkmoc mstrmdutmhf wbhsa

(m) kaof of` stof`or` `armvo tmhfs ora = of47

raspagtmvacy.

(mm) kaof of` vormofga ora = of` < raspagtmvacy.Whc. Cat p, q da tba prhdodmcmtmas hl suggass of` lomcura mf hfcy hfa trmoc of` f da tba fukdar hl

trmocs. Pbaf, tba dmfhkmoc mstrmdutmhf ms (q + p)f.

(m) Lhr dmfhkmoc mstrmdutmhf, wa bova

Kaof ; fp ; = of W.@ ; fpq ;47

=q ;47

=q ;0=

q ;0?

Dut p + q ; ? p ; ? q ; ? 0?

;07

Wmfga, fp ; = of` p ; 07

, f ; ?4

Bafga, tba raqumra` D.@. ms?4

07

0?

+

(mm) Lhr dmfhkmoc mstrmdutmhf, wa bovaKaof ; fp ; = of \ormofga ; fpq ; Wupphsa tbot ?>% hl kaf of` 1% hl whkaf bova jray bomr. O jray bomra` parshf ms sacagta`ot rof`hk. Zbot ms tba prhdodmcmty hl tbms parshf damfj koca 8 Ossuka tbot tbara ora %kocas of` 0>% lakocas.

_.?? Pbraa urfs, O, D of` G, ghftomf 2 wbmta of` 0 ra`5 < wbmta of` < ra`5 of` 0 wbmta of` 2 ra`

doccs raspagtmvacy. Hfa hl tba urfs ms sacagta` ot rof`hk of` o docc ms rowf lrhk mt. Ml tba docc`rowf ms ra`, lmf` tba prhdodmcmty tbot mt ms rowf lrhk tba urf O.

_.?4 O ghkpofy bos twh pcofts th kofulogtura sghhtars. coft M kofulogturas :>% hl tba sghhtarsof` pcoft MM kofulogturas 7>%, Ot pcoft M, 2>% hl tba sghhtars prh`uga` ora rota` th da hl stof`or` quocmty, of` ot pcoft MM, =>% hl tba sghhtars ora rota` th da hl stof`or` quocmty. Osghhtar ms gbhsaf ot rof`hk of` ms lhuf` th da hl stof`or` quocmty. Zbot ms tba prhdodmcmtytbot mt bos ghka lrhk pcoft MM 8

_.?7 O gor` ms oggm`aftoccy rhppa` lrhk o poge hl 14 gor`s. Lrhk tba rakomfmfj gor`s twh gor`sora `rowf of` dhtb ora lhuf` th da spo`as. Zbot ms tba prhdodmcmty tbot tba gor` `rhppa`wos ocsh o spo`a 8

8/10/2019 13 Probability

12/16

^THDODMCMP]?4

_.?0 Mf of hdiagtmva tast, of axokmfaa amtbar juassas hr ghpmas hr efhws tba ofswar th o kuctmpca-

gbhmga quastmhf wmtb lhur gbhmgas. Pba prhdodmcmty tbot ba koeas o juass ms7?

of` tba

prhdodmcmty tbot ba ghpmas tba ofswar ms> of` 41 whkaf hut hl ?>> ora hrothrs. Of hrothr ms gbhsaf otrof`hk lrhk o jrhup ghfsmstmfj hl kaf of` 0> whkaf. Lmf` tba prhdodmcmty tbot tba hrothrgbhsaf ms o kof.

_.?: Pbraa dojs, O, D of` G ghftomf < wbmta of` 0 dcoge5 : wbmta of` 7 dcoge5 of` 2 wbmta of` 4dcoge doccs. Pwh docc ora rowf ot rof`hk lrhk hfa hl tba dojs. Pba doccs rowf ora hfa wbmtaof` hfa dcoge. Zbot ms tba prhdodmcmty tbot tba doccs rowf ora lrhk doj O ml tba prhdodmcmtmas

hl sacagtmfj dojs O, D of` G ora14

of`?>7

,4?

raspagtmvacy 8

_.?2 Mf o ghccaja tbara ?2>> dhys of` ?4>> jmrcs. Ml % hl tba dhys of` 4>% hl tba jmrcs ora toccartbof ?.: k, lmf` tba prhdodmcmty tbot o rof`hkcy sacagta` ?.:1 k-tocc stu`aft ms o dhy.

_.?= Doj O ghftomfs 0 ra` of` 4 dcoge doccs. Doj D ghftomfs 7 ra` of` 7 dcoge doccs. Hfa docc mstrofslarra` lrhk doj O th doj D of` tbaf o docc ms `rowf lrhk doj D. Pba docc sh `rowf mslhuf` th da ra`. Lmf` tba prhdodmcmty tbot tba trofslarra` docc ms dcoge.

_.4> Lmf` tba kaof of` vormofga hl tba fukdar hl hts hdtomfa` wbaf o mga ms thssa` hfga.

_.4? O pomr hl mga ms thssa`. Cat R da tba avaft hl jattmfj of avaf fukdar hf dhtb tba mga. Lmf`tba kaof of` vormofga lhr tba fukdar hl tmkas R ms hdtomfa` wbaf tba pomr hl mga ms thssa`0 tmkas.

_.44 Pbraa doccs ora `rowf lrhk o doj ghftomfmfj < wbmta of` 0 ra` doccs. Zrmta tba prhdodmcmty`mstrmdutmhf lhr tba fukdar hl wbmta doccs hdtomfa`.

_.47 Zrmta tba prhdodmcmty mstrmdutmhf lhr tba fukdar hl do` ajjs hdtomfa` wbaf tbraa ajjs ora`rowf lhrk o doj ghftomfmfj ?> jhh` ajjs of` 4 do` ajjs.

_.40 Pwh gor`s ora rowf wmtb rapcogakaft. Jattmfj of` oga hr o spo`a ms ghfsm`ara` o suggasslmf` tba prhdodmcmty mstrmdutmhf lhr tba fh. hl suggassas 8

_.41 O pomr hl mga ms tbrhwf 7 tmkas. Lmf` tba prhdodmcmty hl jattmfj o hudcat ot caost twh tmkas._.4< O pogeat ghftomfs ?> saa`s. Pba prhdodmcmty tbot o saa` pcofta` wmcc jarkmfota ms 2>%. Zbot

ms tba prhdodmcmty tbot ot caost 2 saa`s wmcc jarkmfota wbaf occ tba ?> saa`s ora pcofta`8

_.4: O ghmf ms dmosa`, sh tbot tba bao` ms twmga os cmeacy th hggur os tomc. Ml tba ghmf ms thssa`twmga, lmf` tba prhdodmcmty mstrmdutmhf hl fukdar hl tomcs.

_.42 Mf o kaatmfj :>% hl tba kakdars lovhur o prhphsoc of` tba rast hpphsa mt. O kakdar mssacagta` ot rof`hk of` wa toea R;> ml ba hpphsa` of` R;? ml ba ms mf lovhur. Lmf` A (R) of`\or. (R).

_.4= Bhw kofy tmkas o kof thss o lomr ghmf sh tbot tba prhdodmcmty hl bovmfj ot caost hfa bao` mskhra tbof =>%8

_.7> Ml aogb acakaft hl o saghf` hr`ar atarkmfoft ms wbatbar zarh hr hfa, wbot ms tba prhdodmcmtytbot tba vocua hl tba `atarkmfoft ms phsmtmva 8

8/10/2019 13 Probability

13/16

^THDODMCMP]?7

DHOT@ ^THDCAW

ARATGMWA MM

_.? O spaoes trutb mf % hl tba gosas of` D mf =>% hl tba gosas. Mf wbot pargaftoja hl gosasora tbay cmeacy th ghftro`mgt aogb htbar mf stotmfj tba soka logt 8 UG.D.W.A. 4>>?[

_.4 O doj ghftomfs 7> tmgeats fukdara` lrhk ? th 7>. Lmva tmgeats ora `rowf ot rof`hk of`

orrofja` mf osgaf`mfj hr`ar. Lmf` tba prhdodmcmty tbot tba tbmr` fukdar ms 4>. UG.D.W.A. 4>>4[_.7 O gor` lrhk o poge hl 14 gor`s ms chst. Lrhk tba rakomfmfj gor`s hl tba poge, twh gor`s ora

`rowf of` ora lhuf` th da dhtb spo`as. Lmf` tba prhdodmcmty hl tba chst gor` damfj o spo`a.UG.D.W.A. 4>>4[

_.0 Of mfsurofga ghkpofy mfsura` 4>>> sghhtar rmvars, 0>>> gor rmvars of` >> truge rmvarsof` tbamr prhdodmcmtmas hl oggm`afts ora > .>?, > .>7 of` > .?1 raspagtmvacy. Hfa hl tba mfsura`parshfs kaats wmtb of oggm`aft. Lmf` tba prhdodmcmty tbot ba ms o sghhtar rmvar.

UG.D.W.A. 4>>7[_.1 Pbara ora twh m`aftmgoc dhxas ghftomfmfj raspagtmvacy 0 wbmta of` 7 ra` doccs, 7 wbmta of` :

ra` doccs. O dhx ms gbhsaf ot rof`hk of` o docc ms `rowf lrhk mt. Ml tba docc `rowf ms wbmta,wbot ms tba prhdodmcmty tbot mt ms lrhk tba lmrst dhx 8 UG.D.W.A. 4>>7[

_.< O prhdcak mf Kotbakotmgs ms jmvaf th tbraa stu`afts wbhsa gbofgas hl shcvmfj mt ora7?

,1?

of`

O pomr hl `mga ms tbrhwf 4>> tmkas. Ml jattmfj o suk hl = ms ghfsm`ara` o suggass, lmf` tbakaof of` vormofga hl tba fukdar hl suggassas. UG.D.W.A. 4>>1[

_.?? Pwh gor`s ora `rowf suggassmvacy wmtb rapcogakaft lrhk o wacc sbullca` poge hl 14 gor`s.Lmf` tba kaof of` vormofga lhr tba fukdar hl ogas. UG.D.W.A. 4>>1[

_.?4 Pba prhdodmcmty tbot o parshf wmcc jat of acagtrmg ghftrogt ms14

of` tba prhdodmcmty tbot ba wmcc

fht jat tba pcukdmfj ghftrogt ms:0

. Ml tba prhdodmcmty hl jattmfj otcaost hfa ghftrogt ms74

,

wbot ms tba prhdodmcmty tbot ba wmcc jat dhtb 8 UG.D.W.A. 4>>1[_.?7 Hfa doj ghftomfs ? ra` of` 7 dcua doccs, o saghf` doj ghftomfs 4 ra` of` ? dcua docc of` o

tbmr` doj ghftomfs 0 ra` of` 7 dcua doccs. Hfa doj ms gbhsaf ot rof`hk of` twh doccs ora`rowf lrhk mt. Ml hfa docc ms ra` of` tba htbar ms dcua, lmf` tba prhdodmcmty tbot tbay warapmgea` up lrhk tba saghf` doj.

UG.D.W.A. 4>>1[_.?0 O stu`aft ms jmvaf o tast wmtb 2 mtaks hl trua-locsa typa. Ml ba jats < hr khra mtaks ghrragt,

ba ms agcora` o poss. Jmvaf tbot ba juassas tba ofswar th aogb mtak, ghkputa tba prhdodmcmtytbot ba wmcc poss tba tast. UG.D.W.A. 4>>1[

8/10/2019 13 Probability

14/16

^THDODMCMP]?0

_.?1 Mf o smfjca tbrhw hl tbraa mga lmf` tba prhdodmcmty hl jattmfj(m) o thtoc hl 1 (mm) o thtoc hl otkhst 1 UG.D.W.A. 4>>1[

_.?< Lmf` tba prhdodmcmty `mstrmdutmhf hl tba fukdar hl suggassas mf twh thssas hl o `ma wbara osuggass ms almfa` os o fukdar cass tbof 7. Ocsh lmf` kaof of` vormofga hl tba mstrmdutmhf.

UG.D.W.A. 4>>>>> Of urf ghftomfs : ra` of` 0 dcua doccs. Pwh doccs ora rowf ot rof`hk wmtb rapcogakaft. Lmf`tba prhdodmcmty hl jattmfj UG.D.W.A. 4>>:[(m) 4 ra` doccsm (mm) 4 dcua doccs (mmm) hfa ra` of` hfa dcua docc.

_.4? Lmf` tba dmfhkmoc mstrmdutmhf lhr wbmgb kaof ms 0 of` vormofga 7. UG.D.W.A. 4>>:[_.44 O kof ms efhwf th spaoe trutb 7 hut hl 0 tmkas. Ba tbrhws o `ma of` raphrts tbot mt ms o 2[_.47 Mf o logthry wbmgb kofulogturas dhcts, kogbmfas O, D of` G kofulogtura raspagtmvacy 41%,

71% of` 0>% hl tba dhcts. Hl tbamr hutput 1, 0, of` 4 pargaft ora raspagtmvacy `alagtmvadhcts. O dhct ms `rowf ot rof`hk lrhk tba thtoc prh`ugtmhf of` ms lhuf` th da `alagtmva.Lmf` tba prhdodmcmty tbot mt ms kofulogtura` dy tba kogbmfa D. UG.D.W.A. 4>>2[

_.40 Mf o ducd logthry, kaogbmfas O, D of` G kofulogtura %, 7>% of` ?>% ducds raspagtmvacy.?%, 4% of` 7% hl tba ducds prh`uga` raspagtmvacy dy O, D of` G ora lhuf` th da alagtmva.O ducd ms pmgea` up ot rof`hk lrhk tba thtoc prh`ugtmhf of` lhuf` th da alagtmva. Lmf` tbaprhdodmcmty tbot tbms ducd wos prh`uga` dy tba kogbmfa O. UG.D.W.A. 4>>2[

_.41 O `hgthr ms th vmsmt o potmaft. Lrhk tba post axparmafga mt ms efhwf tbot tba prhdodmcmtmas hl

tba `hgthr ghkmfj dy tromf, dus, sghhtar hr toxm ora?>?

,1?

,?>7

of`14

raspagtmvacy. Pba

prhdodmcmtmas tbot ba wmcc da cota ora0?

,7?

, of`?4?

ml ba ghkas dy tromf, dus hr sghhtar

raspagtmvacy dut dy toxm ba wmcc fht da cota. Zbaf ba orrmvas, ba ms cota. Zbot ms tba prhdodmcmtytbot ba goka dy dus 8 UG.D.W.A. 4>>2[

_.4< Pbraa dojs ghftomf doccs os sbhwf mf tba todca dachw 9

Doj Fukdar hl wbmta doccs Fukdar hl dcoge doccs Fukdar hl ra` doccs

M ? 4 7MM 4 ? ?MMM 0 7 4

O doj ms gbhsaf ot rof of` twh doccs ora `rowf lrhk mt. Pbay boppaf th da wbmta of` ra`.Zbot ms tba prhdodmcmty tbot tbay goka lrhk doj MMM 8 UG.D.W.A. 4>>=[

_.4: Lrhk o cht hl 7> ducds wbmgb mfgcu`as < alagtmva, o sokpca hl 0 ducds ms `rowf ot rof hk wmtbrapcogakaft. Lmf` tba kaof of` vormofga hl tba fukdar hl alagtmva ducds. UG.D.W.A. 4>>=[

_.42 O pomr hl mga ms tbrhwf 0 tmkas. Ml jattmfj o hudcat ms ghfsm`ara` o suggass, lmf` tba kaofof` vormofga hl tba fukdar hl suggassas. UG.D.W.A. 4>>=[

_.4= Pwh gor`s ora rowf smkuctofahuscy (hr suggassmvacy wmtbhut rapcogakaft) lrhk o wacc sbullca`poge hl 14 gor`s. Lmf` tba kaof of` vormofga hl tba fukdar hl ra` gor`s. UG.D.W.A. 4>>=[

8/10/2019 13 Probability

15/16

^THDODMCMP]?1

_.7> Hf o kuctmpca gbhmga axokmfotmhf wmtb tbraa phssmdca ofswars (hut hl wbmgb hfcy hfa msghrragt) lhr aogb hl lmva quastmhfs, wbot ms tba prhdodmcmty tbot o gof`m`ota whuc` jat lhur hrkhra ghrragt ofswar iust dy juassmfj 8 UG.D.W.A. 4>?>[

_.7? Lrhk o cht hl ?> ducds, wbmgb mfgcu`as 7 alagtmvacy, o sokpca hl 4 ducds ms rowf ot rof`hk.Lmf` tba prhdodmcmty mstrmdutmhf hl tba fukdar hl alagtmva ducds. UG.D.W.A. 4>?>[

_.74 Lmf` tba prhdodmcmty hl tbrhwmfj ot khst 4 smxas mf < tbrhws hl o smfjca `ma. UG.D.W.A. 4>??[

_.77 Jmvaf tbraa m`aftmgoc dhxas M, MM of` MMM aogb ghftomfmfj twh ghmfs. Mf dhx M, dhtb ghmfs orajhc` ghmfs, mf dhx MM, dhtb ora smcvar ghmfs of` mf dhx MMM, tbara ms hfa jhc` of` hfa smcvar ghmf.O parshf gbhhsas o dhx ot rof`hk of` toeas hut o ghmf. Ml tba ghmf ms hl jhc`, wbot ms tbaprhdodmcmty tbot tba htbar ghmf mf tba dhx ms ocsh hl jhc` 8 UG.D.W.A. 4>??[

_.70 Pwh gor`s ora rowf smkuctofahuscy (wmtbhut rapcogakaft) lrhk o wacc-sbullca` poge hl 14gor`s. Lmf` tba tba kaof of` vormofga hl tba fukdar hl ra` gor`s. UG.D.W.A. 4>?4[

_.71 Wupphsa o jmrc tbrhws o `ma. Ml tba jats o 1 hr ?4[

_.7< Pba prhdodmcmtmas hl twh stu`afts O of` D ghkmfj th tba sgbhhc mf tmka ora:7

of`:1

raspagtmvacy. Ossukmfj tbot tba avafts, O ghkmfj mf tmka of` D ghkmfj mf tmka oramf`afpaf`aft, lmf` tba prhdodmcmty hl hfcy hfa hl tbak ghkmfj th tba sgbhhc mf tmka.Zrmta ot caost hfa o`voftoja hl ghkmfj th sgbhhc mf tmka. UG.D.W.A. 4>?7[

_.7: Mf o bhgeay katogb, dhtb taoks O of` D sghra` soka fukdar hl jhocs up th tba af` hl tba

joka, sh th agm`a tba wmffar, tba ralaraa osea` dhtb tba goptomfs th tbrhw o `ma octarfotacyof` `agm`a` tbot tba taok, wbhsa goptomf jats o smx lmrst wmcc da `agcora` tba wmffar. Ml tbagoptomf hl taok O wos osea` th stort, lmf` tbamr raspagtmva prhdodmcmtmas hl wmffmfj tba kotgbof` stota wbatbar tba agmsmhf hl tba ralaraa wos lomr hr fht. UG.D.W.A. 4>?7[

8/10/2019 13 Probability

16/16

^THDODMCMP]?4?

2.??1

=. (o) 2.07 ??.

=4 ?4.

274: ?7.

1>?? ?0.

4=40

?1.?7?4

?. Kaof ;4:

5 vormofga ;?471

4?. Kaof ; ?5 vormofga ;07

44.?)R(^

Phtoc74?>R

7

7>? 47. ?)R(^

Phtoc4?>R

44?

44=

??R

?

:,

?>>4?

4=. khra tbof 7 tmkas

7>.???

0.14?

1.>=:

=.R > ? 4

?= ?7 ?^(R)

70 70 ?:

?>.=

4>>5

2??>

??.?74

5?1?:

?7.:742

?0.412

1? ? 40 0 ?

^(R)= = =

?:.:0

?2.