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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)
8/12/2019 Probability Theory E
2/25
DHYJR
"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}
8/12/2019 Probability Theory E
3/25
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.
8/12/2019 Probability Theory E
4/25
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
8/12/2019 Probability Theory E
5/25
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
`.c. W(H) 4ad
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.
8/12/2019 Probability Theory E
6/25
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.
8/12/2019 Probability Theory E
7/25
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
8/12/2019 Probability Theory E
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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