Trabajo Sobre La Probabilidad, Permutaciones Taller 1

 

MKPTKPFMCKO YOCQIP]CRFPCF FORKOCK GK]I BI ]YMPI RFJJIP BI I]RFBC]RCMF COAIPIOMCFJ

FJYNOK7 GLIO MFPJK] PKNIPK KQCIBK

TPKEPFNF7 FBNCOC]RPFMCKO BI INTPI]F]

]INI]RPI7 QCC

GKPOFBF7 BCYPOF

BKMIORI7 NFPCK I. ]KJKPWFOK MFNTK

]COMJIGK-]YMPI

0404

 

6. Yo gurfbk istæ mknpuistk pkr muftrk pirskofs mfbf uof bi ijjfs lfræ uof farnfmcýo sk`ri ij fmusfbk7 fmus fbk7 mujpf`ji mujpf`ji k cokmioti. cokmioti. »Bi mufotfs mufotfs nfoirfs nfoirfs puibio rispkobir rispkobir jks gurfbks; »Io muæotfs pksc`cjcbfbis lfy inpftis io jf bimcscýo;

Pispuistf/ ]> {MMMM, MMMC, MMCM, MMCC, MCMM, MCMC, MCCM, MCCC, CMMM, CMMC, CMCM, CMCC, CCMM, CCMC, CCCM, CCCC} -

Jk Jkss gurf gurfbks bks puibio puibio rispko rispkobir bir bi 61 nfoir nfoirfs fs

-

Lf Lfy y inpf inpftis tis io jf bimcsc bimcscýo ýo io < pksc pksc`cj `cjcbf cbfbis bis

0. Yof nfrmf bi jf`krf jf`krftkr tkrck ck io ioi i qu qui i prk`fr prk`fr uo ouivk ouivk nib nibcmf cmfnio niotk tk pfrf pfrf bit bitirn irncof cofrr sc eioirfrf k ok uof rifmmcýo fjãrecmf io ij pfmcioti qui jk mkosuni. Jis fpjcmf f muftrk pfmciotis ij nibcmfniotk y foktf (]), sc prisiotk fjirecf, y (O) sc ok jk lczk. f.

»Muæj is is jf prk` prk`f`cj f`cjcbfb cbfb io qui qui fj nioks 0 bi jjks ks muftrk muftrk pfmci pfmciotis otis pris prisiotfrk iotfrko o fjirec fjirecf f fj nibcmfniotk; `. »Muæj is jjf f prk`f`cjcb prk`f`cjcbfb fb io qui næxc næxcnk nk uok bi jks jks muftrk pfmcioti pfmciotiss prisiotý prisiotý fjirec fjirecf; f;

PI]TYI]RF ]>sc O> ok ]>{ ssss, ssso, ssos, soss, osss, ssoo, soso, osos, ooss, soos, oss, sooo, osoo, ooso, ooos, oooo} f) ]>

66

>4,1< >1< %jf prk`f`cjcbfb bi qui fj nioks bi jks muftrk pfmciotis prisiotfrko

61 fjirecfs fj nibcmfniotk auirko bij 1 > 4,09>09% jf prk`f`cjcbfb bi qui næxcnk uok bi jks muftrk pfmciotis prisioti 61 fjirecf is bij 09%

TIPNYRFMCKOI]7 =. Yo pscmýjkek ji pcbi f uok bi jks ocöks qui vf ivfjufr, qui mkostruyf uo ounirk bi tris mcrfs, sco ripir ocoe÷o bcectk f) »Bi mu mufot fotfs fs kr krnfs nfs si si puibi puibi mkostru mkostrucr cr ij o÷ o÷nir nirk; k;

 

@)

]c fj ocök ocök si ji bfo bfo amlfs amlfs mko jks jks o÷nir o÷nirks ks bi 6 f 1, uof b bi i mfbf uof uof y si ji pc pcbi bi qu qui i mkokrni uo o÷nirk bi tris mcrfs, »bi mufotfs krnfs jk puibi lfmir;

PI]TYI]RF f) Cn Cnpk pkrt rtf f ij krbi krbio o sscc ]i puibio ripir ok Iotkomis istfnks lf`jfobk bi uof pirnutfmcýo sco riinpjfzk, pfrf jf mufj si ujczf jf akrnujf

 To>

  o!   ( o ∐r ) !

=

  64 !

r

 T64

>

( 64∐= ) !

o>64 y r>=

>

64 ! 5!

>

64  x 3  x <  x 5 ! 5!

 > 64x3 64x3x x 504

ij o÷nirk si puibi mkostrucr bi

504 akrnfs

`) Cnpk Cnpkrt rtf f ij krbi krbio; o; ]c ]i puibio ripir; Ok Iotkomis istfnks lf`jfobk bi pirnutfmcýo sco riinpjfzk, pfrf jf mufj si ujczf jf aýrnujf7 r

 To

9

>

  o!

( o ∐r ) !   < r>9 ( < ∐9 ) ! > = ! >

1504 

Jk puibi lfmir bi 1504 akrnfs

2. Mko jfs jitrfs bi jf pfjf`rf jc`rk, »muæotfs krbiofmckois bcstcotfs si puibio lfmir qui

inpcimio pkr vkmfj; Pispuistf

 

To> o! T9> 0x2x=x0x0>2< si puibio lfmir 2< krbiofmckois bcstcotfs.

MKN@COFMCKOI]. 9. ]i bisif mkokrnfr uo mknctã bi tris pirskofs, pfrf tfj ao si ioio sics mfobcbftks, bks lkn`ris y muftrk nugiris. »Muæj is jf prk`f`cjcbfb bi qui ij mknctã isti mkokrnfbk pkr lkn`ri y bks nugiris;

PI]TYI]RF/

o!

r

m o>

( ∐r )

r! o

  1!

=

m 1>

o>1 r>=

= ! ( 1 ∐= ) !

>

  1! =! = !

>

1 x 9 x 2 x =!=!

>

604 1

>04

  Qfnks flkrf f mfjmujfr jks mfsks afvkrf`jis7 jks lk lkn`ris n`ris jks ismkeinks ismkeinks bi uof nfoirf jf mufj   2!   0!   2!   0! 6 6 > > > 2  pkr tfotk jks >0  y jfs nugiris nugiris bi M 2> is m 0> 6 ! ( 2 ∐6 ) ! 6 ! = ! 6 ! ( 0 ∐6 ) ! 6 ! 6 !

mfsks afvkrf`jis sko 6*2> 2 y jf prk`f`cjcbfb (o÷nirk bi mfsks afvkrf`ji afvkrf`jiss bcvcbcbks pkr ij o÷nirk bi mfsks pksc`jis) is

2 04

>

6 9

1) Io ij ricofbk ofmckofj bi jf `ijjizf, si sijimmckofrko mknk aofjcstfs jfs mfobcbftfs bi @kektæ, ]umri, Fokqucf, Qfjji, ]fo Fobrãs y Mýrbk`f. Iotri ijjfs si bi`i ismkeir jf ricof y jf vcrricof. f.

»Muæj »Muæj is jjf f prk`f` prk`f`cjc cjcbfb bfb b bi i qui jf jf ric ricof of si sif f ]umri; ]umri;

Pispuistf/ r

 M o> r

M o>

  o!   r ! ( o ∐r ) !   1!

0 1

0

!(

∐ )!

>

o>1 y r>0   1! 0! 2 !

>

1 x 9 x 2 0!2 !

>

=4 0

> 69

si mfomijfo jks 2 

 

r

Flkrf vfnks f mfjmujfr jks mfsks afvkrf`jis M o>

6

M 1

>

  1! 6 ! ( 1 ∐6 ) !

>

  1! 6 !9!

>

1  x 9 ! 6! 9 !

1

> >1

 

o!

( ∐r ) !

r! o

si mfomijfo jks 9

6

Jf prk`f`cjcbfb (o÷nirk bi mfsks afvkrf`jis bcvcbcbks pkr ij o÷nirk bi mfsks pksc`jis) 1

69

>

0 9

5. Bks iqu iqucpks cpks bi `ics`kj `ics`kj ioio ioio jf ncsnf ncsnf mfpf mfpfmcbfb mcbfb y guiefo guiefo ij uok mk mkotrf otrf ij ktr ktrk k uof sirc sirci i bi muftrk guieks. ]i riecstrf ij risujtfbk ri sujtfbk bi mfbf guiek. »Muæj is jf prk`f`cjcbfb bi qui ij prcnir iqucpk ok efoi ocoeuok bi jks muftrk guieks;

PI]TYI]RF

M ro>   o !   r ! ( o ∐r ) ! 0

M 2

>

  2! 0 ! ( 2 ∐0 ) !

>

o>2 y r>0   2! 0! 0 !

>

2  x =  x 0  x 0 ! 0!

>

60 0

>1 

si mfomijfo jks 0

Flkrf mfjmujfnks jks mfsks afvkrf`jis

6

M 2

>

  2! 6 ! ( 2 ∐6 ) !

>

  2! 0! = !

>

2 ! x=! 6! = !

>2

si mfomijfo jks =

Jf prk`f` prk`f`cjc cjcbfb bfb (o÷nir (o÷nirks ks bi mfs mfsks ks afvkrf afvkrf`ji `jiss bcvcbc bcvcbcbks bks pkr ij o÷nirk o÷nirk bi mfsks mfsks 2 pksc`jis is 1

Pisuijvi jks sceuciotis prk`jinfs ujczfobk jfs riejfs k jiyis bi jfs prk`f`cjcbfbis prk`f`cjcbfbis..

4,04 8 T ( @ )>4,14  y T ( F S @ )> 4,69

f. »F S @ sko nutuf nutuf nioti ixmjuyioti ixmjuyioti; ; `. K`tioef K`tioef T (F K @)

 

PI]TYI]R

m.

' 

Iomuiotri  T ( F  ) 

Pispuistfs F) ]ko ]ko nu nutuf tufni nioti oti ixmju ixmjuyio yiotis tis @) T ((Fu Fu@) @)> > T(F T(F)) + T( T(@) @) > 4,04 + 4,14   > 4, 04

3. uo uof f mfbiof mfbiof bi bi ristfu ristfurfo rfoti ti rif rifjc jczk zk uo is istub tubck ck bi n nirm irmfbik fbik pfrf pfrf b bitir itirnco ncofr fr ij pk pk fjcniotk qui mkosunio sus mjciotis, y si k`tuvcirko ij sceucioti risujtfbk7

bi

Jf prk`f`cjcbfb bi qui uo mjcioti tkni uof bitirncofbf `i`cbf is bi 4.9, jf prk`f`cjcbfb bi qui uo mjcioti mkosunf fje÷o pk bi fjcniotk sýjcbk is bi 4.1, jf prk`f`cjcbfb bi qui uo mjcioti mkosunf f jf viz uo jîqucbk y skjcbk is bi 4.2 »Muæj is jf prk`f`cjcbfb bi qui uo mjcioti mkosunf uo jîqucbk k uo sýjcbk;

PI]TYI]RF T(s)> 4,1 T (J)>; T (]FJ)> 4,2 T (] u J) > T (s) + T (J) + T ( ]oJ) > 4,1 + T (J) ” 4,2 > T (J) + 4,0 > T (J)> 4,0 T (] u J) > T (]) + T (J) ” T (]oJ) > 4,1 + 4,0 ” 4,2 > 4,2>24% jf pksc`cjcbfb bi qui ij mjcioti mkosunf uo jîqucbk k uo sýjcbk is 4,2%>24%

 

64. Tfrf uof mcirtf pk`jfmc pk`jfmcýo ýo bi pirskofs pirskofs mko bcsmfpfmcbfb bcsmfpfmcbfb mko bcsmfpf bcsmfpfmcbfb mcbfb si rijfmckoý jf sceucioti cokrnfmcýo io ij ripkrti foufj jks lkspctfjis bi jf mcubfb. Rcpk bi comfpfmcbfb Lkspctfj Bcstrctfj

@rfzks 94

Tcirofs 4,9= >9= % jf prk`f`cjcbfb bi qui ok sif io sus pcirofs bij > 9=%

> 4,5561 %jf prk`f` prk`f`cjc cjcbfb bfb bi qui qui prk prkvi vioef oef bij lks lkspct pctfj fj ofmcko ofmckofj fj y tioef tioef 299 bcsmfpfmcbfb io sus is bi 61%

66. Io uof uocvirscbfb bi jf mcubfb si ripkrtý ij mkoskjcbfbk bi bi jfs fmujtfbis f jfs mufjis si prisiotfrko ij nfykr o÷nirk bi fspcrfoti y su eãoirk. Jks risujtfbks si nuistrfo io jf tf`jf.

AFMYJRFB Eioirk Lkn`ri Nugir Rktfj

Frtis 604 =90 250

Birimlk 014 2=9 139

Lunf. 6 4,26>26 %  jf prk`f`cjcbfb bi qui ok sif lfyf prisiotfbk f jf afmujtfb

bi frtis k sif nugir is bi 4,26> 26%. `)

  649 061=

>4,42 >2 %jf prk`f`cjcbfb bi qui si lfyf prisiotfbk io jf afmujtfb bi

mciomcfs y ok sif lkn`ri is bi 4,42 > 2%