probability 2 notes

7/30/2019 Probability 2 Notes

1/5

^rlbebfifxv 2 - Dlxm{ ;

Xjm kldhfxfldei hf{xrfb}xfld lc e redhl` serfebim Z afsmd ed msmdx B$

Imx Z bm e redhl` serfebim hmstdmh ld xjm {e`~im {~ekm [ edh B bm ed msmdx fd [$ Hmdlxm

^*Z 0 zyB& ^*Z0z edh B&

^*B&bv cZyB*z&$ Xjf{ f{ e ~rlbebfifxv `e{{ c}dkxfld$ Qm ked xjmrmclrm

stdh xjm mz~mkxexfld lcZ kldhfxfldei ld B$ M_ZyBP 0 zz cZyB*z&$

Mze`~im Qm xl{{ e klfd xqfkm$ Imx Z kl}dx xjm d}`bmr lc jmeh{) {l Z Bfdl`fei

2) 52

) edh

imx B5 bm xjm msmdx xjex xjm str{x l}xkl`m f{ e jmeh edh B2 bm xjm msmdx xjex xjm str{x l}xkl`m f{ e

xefi$ Xjmd ^*B5& 0 ^*{JX)JJp& 052

edh ^*B2& 0 ^*{X J)X Xp& 052

$

Jmdkm ^*Z 0 >yB5& 0 >) ^*Z 0 5yB5& 0^*{JXp

^*B5&0 5

2edh ^*Z 0 2yB5& 0

^*{JJp^*B5&

0 52

$ Xjmd

M_ZyB5P 0;2

$

Ei{l ^*Z 0 >yB2& 0 ^*{X Xp^*B2& 0 52 ) ^*Z 0 5yB2& 0 ^*{X Jp^*B2& 0 52 edh ^*Z 0 2yB2& 0 >$ Xjmrmclrm

M_ZyB2P 052

$

Qm ked ei{l lbxefd xjm kldhfxfldei hf{xrfb}xfld lcZyB5 edhZyB2 bv kld{fhmrfda xjm f`~ifkexfld{lc xjm mz~mrf`mdx$ Fc B5 lkk}r{ xjmd ZyB5 mu}ei{ 5 / V qjmrm V kl}dx{ xjm d}`bmr lc jmeh{fd xjm {mkldh xl{{ lc xjm klfd) {l V Bmrdl}iif

52

$ Fc B2 lkk}r{ xjmd ZyB mu}ei{ V$ Jmdkm

M_ZyB5P 0 5 /M_VP 0 5 /52

edh M_ZyB2P 0 M_VP 052

$

Qm qfii dlq illo ex e {f`fier ieq xl xjm ieq lc xlxei ~rlbebfifxv qjfkj f{ clr mz~mkxexfld{$ Xjf{

ked bm }{mh xl stdh xjm mz~mkxmh h}rexfld lc xjm {mu}mdkm lc ae`m{ *mz~mkxmh d}`bmr lc ae`m{

~ievmh& clr xjm ae`bimr{ r}fd ~rlbim`$

Xjm ieq lc xlxei ~rlbebfifxv clr mz~mkxexfld{

Crl` xjm ieq lc xlxei ~rlbebfifxv) fcB5)$$$)Bd ~erxfxfld [ xjmd clr edv ~l{{fbim sei}m lcz)

^*Z 0 z& 0d

g05

^*Z 0 zyBg&^*Bg& 0d

g05

cZyBg *z&^*Bg&$

@}ixf~ivfda bv z edh {}``fda qm lbxefd xjm Ieq lc Xlxei ^rlbebfifxv clr Mz~mkxexfld{

M_ZP 0d

g05

M_ZyBgP^*Bg&

Mze`~im$ Kld{fhmr xjm {mx-}~ clr e aml`mxrfk hf{xrfb}xfld$ Qm jesm e {mu}mdkm lc fdhm~mdhmdx

xrfei{ lc ed mz~mrf`mdx) qfxj ~rlbebfifxv ~ lc {}kkm{{ ex mekj xrfei$ Z kl}dx{ xjm d}`bmr lc xrfei{

xfii xjm str{x {}kkm{{$

Imx B5 bm xjm msmdx xjex xjm str{x xrfei f{ e {}kkm{{ edh B2 bm xjm msmdx xjex xjm str{x xrfei f{ e

cefi}rm$


2/5

Qjmd B5 lkk}r{) Z `}{x mu}ei 5$ [l ^*Z 0 5 edh B5& 0 ^*B5& edh ^*Z 0 z edh B5& 0 > fcz 4 5$ Jmdkm xjm hf{xrfb}xfld lcZyB f{ kldkmdxrexmh ex xjm {fdaim sei}m 5 f$m$ ZyB f{ fhmdxfkeiivmu}ei xl 5$

Fc B f{ xjm msmdx xjex xjm str{x xrfei f{ e cefi}rm) xjmd xjm d}`bmr lc xrfei{ }dxfi e {}kkm{{ fd xjm

{}b{mu}mdx xrfei{) V) je{ xjm {e`m hf{xrfb}xfld e{ Z$ Qm ei{l jesm kerrfmh l}x xjm str{x xrfei$

Jmdkm ZyB f{ mu}ei xl 5 /V qjmrm V je{ xjm {e`m hf{xrfb}xfld e{ Z$

Jmdkm M_ZyB5P 0 5 edh M_ZyB2P 0 5 /M_VP 0 5 /M_ZP$ Xjmrmclrm

M_ZP 0 M_ZyB5P^*B5& /M_ZyB2P^*B2& 0 ~5 / u *5 /M_ZP&$

Xjmrmclrm M_ZP 0 5~

$

Xjm ae`bimr{ r}fd ~rlbim`) xjm mz~mkxmh h}rexfld lc xjm ae`m$

Qm }{m xjm {e`m dlxexfld e{ bmclrm$ Xjm ae`bimr ~iev{ e {mrfm{ lc ae`m{ {xerxfda qfxj e {xeom

lc o }dfx{$ Jm {xl~{ ~ievfda qjmd jm rmekjm{ mfxjmr @ lr D }dfx{) qjmrm @ o D$ Imx Xobm xjm redhl` serfebim clr xjm d}`bmr lc ae`m{ ~ievmh *xjm h}rexfld lc xjm ae`m&$ Xjmd) fc B5edh B2 erm xjm msmdx{ xjm ae`bimr qfd{ xjm str{x ae`m edh xjm ae`bimr il{m{ xjm str{x ae`m)

xjm ieq lc xlxei ~rlbebfifxv clr mz~mkxexfld{ f{ g}{x

M_XoP 0 M_XoyB5P^*B5& /M_XoyB2P^*B2&

Fc jm qfd{ xjm str{x ae`m jm je{ o/ 5 }dfx{ {l xjm hf{xrfb}xfld lc Xo afsmd B5 je{ xjm {e`mhf{xrfb}xfld e{ 5 / Xo/5 qjmrm Xo/5 `me{}rm{ xjm h}rexfld lc xjm ae`m {xerxfda crl` o/ 5 }dfx{$Jmdkm M_XoyB5P 0 5 /M_Xo/5P$ [f`fieriv M_XoyB2P 0 5 /M_Xo5P$ Imx Mo hmdlxm M_XoP$ Xjmd

Mo 0 ~*5 /Mo/5& / u*5 /Mo5&

edh jmdkm qm lbxefd xjm hfccmrmdkm mu}exfld

~Mo/5Mo/ uMo5 0 5

Dlxm xjex M@ 0 MD 0 > {fdkm xjm ae`bifda {xl~{ ~ievfda f``mhfexmiv$ Qjmd ~ 052

e ~erxfk-

}ier {li}xfld xl xjf{ mu}exfld f{ Mo 0 Ko qjmrm K05

u~ $ Qjmd ~ 052

e ~erxfk}ier {li}xfld f{

Mo 0 Ko2 qjmrm K0 5$ Dlq) e{ clr hfccmrmdxfei mu}exfld{) xjm amdmrei {li}xfld xl xjm ~erxfk-

}ier hfccmrmdkm mu}exfld f{ xjm ~erxfk}ier {li}xfld g}{x lbxefdmh ~i}{ xjm amdmrei {li}xfld xl xjm

amdmrei mu}exfld ~Mo/5Mo/ uMo5 0 >$

Ke{m qjmd ~ 0 52$

Mo 0o

u~/E /B

u

~

o


3/5

[fdkm > 0 M@ 0@

u~ /E /B

u~

@edh > 0 MD 0

Du~ /E /B

u~

D) B 0 *D@&

*u~&

u~

@

u~

D

edh E 0 @*u~& B

u~

@

u~

@

u~

D $ Fc qm qrfxm Mo e{ Mo*@)D& xl mz~ifkfxiv fdki}hm xjm bl}dh-

erfm{ qm lbxefd

Mo*@)D& 0*o@&

*u~&

*D@&

*u~&

u~

o

u~

@

u~

D

u~

@

Ke{m qjmd ~ 0 52$

Mo 0 o2 /E /Bo

[fdkm > 0 M@ 0 @2 /E /B@ edh > 0 MD 0 D

2 /E /BD) B 0 D/@ edh E 0 @D$Jmdkm qrfxfda Mo e{ Mo*@)D& xl mz~ifkfxiv fdki}hm xjm bl}dherfm{

Mo*@)D& 0 *o@&*D o&

Kldhfxfldei hf{xrfb}xfld lcZyV qjmrm Z edh V erm redhl` serfebim{$

Clr edv sei}m v lc V clr qjfkj ^*V 0 v& 4 > qm ked kld{fhmr xjm kldhfxfldei hf{xrfb}xfld lcZyV 0v edh stdh xjm mz~mkxexfld edh serfedkm lcZ lsmr xjf{ kldhfxfldei hf{xrfb}xfld) M_ZyV 0vPedh Ser*ZyV 0 v&$ Imx cZyV*zyv& 0 ^*Z 0 zyV 0 v&$ Kld{fhmr xjm c}dkxfld lcV qjfkj xeom{ xjmsei}m M_ZyV 0 vP qjmd V 0 v$ Xjf{ f{ e redhl` serfebim qjfkj qm hmdlxm bv M_ZyVP$ [f`fierivqm hmstdm Ser*ZyV& edh M_a*Z&yVP xl bm xjm c}dkxfld{ lc V *{l redhl` serfebim{& qjfkj xeomsei}m Ser*ZyV 0 v& edh M_a*Z&yVP qjmd V 0 v$

Xjmlrm`$ *f& M_ZP 0 M_M_ZyVPP) *ff& Ser*Z& 0 M_Ser*ZyV&P /Ser*M_ZyVP& edh *fff& AZ*x& 0M_M_xZyDPP$

^rllc Qm {jlq xjex M_a*Z&P 0 M_M_a*Z&yVPP$ Dlq

M_a*Z&yV 0 vP 0

z0>

a*z&cZyV*zyv& 0

z0>

a*z&^*Z 0 z)V 0 v&

^*V 0 v&

M_M_a*Z&yVPP 0v0>M_a*Z&yV 0 vP^*V 0 v&

0 v0>z0> a*z& ^*Z0z)V0v&^*V0v& ^*V 0 v&

0 z0> a*z&v0> ^*Z 0 z)V 0 v&

0z0> a*z&^*Z 0 z& 0 M_a*Z&P


4/5

*f& Fc qm imx a*Z& 0 Z qm f``mhfexmiv lbxefd M_ZP 0 M_M_ZyVPP$

*ff& Fc qm imx a*Z& 0 Z2 qm lbxefd M_Z2P 0 M_M_Z2yVPP$

Dlq Ser*ZyV&& 0 M_Z2yVP *M_ZyVP&2 edh jmdkm

M_Ser*ZyV&P 0 M_M_Z2yVPPM_*M_ZyVP&2P 0 M_Z2PM_*M_ZyVP&2P

Ser*M_ZyVP& 0 M_*M_ZyVP&2P *M_M_ZyVPP&2 0 M_*M_ZyVP&2P *M_ZP&2

Xjmrmclrm M_Ser*ZyV&P /Ser*M_ZyVP& 0 M_Z2P *M_ZP&2 0 Ser*Z&$

*fff& Fc qm imx a*Z& 0 xZ qm lbxefd AZ*x& 0 M_xZP 0 M_M_xZyDPP$

Mze`~im Imx Z Bfdl`fei*d)~& edh V Bfdl`fei*`)~& qjmrm Z edh V erm fdhm~mdhmdx$Xjmd R 0 Z/VBfdl`fei*d / `)~&$

^*Z 0 zyR 0 r& 0 ^*Z0z)R0r&

^*R0r&0 ^

*Z0z)V0rz&^*R0r&

0 ^*Z0z&^*V0rz&

^*R0r&

0dKz~

zudz`Krz~rzu`r/z

d/`Kr~rud/`r0

dKz`Krz

d/`Kr

Jmdkm xjm kldhfxfldei hf{xrfb}xfld lcZyR 0 r f{ jv~mraml`mxrfk$ Xjf{ ~rlsfhm{ xjm be{f{ lc xjm22 kldxfdamdkv xebim xm{x lc mu}eifxv lc xql bfdl`fei ~ ~ere`mxmr{ fd {xexf{xfk{$

Mze`~im Xjm d}`bmr lc {~e` `m{{eam{ V fd e hev je{ ^lf{{ld hf{xrfb}xfld qfxj ~ere`mxmr $

Mekj {~e` `m{{eam *fdhm~mdhmdxiv& je{ ~rlbebfifxv ~ lc dlx bmfda hmxmkxmh bv xjm {~e` stixmr$

Imx Z bm xjm d}`bmr amxxfda xjrl}aj xjm stixmr$ Xjmd ZyV 0 v je{ Bfdl`fei hf{xrfb}xfld qfxj~ere`mxmr{ d 0 v edh ~$ Imx u 0 5~$

Jmdkm M_ZyV 0vP 0 ~v) Ser*ZyV 0v& 0 ~uv edh M_xZyV 0vP 0 *~x/ u&v {l xjex M_ZyVP 0 ~V)Ser*ZyV& 0 ~uV edh M_xZyVP 0 *~x/ u&V$ Xjmrmclrm7

M_ZP 0 M_M_ZyVPP 0 M_~VP 0 ~M_VP 0 ~

Ser*Z& 0 M_Ser*ZyV&P /Ser*M_ZyVP& 0 M_~uVP /Ser*~V& 0 ~*5~& / ~2 0 ~

AZ

*x& 0 M_M_xZyVPP 0 M_*~x/ u&VP 0 AV

*~x/ u& 0 m**~x/u&5& 0 m~*x5&

B}x xjf{ f{ xjm ~$a$c$ lc e ^lf{{ld r$s$ qfxj ~ere`mxmr 0 ~$ Jmdkm bv xjm }dfu}mdm{{ lc xjm~$a$c$) Z ^lf{{ld*~&$


5/5

Redhl` [}`{$

Imx Z5)Z2)Z;)$$$$ bm e {mu}mdkm lc fdhm~mdhmdx fhmdxfkeiiv hf{xrfb}xmh redhl` serfebim{ *f$f$h$

redhl` serfebim{&) mekj qfxj xjm {e`m hf{xrfb}xfld) mekj jesfda kl``ld `med ) serfedkm 2

edh ~$a$c$ AZ*x&$ Kld{fhmr xjm redhl` {}` V 0 Dg05Zg qjmrm xjm d}`bmr fd xjm {}`) D f{

ei{l e redhl` serfebim edh f{ fdhm~mdhmdx lc xjm Zg$ Xjmd qm ked }{m l}r rm{}ix{ clr kldhfxfldei

mz~mkxexfld{$

[fdkmM_VyD0 dP 0M_dg05ZgP 0dg05M_ZgP 0 d) qm lbxefd xjm rm{}ix xjexM_VP 0M_M_VyDPP 0

M_DP 0 M_DP$

[f`fieriv Ser*VyD 0 d& 0 d2 {l xjex

Ser*V& 0 M_Ser*VyD&P /Ser*M_VyDP& 0 M_D2P /Ser*D& 0 2M_DP /2Ser*D&

Ei{l qm ked lbxefd ed mz~rm{{fld clr xjm ~$a$c$ lcV$

M_xVyD 0 dP 0 M

mdg05Zg

0

d

g05

AZg *x& 0 *AZ*x&&d

{l xjex

AV*x& 0 M_M_xV

yDPP 0 M

*AZ*x&&D

0 AD*AZ*x&&

Mze`~im

Imx Zg bm xjm e`l}dx lc `ldmv xjm gxj k}{xl`mr {~mdh{ fd e hev fd e {jl~$ Xjm Z{ erm f$f$h$

redhl` serfebim{ qfxj `med 2> edh serfedkm 5>$ Xjm d}`bmr lc k}{xl`mr{ ~mr hev D je{

^lf{{ld hf{xrfb}xfld ~ere`mxmr 5>>$ Xjm xlxei {~mdh V fd xjm hev f{ V 0 Dg05Zg$ [l M_VP 0

*2>&*5>>& 0 2>>> edh Ser*V& 0 *5>&*5>>& / *2>&2*5>>& 0 85>>>$

probability 2 notes

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