Taller de Probabilidades

 

RHAA@Z B@ TZIJHJFAFBHB@^ 8. Ais sfkuf`lt`s bhtis r`pr`s`lthl ais r`suathbis b` ulh flv`stfkhnfþl b` g`rnhbis m`nmh `l `a èr`h G`tripiafthlh b` G`b`aaçl, ah nuha flnauyþ ulh gu`strh b` 8444 chgfafhs. Tis``l Rv nhja`

Li pis``l Rv nhja`

Ritha

Tis``l vfvf`lbh Li pis``l vfvf`lbh

:;4 840

864 2>0

234 024

Ritha

>;0

3:0

8444

Nuha `s ah prijhjfafbhb b` qu` ulh chgfafh `a`kfbh ha`htirfhg`lt`? h) Tis`h Rv nhja`9 j) Tis`h vfvf`lbh y Rv nhja`9 n) Tis`h vfvf`lbh i Rv nhja`9 b) li pis`h vfvf`lbh lf pis`h Rv nhja`9 `) Tis`h Rv nhja` bhbi qu` `s ulh chgfafh qu` li pis`` vfvf`lbh9 c) Tis`h vfvf`lbh bhbi qu` `s ulh chgfafh qu` li pis`` Rv nhja`9

:. ^` `ltr`vfstþ h ul krupi s`a`nti b` `gpa`hbis b` ulh `gpr`sh nil r`sp`nti h ul pahl b` p`lsfil`s. ^` ijtuvi ah sfkuf`lt` flcirghnfþl? Nahsfcfnhnfþl

@v`lti

Lÿg`ri b` `gpa`hbis

^up`rvfsir`s B` ghlt`lfgf`lti B` pribunnfþl K`r`lnfh ^`nr`thrfha

H J N B @

>4 :4 :44 84 :44

h) ^f s` s`a`nnfilh ha hzhr ul `gpa`hbi, ¹Nuèa `s ah prijhjfafbhb b` qu` s`h b` ghlt`lfgf`lti i s`nr`thrfha9 j) ^f s` s`a`nnfilh ha hzhr bis `gpa`hbis, ¹Nuèa `s ah prijhjfafbhb b` qu` hgjis s`hl b` pribunnfþl9

>. ^` mh `ahjirhbi ulh kigh b` ghsnhr qu` hyubh h quf`l`s b`ohl b` cughr. ^f `a 34% b` ahs p`rsilhs qu` ah ushl b`ohl b` cughr, ¹Nuèa `s ah prijhjfafbhb b` qu` `l ul krupi b` tr`s cughbir`s, pir ai g`lis uli b`o` b` cughr9

2. ^` r`nfj` ul ait` b` nigputhbir`s, b` ais nuha`s 80 `stèl ju`lis, > tf`l`l b`c`ntis b` t`nahbi y : tf`l`l b`c`ntis b`c`nt is b` bfsni buri. ^` s`a`nnfilhl s`a`nnfilhl tr`s ha hzhr sfl r``gpahzi. r``gpahzi. Nuha `s ah prijhjfafbhb prijhjfafbhb b` qu` ha g`lis uli t`lkh ul b`c`nti9

 

0. Nhbh v`lb`bir `l ulh `gpr`sh s` nhafcfnh nil r`sp`nti h su mhjfafbhb phrh ahs v`lths, y nil r`sp`nti b` su pisfjfafbhb b` priginfþl h ul nhrki sup`rfir. Ah thjah gu`strh ais r`suathbis.

Mhjfafbhb `l ahs v`lths

Tisfjfafbhb b` priginfþl

Ritha

Z`kuahr

Ju`lh

@xn`a`lt`

83

8:

::

04

Trig`bfi

>4

04

24

8:4

Tir `lnfgh b`a prig`bfi Ritha

0 08

80 ;;

84 ;:

>4 l7:44

h) B`cflh ul `xp`rfg`lti ha`htirfi ` fb`ltfcfqu` bis `v`ltis qu` s`hl `xnauy`lt`s. j) ¹Nuèa `s ah prijhjfafbhb b` qu` ul v`lb`bir s`a`nnfilhbi ha hzhr t`lkh mhjfafbhb phrh ahs v`lths pir  `lnfgh b`a prig`bfi i `xn`a`lt` pisfjfafbhb b` priginfþl9 n) ¹Nuèa `s ah prijhjfafbhb b` qu` ais bis prfg`ris v`lb`bir`s s`a`nnfilhbis ha hzhr s`hl nilsfb`rhbis nigi v`lb`bir`s prig`bfi9 b) ^f s` shj` qu` ul v`lb`bir s`a`nnfilhbi ha hzhr tf`l` `xn`a`lt` pisfjfafbhb b` priginfþl, nuha `s ah prijhjfafbhb b` qu` t`lkh ulh mhjfafbhb `l ahs v`lths pir b`jhoi b`a prig`bfi9 `) B`a titha b` ais qu` tf`l`l r`kuahr pisfjfafbhb b` priginfþl5 quæ % b` ais v`lb`bir`s tf`l` ulh mhjfafbhb pir `lnfgh b`a prig`bfi9

3. ^upþlkhs` qu` b` ul ait` b` 84 gfnris mhy > b`c`ntuisis. B` bfnmi ait` s` `xtrh` ha`htirfhg`lt` ulh gu`strh gu`str h b` tr`s gfnris sfl r``gpahzi. r``gpahzi. @a ait` s` r`nmhzh r`nmhzh sf ha g`lis ul gfnri `stè b`c`ntuisi. b`c`ntuisi. ¹Nuèa `s ah prijhjfafbhb b` qu` s` r`nmhn` `a ait`9

;. Pl ait` b` 04 pribuntis s` nahsfcfnþ s`kÿl su nhafbhb y s`kÿl su ghrnh. Ah thjah gu`strh ais r`suathbis. Nhafbhb  Hn`pthja` Li hn`pthja` Ritha

Ghrnh  H 84 : 8:

J 80 2 86

Ritha 8; : 86

2: = l704

^f s` s`a`nnfilh ha hzhr ul pribunti? h) Mhaa` ah prijhjfafbhb b` qu` s`h b` ghrnh H i ghrnh J. j) ^f r`suath hn`pthja`, ¹Nuèa `s ah prijhjfafbhb b` qu` s`h b` ghrnh N9 n) Mhaa` ah prijhjfafbhb b` qu` s`h hn`pthja` i b` ghrnh N9 b) B`snrfjh (mhkh ulh afsth) `s `sphnfi gu`strha ^ b` hnu`rbi h ah vhrfhja` ghrnh b`a pribunti

 

`) Gu`str` ulh hpafnhnfþl b` Jhy`s* ^f s` s`a`nnfilh ha hzhr bis b` ais pribuntis. Nuèa `s ah prijhjfafbhb b` qu`? c) Ha g`lis uli s` b` ghrnh H sf `a gu`str`i s` mhn` nil r`pisfnfþl9 k) B` qu` ha g`lis uli s`h b` ghrnh N sf `a gu`str`i s` mhn` sfl r`pisfnfþl9

*=. @l nf`rth khsiafl`rh, 24% b` ais naf`lt`s nigprhl khsiaflh r`kuahr sfl paigi, >0% khsiaflh `xtrh sfl paigi y :0% khsiaflh pr`gful pr`gful sfl paigi. B` ais naf`lt`s naf`lt`s qu` nigprhl nigprhl khsiaflh r`kuahr, sþai `a >4% aa`lhl sus thlqu`s. B` ais qu` nigprhl khsiaflh `xtrh, `a 34% aa`lhl sus thlqu`s, gf`ltrhs qu`, b` ais qu` nigprhl pr`gful, 04% aa`lhl sus thlqu`s. h) ¹Nuèa `s ah prijhjfafbhb b` qu` `a sfkuf`lt` naf`lt` pfbh khsiaflh `xtrh y aa`l` `a thlqu`9 j) ¹Nuèa `s ah prijhjfafbhb b` qu` `a sfkuf`lt` naf`lt` aa`l` `a thlqu`9 n) ^f `a sfkuf`lt` naf`lt` aa`lh `a thlqu`, ¹Nuèa `s ah prijhjfafbhb b` qu` ai mhyh m`nmi nil kh khsiaflh siaflh `xtrh9

6. Plh `gpr`sh b` t`aæcilis ip`rh tr`s `sthnfil`s r`a`vhbirhs fbæltfnhs `l bfc`r`lt`s sftfis. Burhlt` ul p`rçibi b` ul høi, `a lÿg`ri b` b`sp`rc`ntis r`pirthbis pir nhbh `sthnfþl y ahs nhushs s` bhl `l ah sfkuf`lt` thjah?

Nhushs

@sthnfil`s

Ritha

Trija`ghs nil `a sugflfstri b` `a`ntrfnfbhb B`sp`rc`nti b`a nigputhbir Chaahs b` `qufpi `a`ntrþlfni @rrir`s mughlis

 H : > 2 ;

J 8 > 2 3

N 8 : : 0

Ritha

83

82

84

 

2

 

=

 

84

 

8= l724

h) ^f inurr` ul r`pirt`, ¹Nuèa `s ah prijhjfafbhb b` qu` s`h pir b`sp`rc`ntis b`a nigputhbir i qu` priv`lkh b` ah `sthnfþl N9 j) ^f s` s`a`nnfilh ha hzhr bis b` ais r`pirt`s, ¹Nuèa `s ah prijhjfafbhb b` qu` uli s`h b` ah `sthnfþl H9

* 84. @l nf`rth pahlth b` giltho`, tr`s gèquflhs H, J y N, gilthl 20, :0 y >4% b` ais pribuntis r`sp`ntfvhg`lt`. ^` shj` b` ah `xp`rf`lnfh phshbh qu` 63%, 6=% y 6;% b` ais giltho`s m`nmis pir nhbh gèquflh, r`sp`ntfvhg`lt` sil li b`c`ntuisis. ^f s` s`a`nnfilh ha hzhr uli b` ais giltho`s. h) Nuèa `s ah prijhjfafbhb b` qu` `st` s`h b`c`ntuisi9 j) ^f `a pribunti s`a`nnfilhbi r`suatþ b`c`ntuisi, ¹Nuèa `s ah prijhjfafbhb b` qu` v`lkh b` ah gèquflh H9

 

88. Ah `xp`rf`lnfh b`gu`strh qu` ais p`rlis pribunfbis `l ulh gèquflh sil b`ghsfhbi krhlb`s `a 84% b` ahs v`n`s y b`ghsfhbi p`qu`øis `a 0 % b` ahs v`n`s. ^f ul naf`lt` nigprh ul p`rli, ¹Nuèa `s ah prijhjfafbhb b` qu` li s`h lf b`ghsfhbi p`qu`øi lf b`ghsfhbi krhlb`9

8: 8:.. ^upi ^upilk lkh h qu qu` ` ai aiss giti gitir`s r`s b` ul h` h`ri ripa pahl hli i ip`r ip`rhl hl `l ci cirg rgh h fl flb` b`p` p`lbf lbf`l `lt` t` y b` qu qu` ` ch chaaaahl hl ni nil l prijhjfafbhb b` 4.:. ^upilf`lbi qu` ul hvfþl r`hafzh ul vu`ai s`kuri `l thlti s` ghlt`lkh culnfilhlbi nuhlbi g`lis ah gfthb b` sus gitir`s. B`t`rgfl` quæ hvfþl, uli b` nuhtri gitir`s i uli b` bis gitir`s, tf`l`l ghyir prijhjfafbhb b` t`rgflhr su vu`ai `xftishg`lt`9

8>. Plh nigphøçh nigphøçh `stè nilsfb`rhlbi nilsfb`rhlbi fltribunfr fltribunfr tr`s lu`vis pribuntis `l `a g`rnhbi g`rnhbi ainha. @a k`r`lt` k`r`lt` nr`` qu` ah prijhjfafbhb b` æxfti `s b`a 04% phrh `a prfg`ri, 24% phrh `a s`kulbi y 34% phrh `a t`rn`ri. h) B`cflh ais `v`ltis y b`snrfjh (mhkh ulh afsth) `a `sphnfi gu`strha ^. j) Nu Nuha ha `s ah pr prij ijhj hjfafafb fbhb hb b` qu` qu` ha g`li g`liss uli uli b` ai aiss pr prib ibun unti tiss t` t`lk lkh h æx æxft fti, i, sf s` ni nils lsfb fb`r `rhl hl flb`p`lbf`lt`s9

82. ¹Nuèa `s ah prijhjfafbhb b` qu` ulh gèquflh qu` pribun` 3 ft`gs b`c`ntuisis b` nhbh 8444, pribuznh 2 ft`gs b`c`ntuisis s`kufbis.

80. Plh gèquflh nilsth b` bis .cusfja`s b` s`kurfbhb H y J nil`nthbis `l s`rf`5 `s b`nfr, sf uli b` ais bis chaah, ah gèquflh s` phrh, ^f ah prijhjfafbhb b` qu` `a cusfja` H y J chaa`l `s b` 4.:4 y 4.>4 r`sp`ntfvhg`lt`. h) B`cflh ais `v`ltis y hsçkl`a`s ahs prijhjfafbhb`s nirr`spilbf`lt`s. j) ¹Nuèa `s ah prijhjfafbhb b` qu` ais bis cusfja`s b` qu` ais bis cusfja`s chaa`l ha gfsgi tf`gpi9 (Z? 4.43) n) Nuèa `s ah prijhjfafbhb b` qu` ah gèquflh chaa` pir nhush b`a cusfja` H9(Z/4.82) b) Nuèa `s ah prijhjfafbhb b` qu` ah gèquflh chaa` pir nhush b` nuhaquf`rh b` ais bis cusfja`s9 Z/4.>= `) Nuèa `s ah prijhjfafbhb b` qu` ah gèquflh s` phr`9 Z/4.22

83. Nuèa `s ah prijhjfafbhb b` ijt`l`r `xhnthg`lt` bis nhrhs ha ahzhr tr`s gil`bhs9 Z/4.>;0

8;. Nuha `s ah prijhjfafbhb b` ijt`l`r 3 pultis ha ahlzhr bis bhbis sfgætrfnis9 Z/4.8>6

8=. Pl bfspisftfvi nilsfst` b` tr`s nigpil`lt`s flb`p`lbf`lt`s H, J y N. @a bfspisftfvi s` nilsfb`rh b`c`ntuisi sf uli i gès b` ais nigpil`lt`s ai sil. Ah prijhjfafbhb b` qu` nhbh nigpil`lt` s`h b`c`ntuisi `s 4.48, 4.4: y 4.40 r`sp`ntfvhg`lt`. h) ¹Nuèa `s ah prijhjfafbhb b` qu` `a bfspisftfvi s`h b`c`ntuisi b`jfbi ha nigpil`lt` H9 Z/I.446>8 j) Nuèa `s ah prijhjfafbhb b` qu` `a bfspisftfvi s`h b`c`ntuisi9 Z/ 4.4;=>8

 

86. Pl bhbi sfgætrfni tf`l` ul ahbi pflthbi b` rioi, bis b` v`rb` y `a r`sti b` l`kri. ^f s` tfrh `a bhbi 2 v`n`s, ¹Nuèa `s ah prijhjfafbhb b` qu`? h) Ahs tr`s prfg`rhs v`n`s s` ijt`lkh rioi y ah ÿatfgh v`rb`9 j) ^þai ahs tr`s prfg`rhs v`n`s s` ijt`lkh rioi9 n) Ahs tr`s prfg`rhs v`n`s s` ijt`lkh rioi9

*:4. Ah prijhjfafbhb b` qu` ul ohrbfl`ri b` ulh nhsh iavfb` r`khr `a ohrbçl `s :/>. ^f ai rf`kh, ah prijhjfafbhb b` qu` s` ghrnmft` `s b` 8/:. ^f li ai rf`kh ah prijhjfafbhb b` qu` s` ghrnmft` `s b` >/2. @a bu`øi b` ah nhsh, h su r`kr`si mhaah `a ohrbçl ghrnmfti. ¹Nuèa `s ah prijhjfafbhb b` qu` `a ohrbfl`ri li ai mhyh r`khbi9

:8. @l ah chjrfnhnfþl chjrfnhnfþl b` nf`rti nf`rti hrtçnuai s` `lnu`ltrh qu` s` pr`s`lth pr`s`lth ul b`c`nti b`c`nti tfpi 8 nil ulh prijhjfafbhb prijhjfafbhb b` 4.8 y ul b`c`nti tfpi : nil prijhjfafbhb b` 4.40. ^` supil` flb`p`lb`lnfh `ltr`s ais bis tfpis b` b`c`ntis. B`t`rgflhr?  h) Ah prijhjfafbhb b` qu` ul hrtçnuai s`h b`c`ntuisi. j) Ah prijhjfafbhb b` qu` ul hrtçnuai t`lkh sþai ulh nahs` b` b`c`nti n) Ah prijhjfafbhb b` qu` ul hrtçnuai qu` s` shj` tf`l` b`c`nti 8, li t`lkh b`c`nti :.

::. @a bfr`ntir b` ah pahlth b` ah `gpr`sh jhahlzhs s.h, s` flt`r`sh pir nilin`r `a pirn`ltho` b` pf`zhs ju`lhs y b`c`ntuishs pribunfbhs `l tr`s s`nnfil`s b` ah pahlth b` pribunnfþl, nil `a cfl b` fgpa`g`lthr  sfst`ghs b` g`oirhgf`lti qu` prigu`vhl ah nhafbhb, s`kÿl ais r`suathbis. Ah flv`stfkhnfþl hrrioþ ais sfkuf`lt`s bhtis?

Nhafbhb PLFBHB@^ JP@LH^ PLFBHB@^ B@C@NRPI^H^ Ritha

^@NNFÞL

 

^8 6=4 :4

^: 2;4 >4

^> 8034 =4

8444

044

8324

Ritha

 

>484

 

8>4 >824

h) Xuæ s`nnfþl `s susn`ptfja` phrh ghyir hnnfþl k`r`lnfha qu` prip`lbh pir ulh g`oir nhafbhb9 ^f ah cr`nu`lnfh r`ahtfvh, s` tigh nigi prijhjfafbhb `lniltrhr? j) T(^ 8 P ^:/ J) n) T(^ 8 P ^>) b) T(^: / J)

 

::. @l ah nilstrunnfþl b` ulh nhrr`t`rh s` mhl niltrhthbi ahs nigprhs y `ltr`khs b` hr`lh, krhvh y n`g`lti. Nigi `s shjfbi, li sf`gpr` ais ght`rfha`s sil `ltr`khbis h tf`gpi. Ah `xp`rf`lnfh flbfnh qu` ahs prijhjfafbhb`s b` ul `ltr`kh ipirtulh phrh ah hr`lh, krhvh y `a n`g`lti sil 4.>, 4.3 y 4.= r`sp`ntfvhg`lt`. ^f ah `ltr`kh b` ul ght`rfha `s flb`p`lbf`lt` b` ah b` ais itris bis ght`rfha`s? h) @lnu`ltr` ah prijhjfafbhb b` qu` ais tr`s ght`rfha`s s`hl `ltr`khbis h tf`gpi. Z/4.822 j) @lnu`ltr` ah prijhjfafbhb b` qu` ha g`lis uli b` ais ght`rfha`s s`h `ltr`khbi h tf`gpi. Z/4.622.

:>. Pl hk`lt` v`lb`bir `stfgh qu` ah prijhjfafbhb b` mhn`r ulh v`lth `l su prfg`rh vfsfth 4.>, p`ri qu` hug`lth h 4.30 `l ah s`kulbh vfsfth sf `l ah prfg`rh li `c`ntuþ ah v`lth. Tir `strht`kfh `a v`lb`bir sþai vfsfth ulh i, sf `l ah prfg`rh li v`lb`, bis v`n`s. Nhanua` ah prijhjfafbhb b` qu` `a v`lb`bir mhkh ulh v`lth. Z/4.;00

:2. Plh `gpr`sh qu` pribun` pf`zhs g`tèafnhs b` pr`nfsfþl utfafzh ulh gèquflh `sp`nfha phrh `a hnhjhbi b` ahs pf`zhs. Ah gèquflh tf`l` bis niltria`s hutigètfnis nil`nthbis `l s`rf` b` gibi qu` ah gèquflh culnfilh nirr`nthg`lt` sþai sf hgjis niltria`s culnfilhl nirr`nthg`lt`. Ah prijhjfafbhb b` culnfilhgf`lti b` ul niltria `s b` 66% y `s flb`p`lbf`lt` b`a culnfilhgf`lti b`a itri. @lnu`ltr` ah prijhjfafbhb b` qu`? h) Lflkuli b` ais niltria`s culnfil` nirr`nthg`lt`. Z/4.4448 j) Pli b` ais niltria`s culnfil` nirr`nthg`lt`. Z/4.486= n) Ah gèquflh culnfil` nirr`nthg`lt`. Z/4.6=48

:0. ^upilf`lbi qu` phrh hug`lthr ah prijhjfafbhb b` qu` ah gèquflh culnfil` nirr`nthg`lt` (v`r prija`gh hlt`rfir), ul phr hbfnfilha b` niltria`s a`s sil flsthahbis thgjfæl `l s`rf`. Ah cirgh `l qu` `st` phr  hbfnfilha ip`rh `l r`ahnfþl ha phr hbfnfilha `s qu` hmirh ah gèquflh culnfilhrè nirr`nthg`lt` sf ul phr i ais bis bis ccu ulnfi lnfilh lhl l nir nirr` r`nt nthg hg`l `lt` t`.. @ @ln lnu` u`lt ltr` r` ah pr prijhj ijhjfafaffbhb bhb b b` ` qu` qu`?? ' h) Lflkuli b` ais phr`s culnfil` nirr`nthg`lt`. Z/4.444>6348 j) Pli b` ais phr`s culnfil` nirr`nthg`lt`. Z/4.4>644;6= n) Hgjis phr`s culnfil`l nirr`nthg`lt`. Z/ 4.63 b) Ah gèquflh culnfil` nirr`nthg`lt`. Z/4.6666

:3. Rr`s niltrhtfsths nigpft`l pir ah nilstrunnfþl b` nf`rth ijrh. Pl `xp`rti nr`` qu` `a niltrhtfsth H tf`l` ah gfthb b` pisfjfafbhb`s qu` J5 J h su v`z tf`l` :/> b` ahs pisfjfafbhb`s b` N b` ijt`l`r `a niltrhti. ^f ahs hcfrghnfil`s b`a `xp`rti sil hn`rthbhs, ¹Nuèa `s ah prijhjfafbhb phrh nhbh uli b` khlhr `a niltrhti9. Z? 8/3, 8/>, 8/:.

:=. ^`hlb` H,niloultis. J y N tr`s `v`ltis hsinfhbis nil ul `xp`rfg`lti. @xpr`shr ahs sfkuf`lt`s pripisfnfil`s `l tærgflis

 

h) Ha g`lis inurr`l bis `v`ltis j) inurr`l `xhnthg`lt` bis `v`ltis n) H ai gès inurr`l bis `v`ltis b) Li inurr`l ais tr`s. `) Li inurr` lflkuli b` `aais.

:6. ^`hl H, J y N sun`sis li gutuhg`lt` `xnauy`lt`s, b` gibi qu`? T(H l J)7 4.:, T(Hl N)7 4.8:, T(J l N)7 4.83, T(H l J l N )7 4.42, T(H)7 4.2, T(J)7 4. 0:, T(N)7 4.>:. @l ul bfhkrhgh b` v`ll gu`str` ahs prijhjfafbhb`s bhbhs y mhaa` ah prijhjfafbhb b` qu`? h) Inurr` ha g`lis uli b` ais tr`s `v`ltis j) Inurr`l h ai sugi bis n) Inurr`l `xhnthg`lt` bis b) Inurr`l `xhnthg`lt` tr`s `) Inurr`l h ai sugi bis c) Li inurr` lflkuli b` ais tr`s.

*>4. Ha cflha b` ah açl`h b` `lshgja`, `l ah `ahjirhnfþl b`a pribunti ], `a sup`rvfsir s`a`nnfilh ais pribuntis qu` vhl h ulh flsp`nnfþl 844%. ^f `a 80% b` ais pribuntis `ahjirhbis sil b`c`ntuisis, y `a >4% b` ais pribuntis ju`lis y `a ;4% b` ais b`c`ntuisis vhl h ulh flsp`nnfþl nigpa`th, b`t`rgflhr ah prijhjfafbhb b` qu`? h) Pl pribunti qu` phsþ pir flsp`nnfþl flnigpa`th s`h ju`li. j) Pl pribunti s`h b`c`ntuisi bhbi qu` phsþ pir ulh flsp`nnfþl nigpa`th.