wikimama Class 12 Ch 11 Probability

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  • 8/4/2019 wikimama Class 12 Ch 11 Probability

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    [PG@F@CICZS

    JAECHCZCGH]

    Pfhjgd A|{apcdahz

    Pfhjgd A|{apcdahz c} fh a|{apcdahz rlg}a fii {g}}c`ia gqzmgda} ,pa}qiz}# fpa nhgrh

    ch fjufhma) `qz zla pa}qiz ge fhs }{amcecm {apegpdfhma mfhhgz `a {pajcmzaj `aegpa

    mgd{iazcgh ge zla a|{apcdahz(

    ]fd{ia ]{fma

    Zla }az ge fii {g}}c`ia gqzmgda} ge f pfhjgd a|{apcdahz c} }fcj zg `a zla }fd{ia }{fma

    ge zla a|{apcdahz( Cz c} q}qfiis jahgzaj `s ](

    Auahz

    Fhs }q`}az ge zla }fd{ia }{fma c} mfiiaj fh auahz( Fhs auahz f}}gmcfzaj rczl f pfhjgd

    a|{apcdahz mfh `a pa{pa}ahzaj `s f hgh ad{zs }q`}az A ge ]( Zla auahz A c} }fcj zg

    lfua gmmqppaj ce zla gqzmgda a ge zla a|{apcdahz }fzc}eca} a A(

    Avqfiis icnais auahz}'gqzmgda}

    Ch f pfhjgd a|{apcdahz) zrg gp dgpa auahz}' gqzmgda} fpa }fcj zg `a avqfiis icnais) ce hg

    gha ge zlad lf} dgpa mlfhma ge gmmqppahma zlfh gzlap}( Ch gzlap rgpj} afml auahz ge ]

    lf} }fda mlfhma ge gmmqppchb(

    Dqzqfiis A|miq}cua auahz}

    Ch fh a|{apcdahz) zrg gp dgpa auahz} fpa }fcj zg `a dqzqfiis a|miq}cua) ce zla gmmqppchb

    ge gha ge zlad pqia} gqz zla {g}}c`ciczs ge zla gmmqppchb ge gzlap}( c(a( zlas mfhhgz gmmqp

    }cdqizfhagq}is

    A|lfq}zcua }az ge auahz}

    F }az ge gqzmgda} ' auahz} c} }fcj zg `a a|lfq}zcua ce cz mguap} afml fhj auaps {g}}c`ia

    gqzmgda ge zla }fd{ia }{fma ge zla a|{apcdahz( c(a( fz iaf}z gha ge zlad lf} zg gmmqp ch

    f {apegpdfhma ge zla a|{apcdahz( Ce F6) F0) (((( ) Fh fpa auahz} jaechaj gh f }fd{ia }{fma

    ]) fhj F6 F0 F= ((( Fh 4 ]) zlah zla}a auahz} fpa mfiiaj f} a|lfq}zcua auahz} , c(a

    gha ge zla}a dq}z gmmqp#(

    Fh ad{zs }az c} firfs} f }q`}az ge f }az ]( Lahma) zla ad{zs }az mfh `a

    mgh}cjapaj f} pa{pa}ahzchb fh auahz ge fh a|{apcdahz( @qz zlapa c} hg a|{apcdahz

    rczlgqz fh gqzmgda( ) zlq} pa{pa}ahz} fh cd{g}}c ia auahz(

    Ra jaecha zla auahz hgz F ) f} zla auahz rlcml gmmqp} rlah F jga} hgz gmmqp(

    Gezah hgz F c} mfiiaj zla mgd{iadahzfps auahz ge F) gp habfzcgh ge F( ]chma F Fm

    4 ) auahz F fhj cz} mgd{iadahzfps Fm fpa dqzqfiis a|miq}cua(

    [PG@F@CICZS

    F}}gmcfzaj rczl afml gqzmgda ch f }fd{ia }{fma c} f pafi hqd`ap {,; { 6# mfiiaj zla

    {pg`f`ciczs ge zlfz gqzmgda rlcml pa{pa}ahz} zla mlfhma ge lf{{ahchb ge zla auahz(

    Jaechczcgh?

    Fh auahz c} }fcj zg lfua fh f}}gmcfzaj {pg`f`ciczs { jaechaj `s

    { 4hqd`ap

    Zgzfi

    ge gqzmgda} efugqpf`ia zg zla auahz

    hqd`ap ge gqzmgda}(

  • 8/4/2019 wikimama Class 12 Ch 11 Probability

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    Zlapaegpa ce fh auahz mfh lf{{ah ch f rfs} fhj efci ch ` rfs} ,zla zgzfi hg( `achb

    f . `# fhj ce fii zla rfs} fpa avqfiis icnais) zlah zla {pg`f`ciczs ge lf{{ahchb ge zla auahz

    4f

    {f `

    fhj zla {pg`f`ciczs ge hgh$gmmqppahma ge zla auahz 4`

    vf `

    (

    Hgza? { . v 4 6) Fi}g ; {) v 6( Zla {pg`f`ciczs ge fh auahz F c} jahgzaj s [ ,F#(

    Pqia 6?

    Ch zla mf}a ge dqzqfiis a|miq}cua auahz} zla mlfhma ge lf{{ahchb ge gha gp gzlap ge zlad c}

    zla }qd ge zla mlfhma} ge zla }a{fpfza auahz}(

    Mghjczcghfi [pg`f`ciczs

    Pqia 0?

    Ce ra rc}l zg echj zla {pg`f`ciczs ge zla pfhjgd auahz qhjap zla mghjczcgh zlfz zla pfhjgd auahz @ lf}

    gmmqppaj [,F'@# jahgza} zla {pg`f`ciczs ge auahz F lf{{ahchb) bcuah zlfz auahz @ lf} lf{{ahaj( Cz c}

    zla mghjczcghfi {pg`f`ciczs ge F) bcuah @( Rlcia mfimqifzchb [,F'@#) ra f}}qda zlfz auahz @ lf}

    gmmqppaj( Cz cd{ica} zlfz zla gqzmgda} efugqpf`ia zg @ `amgda zla zgzfi gqzmgda} fhj lahma

    gqzmgda} efugqpf`ia zg [,F'@# fpa gqzmgda} mgddgh zg F fhj @(

    Iaz h `a zla hqd`ap ge auahz} ch zla }fd{ia }{fma( Iaz d6 jahgza zla hqd`ap ge aiadahzfps

    auahz} efugqpf`ia zg @ rlcml fpa fi}g efugqpf`ia zg F fhj d 0 jahgza zla hqd`ap ge

    aiadahzfps auahz} efugqpf`ia zg @ ghis( G`ucgq}is d0 d6 fhj6

    0

    d[ F ' @

    d(

    [,F'@# 4 [,F bcuah @#

    4

    6 6

    0 0

    h F @ [ F @d ,d ' h#Zgzfi [,F @#

    Zgzfi h @ d ,d ' h# [,@# [ @

    hg( ge efu mf}a} fhj

    hg( ge mf}a}(

    Zlq} [,F'@# 4

    [ F @

    [ @ ) [ ,@# ;(

    ]cdcifpis[,@ [,@ F# [,F @#

    [ @ ' F ) [,F# ;[,F# [,F#

    fhj F#

    [,F#

    ,[,F @# 4 [,@#( ,[,F'@# 4 [,F#( ,[,@'F##(

    Hgza zlfz ce F fhj @ fpa chja{ahjahz auahz} zlah

    [,F @# 4 [,F# [,@#

    [,F'@# 4[,F @# [,F#[,@#

    [,F#[,@# [,@#

    rlcml }lgqij `a zla mf}a f} gmmqppahma ge F jga} hgz ja{ahj q{gh @(

    ]cdcifpis [,@'F# 4 [,@#(Chja{ahjahz Auahz}

    Auahz} fpa }fcj zg `a chja{ahjahz rlah zla lf{{ahchb ge fhs gha ge zlad jga} hgz feeamz

    zla lf{{ahchb ge fhs ge zla gzlap}(

    ,c#( F fhj @ fpa chja{ahjahz ce

    [,@'F# 4 [,@# fhj [,F'@# 4 [,F#(

    [,F fhj @# 4 [,F @# 4 [ F '@ [,@# [,F#[,@# (

    ,cc#( Zla {pg`f`ciczs ge zla mghmqppahma ge }auapfi chja{ahjahz auahz} c} zla {pgjqmz ge

    zlacp chjcucjqfi {pg`f`ciczca}(

    Padfpn?

  • 8/4/2019 wikimama Class 12 Ch 11 Probability

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    Zla mghuap}a ge zlc} c} fi}g zpqa c(a( ce h bcuah auahz} }fzc}es zla f`gua mghjczcgh

    zlah zlas rcii `a chja{ahjahz(

    Ce zla auahz} F fhj @ fpa chja{ahjahz) zlah F fhj @ fpa fi}g chja{ahjahz([fcprc}a Chja{ahjahz Auahz}

    Zlpaa auahz} A6) A

    0fhj A

    =fpa }fcj zg {fcprc}a chja{ahjahz ce [,A

    6A

    0# 4 [,A

    6# [,A

    0#)

    [,A0 A=# 4 [,A0# [,A=# fhj [,A= A6# 4 [,A=# [,A6#

    Zlpaa auahz} fpa }fcj zg `a dqzqfiis chja{ahjahz ce [,A6 A0# 4 [,A6#[,A0#) [,A0 A=# 4

    [,A0# [,A=# fhj [,A= A6# 4 [,A=# [,A6# fhj [,A6 A0 A=# 4 [,A6# [,A0# [,A=#(

    Ce zrg auahz} F fhj @ fpa dqzqfiis a|miq}cua)[,F @# 4 ; qz [,F# [,@# ; , Ch bahapfi#(

    [,F @# [,F# [,@# dqzqfiis a|miq}cua auahz} rcii hgz `a chja{ahjahz(

    Ra mghmiqja zlfz ce zrg auahz} fpa chja{ahjahz) zlas lfua zg lfua }gda mgddgh

    aiadahz `azraah zlad c(a zlas mfhhgz `a dqzqfiis a|miq}cua( Dqzqfiis a|miq}cuaha}} c}

    q}aj rlah zla auahz} fpa zfnah epgd zla }fda a|{apcdahz fhj chja{ahjahma c} q}aj

    rlah zla auahz} fpa zfnah epgd jceeapahz a|{apcdahz}(

    Zgzfi [pg`f`ciczs Zlagpadce F6) F0) ((() Fh `a f zla }az ge dqzqfiis a|miq}cua fhj a|lfq}zcua auahz} fhj A `a }gda

    auahz rlcml c} f}}gmcfzaj rczl F6) F0) ((() Fh( Zlah {pg`f`ciczs zlfz A gmmqp} c} bcuah `s

    [,A# 4h

    c cc 6

    [ F [ A ' F (

    @fsa"} Zlagpad

    ]q{{g}a F6) F0) ((( ) Fh fpa dqzqfiis a|miq}cua fhj a|lfq}zcua }az ge auahz}( Zlq}) zlas

    jcucja zla }fd{ia }{fma chzg h {fpz} fhj fh auahz @ gmmqp}( Zlah zla mghjczcghfi

    {pg`f`ciczs zlfz F6 lf{{ah} ,bcuah zlfz @ lf} lf{{ahaj# c} bcuah `s [,F6 '@# 4

    c c ch

    c 6c 6

    [ F @ [ F ([ @ ' F[ @

    [ F ([ @ ' F

    (

    Hgza? Ch mghjczcghfi {pg`f`ciczs zla }fd{ia }{fma c} pajqmaj zg zla }az ge }fd{ia}'

    gqzmgda} ch zla auahz rlcml c} bcuah zg lfua lf{{ahaj(

    [pg`f`ciczs jc}zpc`qzcgh ge f pfhjgd ufpcf`ia

    Mgh}cjap zla pfhjgd a|{apcdahz ge zg}}chb zrg mgch}( Zla }fd{ia }{fma ge zla gqzmgda

    c} ] ~LL) LZ) ZL) ZZt(

    Fezap mgd{iazchb zla a|{apcdahz) ra mgqhz zla hqd`ap ge zfci} zlfz lfua zqphaj q{( Iaz

    zlc} hqd`ap `a \(Ce \ 4 ;) hg zfci} lf} zqphaj q{ c(a( L) L lfua gmmqppaj(

    Ce \ 4 6) ghaa zfci lf} zqphaj q{ c(a( aczlap L) Z gp Z) L lfua gmmqppaj(

    Ce \ 4 0) zrg zfci} lfua zqphaj q{ c(a( Z) Z lfua gmmqppaj(

    Ra rpcza zla}a jazfci} f}

    ]fd{ia {gchz Hqd`ap ge zfci}

    LL ;

    LZ 6

    ZL 6

    ZZ 0

    Ra hgza zlfz f hqdapcmfi ufiqa ;) 6 gp 0 c} f}}cbhaj zg afml }fd{ia {gchz ge zla

    }fd{ia }{fma( Zlq}) hqd`ap} ;) 6) 0 fpa zla ufiqa} ge zla ufpcf`ia \ jahgzchb

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    hqd`ap ge zfci} ch afml ge zla gqzmgda ge zla a|{apcdahz( Ra mfii zlc} ufpcf`ia f

    pfhjgd ufpcf`ia(

    F pfhjgd ufpcf`ia c} zlq} f ufpcf`ia rlg}a ufiqa} fpa hqd`ap} jazapdchaj `s zla

    gqzmgda} ge f pfhjgd a|{apcdahz( Iaz q} hgr rpcza zla {pg`f`ciczs ge afml gqzmgda

    ch zla f`gua a|fd{ia( Ra lfua

    [ ,LL# 4 6

    :) [ ,LZ# 4 6

    :) [ ,ZL# 4 6

    :) [ ,ZZ# 4 6

    :

    [ ,\ 4 ;# 46

    :) [ ,\ 4 6# 4

    6

    :.

    6

    :4

    6

    0) [ ,\ 4 0# 4

    6

    :(

    Zla }s}zad mgh}c}zchb ge f pfhjgd ufpcf`ia \) fighb rczl [ ,\# c} mfiiaj zla

    {pg`f`ciczs jc}zpc`qzcgh ge \( Iaz zla pfhjgd ufpcf`ia \ zfna ufiqa} |6) |0) () |h rczl

    {pg`f`ciczca}

    [ ,\ 4 |6# 4 {6) [ ,\ 4 |0# 4 {0( [ ,\ 4 |d# 4 {d(

    Zla {pg`f`ciczs jc}zpc`qzcgh c}\ |6 |0 |=|d

    [ ,\# {6 {0 {={dZla eqhmzcgh [,\# c} mfiiaj zla {pg`f`ciczs eqhmzcgh \( Lapa { c8 ;) c 4 6) 0) ) d fhj

    d

    cc 6

    { 6(

    @chgdcfi Jc}zpc`qzcgh egp ]qmma}}cua Auahz}

    ]q{{g}a { fhj v fpa zla pa}{amzcua mlfhma} ge zla lf{{ahchb fhj efcichb ge fh auahzfz f }chbia zpcfi ,v 4 6 $ {#( Zlah zla mlfhma ge cz} lf{{ahchb p zcda} ,a|fmzis# ch hzpcfi} c} hMp {

    pvh$p `amfq}a zla mlfhma ge cz} lf{{ahchb p zcda} fhj efcichb h $ p zcda} chf bcuah gpjap c} {pvh $ p fhj zlapa fpa hMp }qml gpjap}) rlcml fpa dqzqfiis a|miq}cuafhj egp fhs }qml gpjap) zla {pg`f`ciczs c} {pvh $ p( Zlapaegpa) zla pavqcpaj {pg`f`ciczs c}

    hMp {pvh$p(Hgza? Zla {pg`f`ciczca} [,|# fpa bcuah `s zla zapd} ch zla `chgdcfi a|{fh}cgh ge,{ . v#h ( @amfq}a ge zlc}) zla {pg`f`ciczs jc}zpc`qzcgh c} mfiiaj f} f `chgdcfi jc}zpc`qzcgh(