# probability theory e

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• 8/12/2019 Probability Theory E

1/25

DHYJR

"dha`sjludhrpjys`bs.`a" 8

Wreoho`n`ty

Yjcrc hrc vhr`eus pjcaedcah `a ahturc, nchi`ak te ha eutbedc, wj`bj bhaaet oc prci`btci hpr`er`

c.k. `a tess`ak ef h be`a, h jchi er h th`n dhy rcsunt. Wreoho`n ty tjcery h`ds ht dchsur`ak tjcuabcrth`at`cs ef subj eutbedcs.

@ @dper that tcrd`ae neky =(`) _haied cxpcr `dcat =@t `s h prebcss wj`bj rcsunts `a ha eutbedc wj`bj `s eac ef tjc vhr`eus pess`onc eutbedcs tjht hrc

laewa te us ocferc jhai c.k. tjrew`ak ef h i`c `s h rhaied cxpcr`dcat hs `t nchis te fhnn ef eac

ef tjc eutbedc fred {8, ?, 2, 9, 3, 6}. R`d`nhrny thl`ak h bhri fred h phbl ef 3? bhris `s hnse h rhaied

cxpcr`dcat.

(``) Rhdpnc sphb c =@t `s tjc sct ef hnn pess`onc eutbedcs ef h rhaied cxpcr`dcat c.k. {J, Y} `s tjc shdpnc sphbc hsseb`htci

w`tj tess`ak ef h be`a.

@a sct aetht`ea `t bha oc `atcrprctci hs tjc ua`vcrshn sct.

Cxhdpnc # 8 = [r`tc tjc shdpnc sphbc ef tjc cxpcr`dcat H be`a `s tessci hai h i`c `s tjrewa.

Renut`ea = Yjc shdpnc sphbc R 4 {J8, J?, J2, J9, J3, J6, Y8, Y?, Y2, Y9, Y3, Y6}.

Cxhdpnc # ? = [r`tc tjc shdpnc sphbc ef tjc cxpcr`dcat H be`a `s tessci, `f `t sjews jchi h be`a tessci

hkh`a cnsc h i`c `s tjrewa.

Renut`ea = Yjc shdpnc sphbc R 4 {JJ, JY, Y8, Y?, Y2, Y9, Y3, Y6}

Cxhdpnc # 2 = F`ai tjc shdpnc sphbc hsseb`htci w tj tjc cxpcr`dcat ef renn`ak h ph`r ef i bc (pnurhn ef i`c) eabc.

Hnse f`ai tjc audocr ef cncdcats ef tjc shdpnc sphbc.

Renut`ea = Nct eac i`c oc onuc hai tjc etjcr oc krcca. Ruppesc 8 hppchrs ea onuc i`c hai ? hppchrs ea krcca

i`c. [c icaetc tj`s eutbedc oy ha ericrci ph`r (8, ?). R`d`nhrny, `f 2 hppchrs ea onuc i`c hai 3

hppchrs ea krcca i`c, wc icaetc tj`s eutbedc oy (2, 3) hai se ea. Yjus, chbj eutbedc bha oc

icaetci oy ha ericrci ph`r (x, y), wjcrc x `s tjc audocr hppchrci ea tjc f`rst i`c (onuc i`c) hai

y hppchrci ea tjc scbeai i`c (krcca i`c). Yjus, tjc shdpnc sphbc `s k`vca oy

R 4 {(x, y) x `s tjc audocr ea onuc i`c hai y `s tjc audocr ea krcy i`c}

[c aew n`st hnn tjc pess`onc eutbedcs (f`kurc)

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"dha`sjludhrpjys`bs.`a" ?

8 ? 2 9 3 6

8 (8, 8) (8, ?) (8, 2) (8, 9) (8, 3) (8, 6)

? (?, 8) (?, ?) (?, 2) (?, 9) (?, 3) (?, 6)

2 (2, 8) (2, ?) (2, 2) (2, 9) (2, 3) (2, 6)9 (9, 8) (9, ?) (9, 2) (9, 9) (9, 3) (9, 6)

3 (3, 8) (3, ?) (3, 2) (3, 9) (3, 3) (3, 6)

6 (6, 8) (6, ?) (6, 2) (6, 9) (6, 3) (6, 6)

F`kurc

Audocr ef cncdcats (eutbedcs) ef tjc hoevc shdpnc sphbc `s 6 6 `.c., 26

Rcnf prhbt`bc preoncds =

(8) H be`a `s tessci tw`bc, `f tjc scbeai tjrew rcsunts `a jchi, h i`c `s tjrewa tjca wr`tc shdpnc

sphbc ef tjc cxpcr`dcat.

(?) Ha ura beath`as 2 rci ohnns hai ? onuc ohnns. [r`tc shdpnc sphbc ef tjc cxpcr`dcat Rcncbt`ea

ef h ohnn fred tjc ura ht rhaied.

Haswcrs = (8) {JY, YY, JJ8, JJ?, JJ2, JJ9, JJ3, JJ6, YJ8, YJ?, YJ2, YJ9, YJ3, YJ6}.

(?) {_8, _

?, _

2, O

8, O

?}. (Jcrc tjc ohnns hrc i`st`aku`sjci fred eac hai etjcr oy

ahd`ak rci ohnns hs _8, _

?hai _

2 hai tjc onuc ohnns hs O

8hai O

?.)

(```) Cvcat =@t `s suosct ef shdpnc sphbc. c.k. kctt`ak h jchi `a tess`ak h be`a er kctt`ak h pr`dc audocr `a

tjrew`ak h i`c. @a kcacrhn `f h shdpnc sphbc beas`sts a cncdcats, tjca h dhx`dud ef ?

a

cvcatsbha oc hsseb`htci w`tj `t.

(`v) Be dpnc dcat ef cvc at =Yjc bedpncdcat ef ha cvcat H w`tj rcspcbt te h shdpnc sphbc R `s tjc sct ef hnn cncdcats ef R wj`bj

hrc aet `a H. @t `s usuhnny icaetci oy H, H er HHB.

(v) R`dpnc cvc at =@f ha cvcat bevcrs eany eac pe`at ef shdpnc sphbc, tjca `t `s bhnnci h s`dpnc cvcat c.k. kctt`ak h jchi

fennewci oy h th`n `a tjrew`ak ef h be`a ? t`dcs `s h s`dpnc cvcat.

(v`) B edpeua i cvca t =[jca twe er derc tjha twe cvcats ebbur s`dunthaceusny, tjc cvcat `s sh`i te oc h bedpeuai cvcat.

Rydoen`bhnny H O er HO rcprcscat tjc ebburrcabc ef oetj H & O s`dunthaceusny.

Aetc = H O er H + O rcprcscat tjc ebburrcabc ef c`tjcr H er O.

Cxhdpnc # 9 =[r`tc iewa hnn tjc cvcats ef tjc cxpcr`dcat tess`ak ef h be`a.

Renut`ea = R 4 {J, Y}tjc cvcats hrc , {J}, {Y}, {J, Y}

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DHYJR

"dha`sjludhrpjys`bs.`a" 2

Cxhdpnc # 3 = H i`c `s tjrewa. Nct H oc tjc cvcat ha eii audocr turas up hai O oc tjc cvcat h audocr

i`v`s`onc oy 2 turas up. [r`tc tjc cvcats (h) H er O (o) H hai O

Renut`ea = H 4 {8, 2, 3}, O 4 {2, 6}

H er O 4 H O 4 {8, 2, 3, 6}H hai O 4 H O 4 {2}

Rcnf prhbt`bc preoncds =

(2) H be`a s tessci hai h i`c `s tjrewa. Nct H oc tjc cvcat J turas up ea tjc be`a hai eii audocr

turas up ea tjc i`c hai O oc tjc cvcat Y turas up ea tjc be`a hai ha cvca audocr turas up

ea tjc i`c. [r`tc tjc cvcats (h) H er O (o) H hai O.

(9) @a tess`ak ef twe be`as, nct H 4 {JJ, JY} hai O 4 {JY, YY}. Yjca wr tc tjc cvcats

(h) H er O (o) H hai O.

Haswcrs = (2) (h) {J8, J2, J 3, Y?, Y9, Y6} (o) (9) (h) {JJ, JY, YY} (o) {JY}

(v``) C quhnny n`lc ny c vcat s =@f cvcats jhvc shdc bjhabc ef ebburrcabc, tjca tjcy hrc sh`i te oc cquhnny n`lcny.

c. k

(`) @a h s`aknc tess ef h fh`r be`a, tjc cvcats {J} hai {Y} hrc cquhnny n lcny.

(` ) @a h s`aknc tjrew ef ha uao`hsci i`c tjc cvcats {8}, {?}, {2} hai {9}, hrc cquhnny n`lcny.

(` `) @a tess`ak h o`hsci be`a tjc cvcats {J} hai {Y} hrc aet cquhnny n`lcny.

(v```) Du tuhnn y cxb nus` vc / i` sme `at / ` abedpht `o nc cvca ts =Ywe cvcats hrc sh`i te oc dutuhnny cxbnus`vc `f ebburrcabc ef eac ef tjcd rcmcbts tjc pess`o`n`ty ef

ebburrcabc ef tjc etjcr `.c. oetj bhaaet ebbur s`dunthaceusny.

@a tjc vc`a i`hkrhd tjc cvcats H hai O hrc dutuhnny cxbnus`vc. Dhtjcdht`bhnny, wc wr`tc

H O 4 Cvcats H

8, H

?, H

2, ....... H

ahrc sh`i te oc dutuhnny cxbnus`vc cvcats `ff

H` H

m 4 `, m {8, ?, ..., a} wjcrc ` m

Aetc = @f H` H

m4 `, m {8, ?, ..., a} wjcrc ` m, tjca H

8 H

? H

2 .... H

a 4 out beavcrsc

acci aet te oc truc.

Cxhdpnc # 6 =@a h s`aknc tess ef h be`a f`ai wjctjcr tjc cvcats {J}, {Y} hrc dutuhnny cxbnus`vc er aet.

Renut`ea = R`abc {J} {Y} 4 , tjc cvcats hrc dutuhnny cxbnus`vc.

Cxhdpnc # ; = @a h s`aknc tjrew ef h i`c, f`ai wjctjcr tjc cvcats {8, ?}, {?, 2} hrc dutuhnny cxbnus`vc er aet.

Renut`ea = R`abc {8, ?} {?, 2} 4 {?} tjc cvcats hrc aet dutuhnny cxbnus`vc.

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DHYJR

"dha`sjludhrpjys`bs.`a" 9

Rcnf prhbt`bc preoncds =

(3) @a tjrew`ak ef h i`c wr`tc wjctjcr tjc cvcats Bed`ak up ef ha eii audocr hai Bed`ak up

ef ha cvca audocr hrc dutuhnny cxbnus`vc er aet.

(6) Ha cxpcr`dcat `avenvcs renn ak h ph`r ef i`bc hai rcberi`ak tjc audocrs tjht bedc up. Icsbr oc tjc

fennew`ak cvcats =H = tjc sud `s krchtcr tjha :.

O = ? ebburs ea c`tjcr i`c.

B = tjc sud `s ht nchst ; hai h dunt`pnc ef 2.

Hnse, f`ai H O, OB hai H B.Hrc (`) H hai O dutuhnny cxbnus`vc 5

(``) O hai B dutuhnny cxbnus`vc 5

(```) H hai B dutuhnny cxbnus`vc 5

Haswcrs = (3) Pcs

(6) H 4 {(2, 6), (9, 3), (3, 9), (6, 2), (9, 6), (3, 3), (6, 9), (3, 6), (6, 3), (6, 6)}

O 4 {(8, ?), (?, ?), (2, ?), (9, ?), (3, ?), (6, ?), (?, 8), (?, 2), (?, 9). (?, 3), (?, 6)}

B 4 {(2, 6), (6, 2), (3, 9), (9, 3), (6, 6)}

H O 4, O B 4, HB 4 {(2, 6), (6, 2), (3, 9), (9, 3), (6, 6)}(`) Pcs (``) Pcs (```) Ae.

(`x) Cxjhus t`vc systcd ef cvcats =@f chbj eutbedc ef ha cxpcr`dcat `s hsseb`htci w`tj ht nchst eac ef tjc cvcats C

8, C

?, C

2, .........C

a,

tjca benncbt`vcny tjc cvcats hrc sh`i te oc cxjhust`vc. Dhtjcdht`bhnny wc wr`tc

C8 C

? C

2.........C

a 4 R. (Rhdpnc sphbc)

Cxhdpnc # : =@a tjrew`ak ef h i`c, nct H oc tjc cvcat cvca audocr turas up, O oc tjc cvcat ha eii pr`dc

turas up hai B oc tjc cvcat h audocrs ncss tjha 9 turas up. F`ai wjctjcr tjc cvcats

H, O hai B ferd ha cxjhust `vc systcd er aet.

Renut`ea = H {?, 9, 6}, O {2, 3} hai B {8, ?, 2}.Bnchrny H O B 4 {8, ?, 2, 9, 3, 6} 4 R. Jcabc tjc systcd ef cvcats `s cxjhust`vc.

Cxhdpnc # 0 = Yjrcc be`as hrc tessci. Icsbr`oc

( ) twe cvcats H hai O wj`bj hrc dutuhnny cxbnus`vc

(` ) tjrcc cvcatsH, O hai B wj`bj hrc dutuhnny cxbnus`vc hai cxjhust`vc.

(` ) twe cvcats H hai O wj`bj hrc aet dutuhnny cxbnus`vc.

( v) twe cvcats H hai O wj bj hrc dutuhnny cxbnus`vc out aet cxjhust`vc.

(v) tjrcc cvcats H, O hai B wj bj hrc dutuhnny cxbnus`vc out aet cxjhust`vc.

Has. (`) H = kctt`ak ht nchst twe jchis O = kctt`ak ht nchst twe th`ns

(` ) H = kctt`ak ht dest eac jchis O = kctt ak cxhbtny twe jchis

B = kctt`ak cxhbtny tjrcc jchis

(```) H = kctt`ak ht dest twe th`ns O = kctt`ak cxhbtny twe jchis

( v) H = kctt`ak cxhbtny eac jchi O = kctt`ak cxhbtny twe jchis

(v) H = kctt`ak cxhbtny eac th`n O = kctt`ak cxhbtny twe th`ns

B = kctt`ak cxhbtny tjrcc th`ns\Aetc = Yjcrc dhy oc etjcr bhscs hnseT

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DHYJR

"dha`sjludhrpjys`bs.`a" 3

Rcnf prhbt`bc preoncds =

(;) @a tjrew`ak ef h i`c wj`bj ef tjc fennew`ak ph r ef cvcats hrc dutuhnny cxbnus`vc 5

(h) tjc cvcats bed`ak up ef ha eii audocr hai bed`ak up ef ha cvca audocr

(o) tjc cvcats bed`ak up ef ha eii audocr hai bed`ak up ef h audocr 9

(:) @a tjrew`ak ef h i`c wj`bj ef tjc fennew`ak systcd ef cvcats hrc cxjhust`vc 5(h) tjc cvcats ha eii audocr turas up, h audocr9 turas up hai tjc audocr 3 turas

up.

(o) tjc cv cats h audocr 9 turas up, h audocr 1 9 turas up.(b) tjc cvcats ha cvca audocr turas up, h audocr i`v`s`onc oy 2 turas up, audocr

8 er ? turas up hai tjc audocr 6 turas up.

Haswcrs (;) (h) (:) (o)

@ @ Bnhss` bh n h p r` er` icf `a `t `ea e f preoho `n `t y =@f ha cxpcr`dcat rcsunts `a h tethn ef (d + a) eutbedcs wj`bj hrc cquhnny n`lcny hai `f d eutbedcs

hrc fhverhonc te ha cvcat H wj`nc a hrc uafhverhonc, tjca tjc preoho`n`ty ef ebburrcabc ef tjc cvcat

H, icaetci oy W(H), `s icf`aci oyad

d

4

eutbedcsefaudocrtethn

eutbedcsfhveurhoncefaudocr

d

.

[c shy tjht eiis `a fhveur ef H hrc d = a, wj`nc eiis hkh`ast H hrc a = d.

Aetc tjht )H(W

er W(H) er W(HB), `.c. preoho n`ty ef aea-ebburrcabc ef H 4ad

a

4 8 W(H)

@a tjc hoevc wc sjhnn icaetc tjc audocr ef eut bedcs fhveurhonc te tjc cvcat H oy a(H) hai tjc tethn

audocr ef eut bedcs `a tjc shdpnc sphbc R oy a(R).

W(H) 4)R(a

)H(a.

Cxhdpnc # 87 = @a tjrew`ak ef h fh`r i`c f`ai tjc preoho`n`ty ef tjc cvcat h audocr 9 turas up.Renut`ea = Rhdpnc sphbc R 4 {8, ?, 2, 9, 3, 6} < cvcat H 4 {8, ?, 2, 9}

a(H) 4 9 hai a(R) 4 6

W(H) 4)R(a

)H(a4

6

94

2

?.

Cxhdpnc # 88 = @a tjrew`ak ef h fh`r i`c, f`ai tjc preoho`n`ty ef tura`ak up ef ha eii audocr 9.Renut`ea = R 4 {8, ?, 2, 9, 3, 6}

Nct C oc tjc cvcat tura`ak up ef ha eii audocr 9tjca C 4 {3}

W(C) 4)R(a

)C(a4

6

8.

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DHYJR

"dha`sjludhrpjys`bs.`a" 6

Cxhdpnc # 8? = @a tjrew`ak h ph`r ef fh`r i`bc, f`ai tjc preoho`n`ty ef kctt`ak h tethn ef :.

Renut`ea = [jca h ph`r ef i`bc `s tjrewa tjc shdpnc sphbc beas`sts

{(8, 8) (8, ?) .......... (8, 6)

(?, 8,) (?, ?,)......... (?, 6)

.... ..... .... ...

.... ... ... ...

(6, 8), (6, ?) ........ (6, 6)}Aetc tjht (8, ?) hai (?, 8) hrc beas`icrci hs scphrhtc pe`ats te dhlc chbj eutbedc hs cquhnny

n`lcny.

Ye kct h tethn ef :, fhveurhonc eutbedcs hrc, (?, 6) (2, 3) (9, 9) (3, 2) hai (6, ?).

Jcabc rcqu`rci preoho`n`ty 426

3

Cxhdpnc # 82 =H feur i`k`t audocr `s ferdci us`ak tjc i`k`ts 7, 8, ?, 2, 9 w tjeut rcpct t`ea. F`ai tjc preoho n ty tjht

`t `s i`v`s`onc oy 9

Renut`ea = Yethn 9 i`k`t audocrs ferdci

Chbj ef tjcsc 06 audocrs hrc cquhnny n`lcny & dutuhnny cxbnus`vc ef chbj etjcr.

Aew, H audocr `s i`v`s`onc oy 9, `f nhst twe i`k`ts ef tjc audocr `s i`v`s`onc oy 9

Jcabc wc bha jhvc f`rst twe pnhbcs bha oc f nnci `a 2 ? 4 6 whys

f`rst twe pnhbcs bha oc f nnci `a ? ? 4 9 whys

6 whys

9 whys

9 whys

6 whys

UUUUUUUUUU

Yethn audocr ef whys 27 whys

preoho`n`ty 4

eutbedcsYethn

eutbedcsfhverhonc4

06

274

86

3Has.

Rcnf prhbt`bc preoncds =

(0) H ohk beath`as 9 wj`tc, 2 rci hai ? onuc ohnns. H ohnn `s irhwa ht rhaied. F`ai tjc preoho`n`ty

ef tjc cvcat (h) tjc ohnn irhwa `s wj`tc er rci (o) tjc ohnn irhwa `s wj tc hs wcnn hs rci.

(87) @a tjrew`ak h ph`r ef fh r i`bc f`ai tjc preoho n`ty ef tjc cvcats h tethn ef ef ncss tjha er cquhn

te 0.

Haswcrs (0) (h) ;/0 (o) 7 (87) 3/26.

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DHYJR

"dha`sjludhrpjys`bs.`a" ;

@ @@ Hii `t `ea tjcercd ef p reoho` n` ty =@f H hai O hrc hay twe cvcats hsseb`htci w`tj ha cxpcr`dcat, tjca

W(HO) 4 W(H) + W(O) W(HO)

Ic Derkhas nhws = @f H & O hrc twe suoscts ef h ua`vcrshn sct S, tjca

(h) (H O)b 4 Hb Ob

(o) (H O)b 4 Hb Ob

I`str`out`vc nhws = (h) H (O B) 4 (H O) (H B)(o) H (O B) 4 (H O) (H B)

Fer hay tjrcc cvcats H, O hai B wc jhvc tjc f`kurc

(`) W(H er O er B) 4 W(H) + W(O) + W(B ) W(H O) W(O B) W(B H) + W(H O B)(` ) W (ht nchst twe ef H, O, B ebbur) 4 W(O B) + W(B H) + W(H O) ?W(H O B)

(` ) W(cxhbtny twe ef H, O, B ebbur) 4 W(O B) + W(B H) + W(H O) 2W(H O B)(`v) W(cxhbtny eac ef H, O, B ebbur) 4W(H) + W(O) + W(B) ?W(O B) ?W(B H) ?W(H O) + 2W(H O B)

Cxhdpnc # 89 = H ohk beath`as 9 wj`tc, 2rci hai 9 krcca ohnns. H ohnn `s irhwa ht rhaied. F`ai tjc preoho`n`ty

ef tjc cvcat tjc ohnn irhwa `s wj`tc er krcca.

Renut`ea = Nct H oc tjc cvcat tjc ohnn irhwa `s wj`tc hai O oc tjc cvcat tjc ohnn irhwa `s krcca.

W(Yjc ohnn irhwa `s wj`tc er krcca) 4 W (H O) 4 W(H) + W(O) W(H O) 488

:

Cxhdpnc # 83 = @a tjrew`ak ef h i`c, nct H oc tjc cvcat ha eii audocr turas up, O oc tjc cvcat h audocri`v`s`onc oy 2 turas up hai B oc tjc cvcat h audocr 9 turas up. Yjca f`ai tjc preoho`n`tytjht cxhbtny twe ef H, O hai B ebbur.

Renut`ea = Cvcat H 4 {8, 2, 3}, cv cat O 4 {2, 6} hai cvcat B 4 {8, ?, 2, 9}

H O 4 {2}, O B 4 {2}, H B 4 {8, 2} hai H O B 4 {2}.Yjus W(cxhbtny twe ef H, O hai B ebbur)

4 W(H O) + W(O B) + W(B H) 2W(H O B)

46

8+

6

8+

6

? 2

6

84

6

8

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DHYJR

"dha`sjludhrpjys`bs.`a" :

Rcnf prhbt`bc preoncds =

(88) @a tjrew`ak ef h i`c, nct H oc tjc cvcat ha eii audocr turas up, O oc tjc cvcat h audocr

i`v`s`onc oy 2 turas up hai B oc tjc cvcat h audocr 9 turas up. Yjca f`ai tjc preoho`n`tytjht htnchst twe ef H, O hai B ebbur.

(8?) @a tjc preoncd audocr 88, f`ai tjc preoho`n`ty tjht cxhbtny eac ef H, O hai B ebburs.

Haswcrs (88)2

8(8?)

2

?

@V Beai` t`eahn preoho` n `ty@f H hai O hrc twe cvcats, tjca W(H/O) 4

W(O)

O)W(H .

Aetc tjht fer dutuhnny cxbnus`vc cvcats W(H/O) 4 7.

Cxhdpnc # 86 = @f W(H/O) 4 7.? hai W(O) 4 7.3 hai W(H) 4 7.?. F`ai W(H O).

Renut`ea = W(H O) 4 W(H) W(H O)

Hnse W(H/O) 4)O(W

)OH(W

W(H O) 4 7.8Fred k`vca ihth,

W(H O) 4 7.8

Cxhdpnc # 8; =@f W(H) 4 7.?3, W(O) 4 7.3 hai W(H O) 4 7.89, f`ai preoho`n ty tjht ac`tjcr H aer O ebburs. Hnse

f`ai W OHRenut`ea = [c jhvc te f`ai W OH 4 8 W(HO) (oy I c-Derkhas nhw)

Hnse, W(H O) 4 W(H) + W(O) W(HO)

putt`ak ihth wc kct, W OH 4 7.20

Yjc sjhici rck`ea icaetcs tjc s`dunthaceus ebburrcabc ef H hai O

Jcabc W OH 4 W(H) W(H O) 4 7.88

Rcnf prhbt`bc preoncd =

(82) @f W(H / O) 4 7.?, W(H O) 4 7.0, tjca f`ai W(H O) 5

Haswcr = 7.9

• 8/12/2019 Probability Theory E

9/25

DHYJR

"dha`sjludhrpjys`bs.`a" 0

V @ aicpcaicat hai icpcaicat cvca ts@f twe cvcats hrc subj tjht ebburcabc er aea-ebburcabc ef eac iecs aet hffcbt tjc bjhabcs ef ebburcabc

er aea-ebburcabc ef tjc etjcr cvcat, tjca tjc cvcats hrc sh`i te oc `aicpcaicat. Dhtjcdht`bhnny = `f

W(H O) 4 W(H) W(O), tjca H hai O hrc `aicpcaicat.

Aetc= ( ) @ f H hai O hrc `aicpcaicat, tjca

(h) Hhai Ohrc `aicpcaicat,(o) H hai O hrc `aicpcaicat hai(b) H hai O hrc `aicpcaicat.

(` ) @f H hai O hrc `aicpcaicat, tjca W(H / O) 4 W(H).

@f cvcats hrc aet `aicpcaicat tjca tjcy hrc sh`i te oc icpcaicat.

@aicpcaicaby ef tjrcc er derc cvcatsYjrcc cvcats H, O & B hrc `aicpcaicat `f & eany `f hnn tjc fennew`ak beai`t`eas jeni =

W(H O) 4 W(H) . W(O) < W(O B) 4 W(O) . W(B)W (B H) 4 W(B) . W(H) < W(H O B) 4 W(H) . W(O) . W(B)

Cxhdpnc # 8: = H ph`r ef fh`r be`as `s tessci y`cni`ak tjc cqu`preohonc sphbc R 4 {JJ, JY, YJ, YY}. Beas`icr

tjc cvcats=

H 4 {jch i ea f`rst be`a} 4 {JJ, JY}, O 4 {jchi ea scb eai be`a} 4 {JJ, YJ}

B 4 {jchi ea cxhbtny eac be`a} 4 {JY, YJ}

Yjca bjcbl wjctjcr H, O, B hrc `aicpcaicat er aet.

Renut`ea = W(H) 4 W(O) 4 W(B) 49

?4

?

8.

Hns e W(H O) 49

84 W(H) W(O), W(H B) 4

9

84 W(H) W(B), W(O B) 4

9

84 W(O) W(B)

out W(H O B) 4 7 W(H) W(O) W(B)

H, O & B hrc aet `aicpcaicat

Cxhdpnc # 80 =@a irhw`ak twe ohnns fred h oex beath`a`ak 6 rci hai 9 wj`tc ohnns w`tjeut rcpnhbcdcat, wj`bj

ef tjc fennew`ak ph`rs `s `aicpcaicat 5

(h) _ci ea f`rst irhw hai rci ea scbeai irhw

(o) _ci ea f`rst irhw hai wj`tc ea scbeai irhw

Renut`ea = Nct C oc tjc cvcat _ci ea f`rst irhw, F oc tjc cvcat _ci ea scbeai irhw hai K oc tjc cvcat

wj`tc ea scbeai irhw.

W(C) 487

6, W(F) 4

87

6, W(K) 4

87

9

(h) W(C F) 4?

87

?6

W

W4

2

8

W(C) . W(F) 43

2

3

24

?3

0

2

8

C hai F hrc aet `aicpcaicat

(o) W(C) . W(K) 487

6

87

94

?3

6

W(C K) 4?

87

89

86

W

WW 4

83

9

W(C) . W(K) W(C K) C hai K hrc aet `aicpcaicat

• 8/12/2019 Probability Theory E

10/25

DHYJR

"dha`sjludhrpjys`bs.`a" 87

Cxhdpnc # ?7 =@f twe sw`tbjcs R8

hai R?

jhvc rcspcbt`vcny 07% hai :7% bjhabcs ef werl`ak. F`ai tjc preoho`n t`cs

tjht chbj ef tjc fennew`ak b`rbu`ts w`nn werl.

Renut`ea = Beas icr tjc fennew`ak cvcats =

H 4 Rw`tbj R8werls,

O 4 Rw`tbj R?

werls,

[c jhvc,

W(H) 4877

074

87

0hai W(O) 4

877

:74

87

:

( ) Yjc b`rbu`t w nn werl f tjc burrcat fnews `a tjc b`rbu`t. Yj`s `s pess`onc eany wjca oetj tjcsw`tbjcs werl tekctjcr. Yjcrcferc,

_cqu`rci preoho n`ty

4 W(HO) 4 W(H) W (O) \H hai O hrc `aicpcaicat cvcatsT

487

0

87

:4

877

;?4

?3

8:

(` ) Yjc b rbu t w nn werl `f tjc burrcat fnews `atjc b rbu t. Yj`s `s pess`onc eanywjca ht nchst eac

ef tjc twe sw`tbjcs R8, R

? werls. Yjcrcferc,

_cqu`rci Wreoho`n ty

4 W(HO) 4 8 W )H( W(O

) \H, Ohrc `aicpcaicat cvcatsT

4 8

87

08

87

:8 4 8

87

8

87

?4

37

90

Cxhdpnc # ?8 =H spchls trutj `a 67% ef tjc bhscs hai o `a 07% ef tjc bhscs. @a wjht pcrbcathkc ef bhscs hrc tjcy

n`lcny te beatrhi`bt chbj etjcr `a stht`ak tjc shdc fhbt5

Renut`ea = Nct C oc tjc cvcat tjht H spchls trutj hai F oc tjc cvcat tjht O spchls trutj. Yjca C hai F hrc

`aicpcaicat cvcats subj tjht

W(C) 4 877

674 3

2hai W(F) 4 877

074 87

0

H hai O w`nn beatrhi`bt chbj etjcr `a ahrrht`ak tjc shdc fhbt `a tjc fennew`ak dutuhnny cxbnus`vc

whys=

(`) H spchls trutj hai O tcnns h n`c `.c. C F

(` ) H tcnns h n`c hai O spchls trutj n`c `.c. C F

W(H hai O beatrhi`bt chbj etjcr)

4 W(@ er @@) 4 (@ @@) 4 W\(C F ) ( C F)T

4 W(C F ) + W ( C F) \C F hai C F hrc dutuhnny cxbnus`vcT

4 W(C) W( F ) + W( C) W(F) \C hai F hrc `a icp.T

• 8/12/2019 Probability Theory E

11/25

DHYJR

"dha`sjludhrpjys`bs.`a" 88

43

2

87

08 +

3

28

87

04

3

2

87

8+

3

?

87

04

37

?8

Cxhdpnc # ?? =H oex beath`as 3 ounos ef wj`bj twe hrc icfcbt vc. Ycst `s bhrr`ci ea ounos eac oy eac uat`nn tjc twe

icfcbt vc ounos hrc feuai eut. F`ai tjc preoho n`ty tjht tjc prebcss steps hftcr

(`) Rcbeai tcst (``) Yj`ri tcstRenut`ea = ( ) Wrebcss w nn step hftcr scbeai tcst. Eany `f tjc f`rst hai scbeai ouno hrc oetj feuai te oc

icfcbt`vc

preoho`n`ty 43

?

9

84

87

8(Eov`eusny tjc ounos irhwa hrc aet lcpt ohbl.)

(` ) Wrebcss w nn step hftcr tj ri tcst wjca c`tjcr

(h) IAI 3

?

9

2

2

84

87

8Jcrc I sthais fer icfcbt vc

er (o) AII 3

2

9

? 2

8

4 87

8

hai A `s fer aet icfcbt`vc.

er (b) AAA 3

2

9

?

2

84

87

8

jcabc rcqu`rci preoho`n ty 487

2

Cxhdpnc # ?2 =@f C8 hai C?hrc twe cvcats subj tjht W(C8) 4 9

8< W(C?) 4 ?

8< W

?

8

C

C

4 9

8, tjca bjeesc tjc berrcbt

ept`eas.

(`) C8

hai C?

hrc `aicpcaicat (``) C8

hai C?

hrc cxjhust`vc

(```) C8hai C

?hrc dutuhnny c xbnus`vc ( v) C

8& C

?hrc icpcaicat

Hnse f`ai W

?

8

C

Chai

8

?

C

C

Renut`ea = R`abc W

?

8

C

C4 W(C

8) C

8hai C

?hrc `aicpcaicat ef chbj etjcr

Hnse s`abc W(C8C

?) 4 W(C

8) + W(C

?) W(C

8) . W(C

?)8

Jcabc cvcats hrc aet cxjhust`vc. @aicpcaicat cvcats bhat oc dutuhnny cxbnus`vc.

Jcabc eany (`) `s berrcbt

Furtjcr s`abc C8& C

?hrc `aicpcaicat< C8 hai ?C er 8C , C?hrc 8C , ?C hrc hnse `aicpcaicat.

Jcabc

?

8

C

CW 4 W 8C 4 9

2hai

8

?

C

CW 4 W (C

?) 4

?

8

Cxhdpnc # ?9 =@f bhris hrc irhwa eac oyeac fred h wcnn sjuffnci phbl ef 3? bhris w`tjeut rcpnhbcdcat, uat n ha hbc

hppchrs, f`ai tjc preoho`n ty tjht tjc feurtj bhri `s tjc f`rst hbc te hppchr.

• 8/12/2019 Probability Theory E

12/25

DHYJR

"dha`sjludhrpjys`bs.`a" 8?

Renut`ea = Wreoho`n`ty ef scncbt`ak 2 aea-Hbc hai 8 Hbc eut ef 3? bhris `s cquhn te9

3?

89

29:

B

BB

R`abc wc what 9tj bhri te oc f rst hbc, wc w nn hnse jhvc te beas`icr tjc hrrhakcdcat, Aew 9 bhris

`a shdpnc sphbc bha oc hrrhakci `a 9! whys hai, fhverhonc tjcy bha oc hrrhakci `a 2 ! whys hs wc

what 9tj pes`t`ea te oc ebbup`ci oy hbc

Jcabc rcqu`rci preoho`n ty 49

3?8

92

9:

BBB

!9!2

Hn`tcr =

AAAH `s tjc hrrhakcdcat tjca wc ics`rc `a thl`ak eut bhris, eac oy eac

Jcabc rcqu`rci bjhabc `s3?

9:

38

9;

37

96

90

9

Rcnf prhbt`bc preoncds =

(89) Ha ura beath`as ; rci hai 9 onuc ohnns. Ywe ohnns hrc irhwa ht rhaied w`tj rcpnhbcdcat. F`ai tjcpreoho`n`ty ef kctt`ak

(`) ? rci ohnns (``) ? onuc ohnns (```) eac rci hai eac onuc ohnn

(83) Wreoho n`t cs ef senv ak h spcb f`b preoncd `aicpcaicatny oy H hai O hrc?

8hai

2

8rcspcbt`vcny. @f

oetj try te senvc tjc preoncd `aicpcaicatny, f`ai tjc preoho`n ty tjht

(`) tjc preoncd `s senvci (` ) cxhbtny eac ef tjcd senvcs tjc preoncd.

(86) @a tjrew`ak h ph`r ef i`cs f`ai tjc preoho`n ty ef kctt`ak ha eii audocr ea tjc f`rst i`c hai h

tethn ef ; ea oetj tjc i`cs.

(8;) @a tjrew`ak ef h ph`r ef i`cs, f`ai tjc preoho n`ty ef kctt`ak h oeuonct er h tethn ef 9.

(8:) H ohk beath`as : dhroncs ef wj`bj 2 hrc onuc hai 3 hrc rci. Eac dhronc `s irhwa ht rhaied, `ts

beneur `s aetci hai tjc dhronc `s rcpnhbci `a tjc ohk. H dhronc `s hkh`a irhwa fred tjc ohk hai `ts

beneur `s aetci. F`ai tjc preoho`n`ty tjht tjc dhroncs w`nn oc

( ) onuc fennewci oyrci (` ) onuc hai rci a hayericr (` `) ef tjc shdc beneur.

(80) H be`a s tessci tjr`bc. @a wj`bj ef tjc fennew`ak bhscs hrc tjc cvcats C hai F `aicpcaicat 5

( ) C = tjc f`rst tjrew rcsunts `a jchi.

F = tjc nhst tjrew rcsunt `a th`n.

(` ) C = tjc audocr ef jchis `s twe.

F = tjc nhst tjrew rcsunt `a jchi.

(` ) C = tjc audocr ef jchis `s eii .

F = tjc audocr ef th`ns `s eii.

Haswcrs = (89) (`)8?8

90(``)

8?8

86(```)

8?8

36(83) (`)

2

?(``)

?

8

(86)8?

8(8;)

0

?(8:) (`)

69

83(``)

2?

83(```)

2?

8;

(80) (`)

• 8/12/2019 Probability Theory E

13/25

DHYJR

"dha`sjludhrpjys`bs.`a" 82

V @ O `aed`h n p reoho` n` ty tjcercd =@f ha cxpcr`dcat `s subj tjht tjc preoho`n`ty ef subbcss er fh`nurc iecs aet bjhakc w`tj tr`hns, tjca

tjc preoho`n`ty ef kctt`ak cxhbtny r subbcss `a a tr`hns ef subj ha cxpcr`dcat `s aBrp r qa r, wjcrc p

`s tjc preoho`n`ty ef h subbcss hai q `s tjc preoho`n`ty ef h fh`nurc `a eac phrt`bunhr cxpcr`dcat. Aetc

tjht p + q 4 8.

Cxhdpnc ?3 = H ph`r ef i`bc `s tjrewa 3 t`dcs. F`ai tjc preoho`n`ty ef kctt`ak h ieuonct tw`bc.

Renut`ea = @a h s`aknc tjrew ef h ph`r ef i`bc preoho`n ty ef kctt`ak h ieuonct `s6

8

beas`icr`ak `t te oc h subbcss, p 46

8

q 4 8 6

84

6

3

audocr ef subbcss r 4 ?

W(r 4 ?) 4 3B?

p? q2 4 87 .

?

6

8

.

2

6

3

4

2:::

6?3

Cxhdpnc # ?6 = H ph`r ef i`bc `s tjrewa 9 t`dcs. @f kctt`ak h tethn ef 0 `a h s`aknc tjrew `s beas`icrci hs h

subbcss tjca f`ai tjc preoho`n`ty ef kctt`ak h tethn ef 0 tjr`bc.

Renut`ea = p 4 preoho`n`ty ef kctt`ak h tethn ef 0 426

94

0

8

q 4 8 0

84

0

:

r 4 2 , a 4 9

W(r 4 2) 4 9B2

p2 q 4 9

2

0

8

.

0

:4

6368

2?

Cxhdpnc # ?; =@a ha cxhd`aht`ea ef 87 dunt`pnc bje`bc qucst`eas (8 er derc bha oc berrcbt eut ef 9 ept`eas). H

stuicat icb`ics te dhrl tjc haswcrs ht rhaied. F`ai tjc preoho`n`ty tjht jc kcts cxhbtny twe

qucst`eas berrcbt.Renut`ea = H stuicat bha dhrl 83 i`ffcrcat haswcrs te h DBX w`tj 9 ept ea `.c.9 B

8+ 9 B

?+ 9B

2+ 9 B

94 83

Jcabc `f jc dhrls tjc haswcr ht rhaied, bjhabc tjht j`s haswcr `s berrcbt 483

8hai oc`ak

`aberrcbt`ak83

89. Yjus p 4

83

8, q 4

83

89.

W (? subbcss) 4 87B?

?

83

8

:

83

89

• 8/12/2019 Probability Theory E

14/25

DHYJR

"dha`sjludhrpjys`bs.`a" 89

Cxhdpnc # ?: =H fhd`ny jhs tjrcc bj`nirca. Cvcat H `s tjht fhd`ny jhs ht dest eac oey, Cvcat O `s tjht fhd`ny jhs

ht nchst eac oey hai eac k`rn, Cvcat B `s tjht tjc fhd`ny jhs ht dest eac k`rn. F`ai wjctjcr cvcats

H hai O hrc `aicpcaicat. Hnse f`ai wjctjcr H, O, B hrc `aicpcaicat er aet.

Renut`ea = H fhd ny ef tjrcc bj`nirca bha jhvc

(`) Hnn 2 oeys (``) ? oeys + 8 k`rn (```) 8 oey + ? k`rns (`v) 2 k`rns

(`) W (2 oeys) 4 2B7

2

?8

4:8 (R`abc chbj bj`ni `s cquhnny n`lcny te oc h oey er h k`rn)

(` ) W (? oeys +8k`rn) 4 2B8

?

?

8

?

84

:

2(Aetc tjht tjcrc hrc tjrcc bhscs OOK, OKO, KOO)

(` `) W (8 oey + ? k rns) 4 2B?

8

?

8

?

?

8

4

:

2

( v) W (2 k`rns) 4

:

8

Cvcat H `s hsseb`htci w`tj (```) & (`v). Jcabc W(H) 4?

8

Cvcat O `s hsseb`htci w`tj (``) & (```). Jcabc W(O) 49

2

Cvcat B `s hsseb`htci w`tj (`) & (``). Jcabc W(B) 4?

8

W(H O) 4 W(```) 4:

24 W(H) . W(O) . Jcabc H hai O hrc `aicpcaicat ef chbj etjcr

W(H B) 4 7W(H) . W(B) . Jcabc H, O, B hrc aet `aicpcaicat

Rcnf prhbt`bc preoncds =

(?7) H oex beath`as ? rci hai 2 onuc ohnns. Ywe ohnns hrc irhwa subbcss`vcny w`tjeut rcpnhbcdcat.

@f kctt`ak h rci ohnn ea f`rst irhw hai h onuc ohnn ea scbeai irhw `s beas`icrci h subbcss,

tjca f`ai tjc preoho`n`ty ef ? subbcsscs `a 2 pcrferdhabcs.

(?8) Wreoho`n ty tjht h ouno preiubci oy h fhbtery w`nn fusc hftcr ha ychr ef usc `s 7.?. F`ai tjc

preoho`n`ty tjht eut ef 3 subj ounos aet derc tjha 8 ouno w`nn fusc hftcr ha ychr ef usc.

Haswcrs (?7) 8:0 (?8)28?3

?279

V @ @ Cxpcbtht`ea =@f tjcrc hrc a pess`o`n`t`cs H

8, H

?, .... H

a `a ha cxpcr`dcat jhv`ak tjc preoho`n`t`cs p

8, p

?, .........p

a

rcspcbt`vcny. @f vhnuc D8, D

?, ....., D

ahrc hsseb`htci w`tj tjc rcspcbt`vc pess`o`n`ty. Yjca tjc cxpcbtci

vhnuc ef tjc cxpcr`dcat `s k`vca oy

a

8`

`` D.p

• 8/12/2019 Probability Theory E

15/25

DHYJR

"dha`sjludhrpjys`bs.`a" 83

Cxhdpnc # ?0 =H fh`r i`c `s tessci. @f ?, 2 er 3 ebburs, tjc pnhycr w`as tjht audocr ef rupccs, out `f 8, 9, er

6 ebburs, tjc pnhycr nescs tjht audocr ef rupccs. Yjca f`ai tjc pess`onc phyeffs fer tjc pnhycr.

Renut`ea =

H` ? 2 3 8 9 6

D` ? 2 3 8 9 6

W` 8/6 8/6 8/6 8/6 8/6 8/6

Yjca cxpcbtci vhnuc C ef tjc khdc phyeffs fer tjc pnhycr

4 ?

6

8+ 2

6

8+ 3

6

8 8

6

8 9

6

8 6

6

84

6

8

R`abc C `s ackht`vc tjcrcferc khdc `s uafhverhonc te tjc pnhycr.

Cxhdpnc # 27 = Yjcrc hrc 877 t`blcts `a h rhffnc (Nettcry). Yjcrc `s 8 pr`zc chbj ef _s. 8777/-, _s. 377/- hai

_s. ?77/-. _cdh`a`ak t`blcts hrc onhal. F`ai tjc cxpcbtci pr`bc ef eac subj t`blct.

Renut`ea = Cxpcbtht`ea 4 p`D

` eutbedc ef h t`blct bha oc

p`

D`

p`D

`

(`) @ pr`zc877

88777 87

(``) @@ pr zc877

8377 3

(```) @@@ pr zc877

8?77 ?

(`v) Onhal877

0;7 7

UUUUUUUUUUUUUUUU

p `D` 4 8;UUUUUUUUUUUUUUUU

Jcabc cxpcbtci pr`bc ef eac subj t`blct _s. 8;

Cxhdpnc # 28 =H pursc beath`as feur be`as chbj ef wj`bj `s c`tjcr h rupcc er twe rupccs be`a. F`ai tjc cxpcbtci

vhnuc ef h be`a `a tjht pursc.

Renut`ea = Vhr`eus pess`o`n`t`cs ef be`as `a tjc pursc bha oc

p`

D`

p`D

`

(`) 9 8 rupcc be`as86

89

86

9

(` ) 2 eac _s. + 8 twe _s.86

93

86

?7

(`` ) ? eac _s. + ? twe _s.86

66

86

26

(`v) 8 eac _s. + 2 twe _s.86

9;

86

?:

(`v) 9 twe _s.86

8:

86

:

UUUUUUUUUUUUUUUU

6 / -

UUUUUUUUUUUUUUUU

Jcabc cxpcbtci vhnuc `s _s. 6/-

• 8/12/2019 Probability Theory E

16/25

DHYJR

"dha`sjludhrpjys`bs.`a" 86

Aetc = (tjht s`abc chbj be`a `s cquhnny n`lcny te oc eac _s. er twe _s. be`a, tjc preoho n`ty `s ictcrd`aci

us`ak O`aed`hn preoho`n`ty< uan`lc tjc bhsc wjca tjc pursc beath`aci tjc be`as w`tj hnn pess`o`n`ty

oc`ak cquhnny n`lcny, wjcrc wc thlc p`4

3

8fer chbj.)

Rcnf prhbt`bc preoncds =

(??) Fred h ohk beath`a`ak ? eac rupcc hai 2 twe rupcc be`as h pcrsea `s hnnewci te irhw ? be`as

`ai`sbr`d`ahtcny< f`ai tjc vhnuc ef j s cxpcbtht`ea.

Haswcr = _s. 2.?7

V @ @ @ Ye th n p reoho` n` ty tjcercd@f ha cvcat H bha ebbur w`tj eac ef tjc a dutuhnny cxbnus`vc hai cxjhust`vc cvcats O

8, O

?, ....., O

a

hai tjc preoho`n`t`cs W(H/O8), W(H/O

?) .... W(H/O

a) hrc laewa, tjca

W(H) 4

a

8`

`` )O/H(W.)O(W

Wreef =

Yjc cvcat H ebburs w`tj eac ef tjc a dutuhnny cxbnus`vc hai cxjhust`vc cvcats

O8, O

?, O

2,........,O

a

H 4 (H O8) (H O

?) (H O

2) ........ (H O

a)

W(H) 4 W(H O8) + W(H O

?) + ....... + W(H O

a) 4

a

8`

` )OH(W

Aew,W(H O

`) 4 W(H) . W(O

`/H) 4 W(O

`) . W(H/O

`)

W(H) 4

a

8`

`` )O/H(W.)O(W

Cxhdpnc # 2? = Oex - beath`as 3 rci hai 9 wj`tc ohnns wj`nc oex - beath`as 9 rci hai ? wj`tc ohnns. Hfh`r i`c `s tjrewa. @f `t turas up h dunt`pnc ef 2, h ohnn `s irhwa fred oex - cnsc h ohnn `s irhwafred oex - . F`ai tjc preoho`n`ty tjht tjc ohnn irhwa `s wj`tc.

Renut`ea = Nct H oc tjc cvcat h dunt`pnc ef 2 turas up ea tjc i`c hai _ oc tjc cvcat tjc ohnn irhwa `s

wj`tctjca W (ohnn irhwa `s wj`tc)

4 W(H) . W(_ / H) + W )H( W(_ / H)

46

?

0

9+

6

?8

6

?4

?;

87

• 8/12/2019 Probability Theory E

17/25

DHYJR

"dha`sjludhrpjys`bs.`a" 8;

Cxhdpnc # 22 = Bhris ef ha eri`ahry icbl ef pnhy`ak bhris hrc pnhbci `ate twe jchps. Jchp - beas`sts efhnn tjc rci bhris hai jchp - beas`sts ef hnn tjc onhbl bhris. H jchp `s bjesca ht rhaied haih bhri `s irhwa, f`ai tjc preoho`n`ty tjht tjc bhri irhwa `s h l`ak.

Renut`ea = Nct hai oc tjc cvcats tjht jchp - hai jchp - hrc bjeesca rcspcbt`vcny. Yjca

W() 4 W() 4?

8

Nct L oc tjc cvcat tjc bhri irhwa `s h l`ak

W (L / ) 4?6? hai W(L / ) 4

?6?

W(L) 4 W () W(L / ) + W() W(L / ) 4?

8

?6

?+

?

8

?6

?4

82

8.

Rcnf prhbt`bc preoncds =

(?2) Oex - beath`as 2 rci hai ? onuc ohnns wj`nc oex - @@ beath`as ? rci hai 2 onuc ohnns. H fh`rbe`a `s tessci. @f `t turas up jchi, h ohnn `s irhwa fred oex - , cnsc h ohnn `s irhwa fredoex - . F`ai tjc preoho`n`ty tjht tjc ohnn irhwa `s rci.

(?9) Yjcrc hrc 3 or nn`hat stuicats `a bnhss Z@ hai : or`nn hat stuicats `a bnhss Z@@. Chbj bnhss jhs

37 stuicats. Yjc eiis `a fhveur ef bjees`ak tjc bnhss Z@ hrc ? = 2. @f tjc bnhss Z@ `s aet bjesca

tjca tjc bnhss Z@@ `s bjesca. F`ai tjc preoho`n`ty ef scncbt`ak h or`nn`hat stuicat.

Haswcrs = (?2)?

8(?9)

8?3

8;.

@ Z Ohycs tjcercd =@f ha cvcat H bha ebbur w`tj eac ef tjc a dutuhnny cxbnus`vc hai cxjhust`vc cvcats O

8, O

?, ....., O

ahai

tjc preoho`n`t`cs W(H/O8

), W(H/O?

) .... W(H/Oa

) hrc laewa, tjca

W(O`/ H) 4

a

8`

``

``

)O/H(W.)O(W

)O/H(W.)O(W

Wreef =

Yjc cvcat H ebburs w`tj eac ef tjc a dutuhnny cxbnus`vc hai cxjhust`vc cvcats

O8, O

?, O

2,........,O

a

H 4 (H O8) (H O

?) (H O

2) ........ (H O

a)

W(H) 4 W(H O8) + W(H O?) + ....... + W(H Oa) 4 a

8`` )OH(W

Aew,

W(H O`) 4 W(H) . W(O

`/H) 4 W(O

`) . W(H/O

`)

W (O`/H) 4

)H(W

)O/H(W.)O(W ``4

a

8`

`

``

)OH(W

)O/H(W.)O(W

W(O`/H) 4

)O/H(W.)O(W

)O/H(W.)O(W

``

``

• 8/12/2019 Probability Theory E

18/25

DHYJR

"dha`sjludhrpjys`bs.`a" 8:

Cxhdpnc # 29 = Whns khricacr `s aet icpcaihonc, tjc preoho`n ty tjht jc w nn ferkct te whtcr tjc resc ousj `s2

?. Yjc

resc ousj `s `a qucst`eahonc beai`t`ea hay jew, `f whtcrci tjc preoho n ty ef `ts w tjcr`ak `s?

8, `f aet

whtcrci, tjc preoho`n`ty ef `ts w`tjcr`ak `s

9

2. Whn wcat eut ef stht`ea hai upea rctura`ak, jc f`ais

tjht tjc resc ousj jhs w`tjcrci, wjht `s tjc preoho`n`ty tjht tjc khricacr i`i aet whtcr tjc ousj.

\Jcrc rcsunt `s laewa tjht tjc resc ousj jhs w tjcrci, tjcrcferc. Ohycss tjcercd sjeuni oc usciT

Renut`ea = Nct H 4 tjc cvcat tjht tjc resc ousj jhs w tjcrci

Nct H84 tjc cvcat tjht tjc khricacr i`i aet whtcr.

H?

4 tjc cvcat tjht tjc khricacr whtcrci.

Oy Ohycss tjcercd rcqu`rci preoho n`ty,

W(H8/H) 4

)H/H(W.)H(W)H/H(W.)H(W

)H/H(W.)H(W

??88

88

.....(`)

K`vca, W(H8) 4

2? W(H

?) 4

28

W(H/H8) 4

9

2, W(H/H

?) 4

?

8

Fred (8), W(H8/H) 4

?

8.

2

8

9

2.

2

?9

2.

2

?

4

?6

6

4

9

2

Cxhdpnc # 23 = Yjcrc hrc 3 or`nn`hat stuicats `a bnhss Z@ hai : or`nn`hat stuicats `a bnhss Z@@. Chbj bnhss jhs

37 stuicats. Yjc eiis `a fhveur ef bjees`ak tjc bnhss Z@ hrc ? = 2. @f tjc bnhss Z@ `s aet bjesca

tjca tjc bnhss Z@@ `s bjesca. H stuicat `s h bjesca hai `s feuai te oc or`nn`hat, f`ai tjc preoho`n`ty

tjht tjc bjesca stuicat `s fred bnhss Z@.

Renut`ea = Nct C hai F oc tjc cvcats Bnhss Z@ `s bjesca hai Bnhss Z@@ `s bjesca rcspcbt`vcny.

Yjca W(C) 43

?, W(F) 4

3

2

Nct H oc tjc cvcat Rtuicat bjesca `s or`nn`hat.

Yjca W(H / C) 437

3hai W(H / F) 4

37

:.

W(H) 4 W(C) . W(H / C) + W(F) . W(H / F) 43

?.

37

3+

3

2.

37

:4

?37

29.

W(C / H) 4)F/H(W.)F(W)C/H(W.)C(W

)C/H(W.)C(W

4 8;3

.

Cxhdpnc # 26 =H phbl ef bhris `s beuatci w`tj fhbc iewawhris. @t `s feuai tjht eac bhri `s d`ss`ak. Eac bhri `s

irhwa hai `s feuai te oc rci. F`ai tjc preoho`n ty tjht tjc d`ss`ak bhri `s rci.

Renut`ea = Nct H oc tjc cvcat ef irhw`ak h rci bhri wjca eac bhri `s irhwa eut ef 38 bhris (cxbnui`ak d`ss`ak

bhri.) Nct H8oc tjc cvcat tjht tjc d`ss`ak bhri `s rci hai H

?oc tjc cvcat tjht tjc d`ss`ak bhri `s

onhbl.

• 8/12/2019 Probability Theory E

19/25

DHYJR

"dha`sjludhrpjys`bs.`a" 80

Aew oy Ohycss tjcercd, rcqu`rci preoho`n ty,

W(H8/H) 4

)H/H(W.)H(W)H/H(W.)H(W

)H/H(W(.)H(W

??88

88

..........(`)

@a h phbl ef 3? bhris ?6 hrc rci hai ?6 hrc onhbl.

Aew W(H8) 4 preoho`n`ty tjht tjc d`ss`ak bhri `s rci 4

8

3?

8?6

B

B4

3?

?64

?

8

W(H?) 4 preoho`n`ty tjht tjc d`ss`ak bhri `s onhbl 4

3?

?64

?

8

W(H/H8) 4 preoho`n`ty ef irhw`ak h rci bhri wjca tjc d`ss`ak bhri `s rci.

438

?3

\Yethn audocr ef bhris ncft `s 38 eut ef wj`bj ?3 hrc rci hai ?6 hrc onhbl hs tjc d`ss`ak bhri `s rciT

Hkh`a W(H/H?) 4 Wreoho`n`ty ef irhw`ak h rci bhri wjca tjc d`ss`ak bhri `s onhbl 4

38

?6

Aew fred (`), rcqu`rci preoho`n ty, W(H8/H) 4

38

?6.

?

8

38

?3.

?

838

?3.

?

8

4

38

?3

Cxhdpnc # 2; =H ohk beath`as 6 wj`tc hai ha ualaewa audocr ef onhbl ohnns ( 2). Ohnns hrc irhwa eac oy eac w tjrcpnhbcdcat fred tj`s ohk tw`bc hai `s feuai te oc wj`tc ea oetj ebbhss`ea. F`ai tjc preoho`n ty tjht

tjc ohk jhi cxhbtny 2 Onhbl ohnns.

Renut`ea = Hpr`er`, wc bha tj`al ef tjc fennew`ak pess`o`n`cs

(`) C8 6[ , 7 O(` ) C

? 6[ , 8 O

(```) C2

6[ , ? O

( v) C9

6[ , 2 O

Bnchrny W(C8) 4 W(C

?) 4 W(C

2) 4 W(C

9) 4

9

8

Nct H oc tjc cvcat tjht twe ohnns irhwa eac oy eac w`tj rcpnhbcdcat hrc oetj wj`tc tjcrcferc wc

jhvc te f`ai W

H

C9

Oy Ohycs tjcercd W

H

C94

)C(W.C

HW)C(W.

C

HW)C(W.

C

HW)C(W

C

HW

)C(WC

HW

99

22

??

88

99

W

9C

H4

0

6

0

6< W

2C

H4

:

6

:

6< W

?C

H4

;

6

;

6< W

8C

H4

6

6

6

6 2) (```) W(Z 1 6) (`v) W(7 > Z > 2)

\J`at =SscW(Z) 4 8 te ictcrd`ac l, W(Z > 2) 4 W(7) + W(8) + W(?), W(Z 1 6) 4 W(;) ctb.T

Cxhdpnc # 98 = H ph`r ef i`bc `s tjrewa 3 t`dcs. @f kctt`ak h ieuonct `s beas`icrci hs h subbcss, tjca f`ai

tjc dcha hai vhr`habc ef subbcsscs.

Renut`ea = @a h s`aknc tjrew ef h ph`r ef i`bc, preoho n`ty ef kctt`ak h ieuonct 46

8

beas`icr`ak `t te oc h subbcss, p 4 6

8

• 8/12/2019 Probability Theory E

23/25

DHYJR

"dha`sjludhrpjys`bs.`a" ?2

q 4 8 6

84

6

3

dcha 4 3 6

84

6

3, vhr`habc 4 3

6

8.

6

34

26

?3

Cxhdpnc # 9? = H ph`r ef i`bc `s tjrewa 9 t`dcs. @f kctt`ak h tethn ef 0 `a h s`aknc tjrew `s beas`icrci hs h

subbcss tjca f`ai tjc dcha hai vhr`habc ef subbcsscs.

Renut`ea = p 4 preoho`n`ty ef kctt`ak h tethn ef 0 426

94

0

8

q 4 8 0

84

0

:

dcha 4 ap 4 9 0

84

0

9

vhr`habc 4 apq 4 9

0

8

0

:4

:8

2?

Cxhdpnc # 92 =I`ffcrcabc octwcca dcha hai vhr`habc ef h O`aed`hn vhr`htc `s 8 hai i`ffcrcabc octwcca tjc`r

squhrcs `s 88. F`ai tjc preoho`n`ty ef kctt`ak cxhbtny tjrcc subbcss

Renut`ea = Dcha 4 ap & vhr`habc 4 apq

tjcrcferc, ap apq 4 8 ..........(`)

a?p? a?p?q? 4 88 ..........(``)

Hnse, wc laew tjht p + q 4 8 ..........(```)

I`v`ic cquht ea (` ) oy squhrc ef (`) hai senvc, wc kct, q 46

3, p 4

6

8& a 4 26

Jcabc preoho`n ty ef 2 subbcss 4 26B2

2

68

22

63

Rcnf prhbt`bc preoncds =

(?:) H oex beath`as ? rci hai 2 onuc ohnns. Ywe ohnns hrc irhwa subbcss`vcny w`tjeut rcpnhbcdcat.

@f kctt`ak h rci ohnn ea f`rst irhw hai h onuc ohnn ea scbeai irhw `s beas`icrci h subbcss,

tjca f`ai tjc dcha hai vhr`habc ef subbcsscs.

(?0) Wreoho`n`ty tjht h ouno preiubci oy h fhbtery w`nn fusc hftcr ha ychr ef usc `s 7.?. @f fus`ak ef

h ouno `s beas`icrci ha fh`nurc, f`ai tjc dcha hai vhr`habc ef subbcsscs fer h shdpnc ef 87

ounos.

(27) H rhaied vhr`honc Z `s spcb`f`ci oy tjc fennew`ak i`str`out`ea nhw =

Z ? 2 9

W(Z 4 x) 7.2 7.9 7.2

Yjca tjc vhr`habc ef tj`s i`str`out`ea `s =

(H) 7.6 (O) 7.; (B) 7.;; (I) 8.33

Haswcrs = (?:) d cha 4 ?.8, ? 4.62 (?0) dcha 4 : hai vhr`habc 4 8.6(27) H

• 8/12/2019 Probability Theory E

24/25

DHYJR

"dha`sjludhrpjys`bs.`a" ?9

Z@ @ Kcedctr`bhn hppn`bht`eas=Yjc fennew`ak sthtcdcats hrc hx`edht b =

(`) @f h pe`at `s thlca ht rhaied ea h k`vca strh`kjt n`ac sckdcat HO, tjc bjhabc tjht t fhnns ea h phrt`bunhr

sckdcat WX ef tjc n`ac sckdcat `s WX/HO. `.c. preoho n`ty 4ncaktjtethn

ncaktjhoncvhrfh

(` ) @f h pe`at `s thlca ht rhaied ea tjc hrch R wj`bj `abnuics ha hrch, tjc bjhabc tjht tjc pe`at fhnns

ea `s /R. `.c.hrchtethn

hrchhoncvhrfh

Cxhdpnc # 99 =H spjcrc `s b`rbudsbr`oci evcr h buoc. F`ai tjc preoho`n ty tjht h pe`at n`cs `as`ic tjc spjcrc, n`cs

euts`ic tjc buoc.

Renut`ea = _cqu rci preoho`n`ty 4venudctethn

venudcfhverhonc

Bnchrny `f cikc ncaktj ef buoc `s h rhi`us ef spjcrc w`nn oc?

2h

Yjus, venudc ef spjcrc 42

9

2

?

2h

4

?

2h2

Jcabc W 4 8

?

2

8

4 8 2

?

Cxhdpnc # 93 = H k`vca n`ac sckdcat `s i`v`ici ht rhaied `ate tjrcc phrts. [jht `s tjc preoho`n`ty tjht tjcy

ferd s`ics ef h pess`onc tr`haknc 5

Renut`ea = Nct HO oc tjc n`ac sckdcat ef ncaktj .

Nct B hai I oc tjc pe`ats wj`bj i`v`ic HO `ate tjrcc phrts.

Nct HB 4 x, BI 4 y. Yjca IO 4 x y.

Bnchrny x + y > tjc shdpnc sphbc `s k`vca oy

tjc rck`ea cabnesci oy EWX, wjcrc EW 4 EX 4

Hrch efEWX 4?

?

• 8/12/2019 Probability Theory E

25/25

DHYJR

Aew `f tjc phrts HB, BI hai IO ferd h tr`haknc, tjca

x + y 1 x y `.c. x + y 1?

...........(`)

x + x y 1 y `.c. y >?

...........(``)

y + x y 1 x `.c. x >? ...........(```)

fred (`), (``) hai (```), wc kct

tjc cvcat `s k`vca oy tjc rck`ea bnesci `a _RY

Wreoho`n`ty ef tjc cvcat 4)EWX(hr

)_RY(hr

4

?

?.

?.

?

8

?

49

8

Cxhdpnc # 96 = Ea h n`ac sckdcat ef ncaktj N twe pe`ats hrc thlca ht rhaied, f`ai tjc preoho`n`ty tjht tjci`sthabc octwcca tjcd `s , wjcrc > 8

Renut`ea = Nct HO oc tjc n`ac sckdcat

Nct B hai I oc hay twe pe`ats ea HO se tjht HB 4 x hai BI 4 y. Yjca x + y > N, y 1

shdpnc sphbc `s rcprcscatci oy tjc rck`ea cabnesci oyEWX.

Hrch efEWX 4?

8N?

Yjc cvcat `s rcprcscatci oy tjc rck`ea, oeuaici oy tjc_RX

Hrch ef_RX 4?

8(N )?

preoho`n`ty ef tjc cvcat 4

?

N

N

Rcnf prhbt`bc preoncds =

(28) H n`ac sckdcat ef ncaktj h `s i`v`ici `a twe phrts ht rhaied oy thl`ak h pe`at ea `t, f`ai tjc

preoho`n`ty tjht ae phrt `s krchtcr tjha o, wjcrc ?o 1 h

(2?) H bnetj ef ncaktj 87 dctcrs s te oc rhaiedny i`str outci hdeak tjrcc oretjcrs, f`ai tjc preoho n`ty

tjht ae eac kcts derc tjha 9 dctcrs ef bnetj.

Haswcrs (28)h

ho? (2?)

?3

8