Ejercicios de Probabilidades 2.

October 5, 2022 | Author: Anonymous | Category: N/A

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C@BHR@_HRDH OF ^RHB@BDCDO@OFQ C@BHR@_HRDH OF ^RHB@BDCDO@OFQ

0. Qf sfc sfcfaa faadhe dhe` ` ue` l`gdcd` l`gdcd` quf phsff ohs ùthgñ ùthgñvdc vdcfs, fs, oheof p`r` fc gàs eufvh y fc gàs vdfkh hbsfrv`ghs sd luf l`brdaòh fe chs Fstòhs Vedohs, Furhp` h @sd`. ¸Auàcfs she chs phsdbcfs rfsuctòhs of fstf fxpfrdgfeth8 *O@_HQ 0 ùthgñvdc eufvh 0 ùthgñvdc vdfkh *^VE_HQ OF L@BRDA@ADHE @7 Fstòhs uedohs B7 Furhp` A7 @sd` *^@R@ VE @V_HGH\DC ;

e ( s )7 { ` , b , a }

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e ( s )7 {``, `b, à, b`, bb, ba, a`, ab, aa}

2. Ve` fxpfrdfead` `cf`thrd` ahesdstf fe prfiuet`r ` trfs pfrshe`s odset`s, fcfido`s `c `z`r, sd she p`ro`rd`s h eh of ahesugdr ue oftfrgdeòh prhouath. prhouath. `) Fsar Fsardb dbf f fc fs fsp` pàd adh h gufs gufstr tr`c `c `sha `shad` dòh oh ` od odam amh h fx fxpf pfrd rdgf gfet eth, h, u ucd cdzè zèoh oh c` cftr cftr` ` "s" "s" p` p`r` r` c`s c`s rfspufst`s `rg`v`s y "e" p`r` c`s efi`v`s. *^FRQHE@ ^frshe`0; –s‗ y –e‗ ^frshe`2; –s‗ y –e‗ ^frshe`1; –s‗ y –e‗ *FQ^@ADH GVFQ_R@C

e ( s ) 7 { sss, ss sse, e, see,se s, eee, ess, ees, es ese e} e ( s )7 =

b) ¸Puä ¸Puä fc fcfg fgfe feth thss of ofcc fs fsp` pàd adh h gufs gufstr tr`c `c ètf ètfrd rdhr hr ahes ahest tuy uyfe fe fc fv fvfe feth th "`c "`c gf gfeh ehss oh ohss of c`s c`s pfrshe`s she p`ro`rd`s of ahesugdr fc prhouath"8 *F\FE_H @; "`c gfehs ohs of c`s pfrshe`s she p`ro`rd`s of ahesugdr fc prhouath"

*FQ^@ADH GVFQ_R@C

@ 7{ sss, sse, ses , ess ess } 0

a) Ofsardbf fc fvfeth ahetr`rdh of "gàs of ue` pfrshe` pfrshe` fs fs p`ro`rd` of ahesugdr fc prhouath" B; gàs of ue` pfrshe` fs p`ro`rd` of ahesugdr fc prhouath

B7{ sss,sse,ses,ess } B — 7 {see ,ese ,ese ,e ,ee es , ee eee e} 1. Qfè @ y B ohs fvfeths au`cfsqudfr` of ue fspàdh gufstr`c. Fsardbdr fe ehtàdñe of ahekueth y rfprfsfet`r iràa`gfetf, chs sdiudfetfs fvfeths. `) Haurrf @

o) @ h B h haaurrfe

^ ( @VB )7 ^ ( @ ) + ^ ( B ) − ^ ( @ ⊑ B ) V

V

@ @

B

b) Hau aurrrf sñc sñch @

^ ( @ − B )7 ^ ( @ ) − ^ ( B ) V

f) Haurrf @ y B

^ ( @ ⊑ B ) 7 ^ ( @ ) ⊑ ^ ( B ) V

@ B

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a) @ h B ha haur urrf rfe, e, pfr pfrh h eh `gbh `gbhss

@ ∌ B7 ( @ − B ) V   ((B − @ ) V

l)

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Haurrf sñ sñch B

^ ( B − @ )7 ^ ( B ) − ^ ( @ ) V

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?. Qf v` fetrfvdst`r ` ue iruph sfcfath of fgpcfòhs ahe rfspfath ` ue eufvh pcè of pfesdhefs, chs fgpcfòhs sf ac`sda`rhe ahgh sdiuf; Ac`sdaàdñe Qupfrvdshrfs Of gètf ètfed edgd gdfe feth th Of prhouaadñe Ifrfead`

Eûgfrh 02> 6> 0?6> 1>2

Qfarft`rç` tht`c

@; c` prdgfr` pfrshe` sfcfaadheò` sf` ue fgpcfòh Qupfrvdshr B; c` prdgf rdgfr` r` pf pfrrshe she` gètfedgdfeth

sfc sfcfaa aadh dhe` eò o`

sf`

ue fgpc gpcf` fòh oh of

A; c` prdgfr` pfrshe` sfcfaadheò` sf` ue fgpcfòh of ^rhouaadñe O; c` prdg prdgfr fr` ` pfrs pfrshe he` ` sf sfcf cfaad aadhe heò ò` ` sf sf` ` ue fgpc fgpcf` fòh oh of Ifrfead` F; c` prdg prdgfr fr` ` pfrs pfrshe he` ` sf sfcf cfaad aadhe heò ò` ` sf sf` ` ue fgpc fgpcf` fòh oh of Qfarft`rç` e

e ( Τ)7 e ( @ )+ e ( B )+ e ( A )+ e ( O )+ e ( F )7004> `) ¸A ¸Auà uàcc fs c` prhb prhb`b `bdc dcdo doò ò of quf quf c` prdg prdgfr fr` ` pfrs pfrshe he` ` sf sfcf cfaa aadh dhe` eò` o` sf sf` ` ue fg fgpc pcf` fòh oh of gètfedgdfeth8

^ ( B )7

e ( B) e (Τ)

7

6> 044>

7>,>260

b) ¸A ¸Auà uàcc fs c` pr prhb hb`b `bdc dcdo doò ò of qu quf f c` pr prdg dgfr fr` ` pf pfrs rshe he` ` sf sfcf cfaa aadh dhe` eò` o` sf` sf` `ciu `ciudf dfe e of if ifrf rfea ead` d` h sfarft`rç`8

^ ( OVF )7 ^ ( OVF )7

e ( O ) e (  F F)

+

e ( Τ) e (Τ) 1>2 044>

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e ( O ∤ F ) e ( Τ)

^ ( OVF )7>,0=64 a) ¸Auàc fs c` c` prhb`bdcdo prhb`bdcdoò ò of quf c` prdgfr prdgfr` ` pfrshe` pfrshe` sfcfaadheò sfcfaadheò` ` eh sf` of prhouaad prhouaadñe8 ñe8

^ ( A )

A

^ ( A )

A

70 − 70 −

e ( A ) e (Τ) 0?6> 044>

A

^ ( A ) 7 >,2901 6. Fe c` L` Làu auct ctò ò of L@ L@AF AF@A @A,, fc 26% 26% of chs chs fs fstu tuod odè ètf tfss of ofs` s`pr prhb hb`r `rhe he g`tf g`tfgà gàa a`, `, fc 06 06% % ofs`prhb`rhe Fstòçsa` y fc 0>% ofs`prhb`rhe c`s ohs `sdie`tur`s. `sdie`tur`s. Qf sfcfaadhe` ue fstuodètf `c `z`r. @7OFQ@^RHB@RHE G@_FG@_DA@ 1

B7OFQ@^RHB@RHE FQ_@ODQ_DA@ A7OFQ@^RHB@RHE @GBHQ AVRQHQ

`. Qd ofs`prhb ofs`prhbñ ñ Fstòçsa`, Fstòçsa`, ¸auàc ¸auàc fs c` prhb`bd prhb`bdcdoò cdoò of quf quf ofs`prhb`r ofs`prhb`r` ` G`tfgàa`8 G`tfgàa`8

^ ( @ ∤ B ) ^ ( @ / B )7 ^ ( B ) ^ ( @ / B )7

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7>,,?

¸Auàc fs c` prhb`b prhb`bdcdo dcdoò ò of quf quf of ofs`prh s`prhb`r` b`r` G`tfgà G`tfgàa` a` h Fstòçs Fstòçsa`8 a`8

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e ( @ ) e ( B )

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e (Τ) e (Τ) 26 % 6> %

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06 % 6> %

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e (Τ) 0> % 6> %

^ ( @VB ) 7>,< .4> of prhb`bdcdoò of lueadhe`r. Auèoh fc sdstfg` fs pufsth fe lueadhe`gdfeth.

@ ; C` `c`rg` 0 Lueadhe` . → ^ ( @ )7 >.4> y ^ ( @ — )70 ‑ p ( @ )7>.0> B ; C` `c`rg` 2 Lueadhe` . → ^ ( B )7>.4> y ^ ( B — )70 ‑ p ( B)7 >.0> A ; C` `c`rg` 1 Lueadhe` .→ ^ ( A )7 )7 >.4> y ^ ( A — )70 ‑ p ( A )7>.0> `. ¸Auàc ¸Auàc fs prhb prhb`bd `bdcdo cdoò ò of tho` tho`ss lueadh lueadhefe efe8 8

F 07_ho`sc`s ho`s c`s `c`rg`slueadhe `c`rg`slueadheè è . ^ ( F 2 )7 ^ ( @ ) . ^ ( B ) . ^ ( A )7 )7(( >.4>)( >.4>)( >.4>)7 >.924 b. ¸Auàc ¸Auàc fs prhb`b prhb`bdc dcdo` doò o quf edeiue edeiue` ` lueadhe lueadhef8 f8

F 27 Edeiue` Edeiue``c`rg` `c`rg` lueadhe` . ^ ( F 2 )7 ^ ( @ — ) . ^ ( B — ) . ^ ( A — )7( >.0> )( >.0> )( >.0> )7 >.>>0 9. Ve` hrièdzàdñe of devfsiàdñe ofc ahesugdohr m` fstuodòh chs sfrvdadhs ofetrh ofc pfrçhoh of i`rè` quf hlrfafe chs 6> odstrdbudohrfs of ùthgñvdcfs eufvhs fe adfrt` aduoò y sus m`cc`zihs sf rfsugfe fe c` t`bc` quf sdiuf;

0> `ðhs h gàs fe fc efihadh (A) Gfehs of 0> `ðhs fe fc efihadh _H_@C

Bufe sfrvdadh ofetrh

G`c sfrvdadh ofetrh

ofc pfrçhoh of

ofc pfrçhoh of

i`rè`(@)

i`rè`

0< 0> 2<

? 2> 2?

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2> 1> 6> ?

@; Fc odstrdbudohr of ùthgñvdcfs hlrfaf ue bufe sfrvdadh ofetrh ofc pfrçhoh of i`rètç` B; Fc odstrdbudohr of ùthgñvdcfs tdfef 0> `ðhs h gàs fe fc efihadh `) Qd ue` pfrshe` pfrshe` sfcfaadhe` sfcfaadhe` `cf`thrd `cf`thrd`gfetf `gfetf ueh of fsths fsths odstrdbudoh odstrdbudohrfs rfs of ùths eufvhs, eufvhs, ¸Auàc fs prhb`bdcdoò of quf fcfidrà ` ueh quf hlrfza` ue sfrvdadh s frvdadh of i`rè` òfauòh8

^ ( @ )7

e ( @ )

2<

7 7>,62 e ( Q ) 6>

b) Qd ue` pfrshe` pfrshe` sfcfaadhe sfcfaadhe` ` `cf`thrd`gfet `cf`thrd`gfetf f ` ueh of chs odstrdbudoh odstrdbudohrfs rfs quf m` fstòh fstòh fe fc efihadh efihadh phr 0> `ðhs h gàs, ¸Auàc fs c` prhb`bdcdoò of quf feaufetrf `ciueh quf hlrfza` bufe sfrvdadh of i`rè`8

^ ( A / @ ) 7

e ( A ∤ @ ) 0< 7 7>,=> e ( @ ) 2>

=. Quphei`ghs quf fe ure` m`y 9 fslfr`s ofc gdsgh t`g`ðh, of chs au`cfs ? she bcèa`s y 1 she rhk`s. Qf fxtr`fe suafsdv`gfet suafsdv`gfetf f au`trh au`trh fslfr`s fslfr`s `c `z`r. ¸Auàc fs c` prhb`bdcdoò prhb`bdcdoò of quf c`s ohs prdgfr`s prdgfr`s sfè bcèa`s y c`s ohs sdiudfetfs rhk`s8 (Qde rffgpc`zh y ahe rffgpc`zh).

@; C` bhc` fxtr`ço` fs of ahchr bcèah B; C` bhc` fxtr`ço` fs of ahchr rhkh Fspàdh gufstr`c(s)79 ? 9

∙1 ∙1

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^ ( @ ∤ @ ∤ B ∤ B )7

6 ?

∙2 7 >,>=6

4. M`y 4> shcdadtètfs of ue fgpcfh ahe fc ofp`rt`gfeth of ehtdad`s of ue` fstàdñe of tfcfvdsdñe.

@ciuehs of fcchs she iròuòhs uedvfrsdt`rdhs y htrhs eh: `ciuehs of fcchs tdfefe auèoh gfehs trfs `ðhs of fxpfrdfead` y htrhs edeiue`, oheof c` sfp`ràdñe fxàt` fs; Iròuòhs Eh iròuòhs uedvfrsdt`rdhs uedvfrsdt`rdhs Auèoh gfehs trfs `ðhs of 0= 4 fxpfrdfead` Gfehs of trfs `ðhs 1< 29 Qd fc hrofe fe fc au`c she fetrfvdstòhs phr fc ifrfetf of c` fstàdñe fs `cf`thrdh, I fs fc fvfeth of quf fc prdgfr `spdrètf fetrfvdstòh sf` iròuòh uedvfrsdt`rdh, y _ of quf fc prdgfr `spdrètf fetrfvdstòh tfei` auèoh gfehs trfs `ðhs of fxpfrdfead`, oftfrgdef aò` ue` of c`s prhb`bdcdoòfs sdiudfetfs odrfat`gfetf of chs rfidstrhs y of chs tht`cfs of rfeichefs y ahcuge`s of c` t`bc`; `) ^(I): b) ^(_a): a) ^(I_): a a a o) ^(I _) f) ^(_/I) l) ^(I /_ ) 0>. Ve` ahgp`ðç` prdvò` of p`quftfrç`, fstà prfhaupòh phr c` phsdbdcdoò of quf `ciuehs of sus

fgpcfòhs v`yè ` mufciè. Fstdg` quf c` prhb`bdcdoò of quf sus pdchths sf v`yè ` mufci` fs of >.96 y c` prhb`bdcdoò of quf sus amhlfrfs mìè mufci` fs of >.% of phsdbdcdoòfs of quf chs pdchths rf`cdafe ue p`rh shcdo`rdh of àtdvdoòfs. `. ¸Auàc fs c` prhb`bdcdoò prhb`bdcdoò of quf `gbhs `gbhs iruphs iruphs sf v`yè ` mufci`8 mufci`8 b. Qd chs pdcht pdchths hs màfe màfe mufci`, mufci`, ¸Auàc fs c` prhb`bd prhb`bdcdcdoò oò of quf chs amhlfrfs amhlfrfs ch ch mìè t`gbdäe ahgh àth of shcdo`rdoò8 00. C`s ofsahgphstur`s of gàqude`s she deofpfeodfetfs fetrf sç. Qf tdfefe au`trh gàqude`s,

auy`s rfspfatdv`s prhb`bdcdoòfs she 0%, 2%, 6% y 0>% fe ue oç` p`rtdauc`r, a`caucf c`s sdiudfetfs prhb`bdcdoòfs; `) _ho`s sf ofsahgphefe fc gdsgh oç`.

b) Edeiue` sf ofsahgphef

02. Ve Fahehgdst` o` ahelfrfead`s ` adfrth pûbcdah phr c` g`ðè` y htr` ahelfrfead` ` htrh pûbcdah

ph phrr c` t`ro t`rof. f. Qf` Qf` @ 7{fc 7{fc fa fahe hehg hgds dst` t` o` ue ue` ` g`c` g`c` ahelf ahelfrf rfea ead` d` g`tut g`tutde de`} `} y B 7{fc 7{fc ahelfrfeads ahelfr feadst` t` o` ue` g`c` ahelfrfead` ahelfrfead` vfspfrtde` vfspfrtde`}. }. Qd ^(@) 7 >.1, ^(B) 7 >.2 y ^(@B) 7 >.0, a`caucf c`s sdiudfetfs prhb`bdcdoòfs. `) ^(B/@) b) ^(Ba/@) a) ^(B/@a) o) ^(Ba/@a) 01. Fe ue chtf of 0> `rtçauchs m`y 1 oflfatuhshs. Qd sf thg` `c `z`r trfs `rtçauchs ueh tr`s htrh.

¸auàc fs c` prhb`bdcdoò of quf chs trfs sfè bufehs8 0?. C` prhb`bdcdoò of quf ue ahgfradètf, vfeo` ofetrh of ue gfs, ue chtf of rflrdifròhr`s fs

0/? 0/? y c` prhb` prhb`bd bdcdcdoò oò of vfeo vfeofr fr ue chtf chtf of ahad ahade` e`s s of ofetr etrh h of ue gfs gfs of prhb`bdcdoò of quf; `. \feof \feof chs ohs ohs chtfs chtfs of `rtç `rtçauc auchs hs ofetrh ofetrh of ue ue gfs. gfs. b. \feof \feof –edeiue –edeiueh‗ h‗ of chs chs chtfs chtfs ofetr ofetrh h of ue gfs. gfs.

0/ 0/1. 1. M`cc` M`cc`rr c`

06. A`cauc`r c` prhb`bdcdoò of quf `c tdr`r ue oòh ohs vfafs ahesfautdv`s, c` sug` of chs pueths

hbtfedohs sf` eh gfehr quf =. 0,,92 y c` of quf m`y` `prhbòh `ciue` of c`s ohs p`rtfs fs >,=2. Qd sf fcdif ue `cugeh `c `z`r, ¸Auàc fs c` prhb`bdcdoò of quf `prufbf fc fx`gfe p`r` hbtfefr cdafead`8 @ 7 fc `cugeh `prufb` fc fx`gfe tfñrdah B 7 fc `cugeh `prufb` fc fx`gfe pràatdah | @ ∤ B|7| @|+|B|−¸ @ ∫ B ∭¸ ^ ( @ ∤ B )7 ^ ( @ )+ ^ ( B )− ^ ( @ ∫ B ) ^ ( @ ∤ B )7>..92 −>.=2 ^ ( @ ∤ B ) 7>.6= FE_HEAFQ; C` prhb`bdcdoò of quf `prufbf fc fx`gfe p`r` hbtfefr c` cdafead` fs of >,6= 0=. @c `rrhk`r ohs oòhs, ueh bcèah y ueh ue h efirh, a`cauc`r c` prhb`bdcdoò of hbtfefr hamh pueths

fetrf chs ohs. <

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EFIRH

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9

2 1 ? 6 <

1 ? 6 < 9

? 6 < 9 =

6 < 9 = 4

< 9 = 4 0>

9 = 4 0> 00

= 4 0> 00 02

@; hbtfefr = pueths fetrf chs ohs. ^(@)7e(@)/e¸) 75 6/1.014 FE_HEAFQ; C` prhb`bdcdoò of hbtfefr hamh pueths fetrf chs ohs fs of >.014. 04. Ve tubh of vàçh pufof prhvfedr prhvfedr of au`cqudfr` of trfs l`brd l`brdaètfs aètfs ahe prhb`bdcdoò; prhb`bdcdoò; ^07>.26,

^27>.6 y ^17>.26 ^27>.6 ^17>.26.. C`s prhb`bdc prhb`bdcdo` doòfs ofs of quf fc tubh lueadhef lueadhef ahrrfat` ahrrfat`gfet gfetf f ourètf ourètf ue pfrçhoh pfr çhoh of tdf tdfgph gph fspfaçlda fspfaçldah h she; she; >.0: >.0: >.2: >.?. Rfspfa Rfspfatdv tdv`gf `gfetf etf p`r` chs 1 l`brda` l`brdaètfs etfs.. A`cauc`r c` prhb`bdcdoò of quf fc tubh fcfidoh `c `z`r lueadhef ahrrfat`gfetf. @0; Fc tubh vàçh lueadhe` ahrrfat`gfetf B0; Fc tubh vàçh prhvdfef ofc l`brdaètf ^0 B2; Fc tubh vàçh prhvdfef ofc l`brdaètf ^2 B1; Fc tubh vàçh prhvdfef ofc l`brdaètf ^1 ^(Bd)

^(@/Bd)

B0

>.26

>.0

B2 B1

>.6 >.26

>.2 >.?

2 ^(@)7∐^(@/Bd) ^(Bd) d70 ^ (@)7 (>.26) (>.0) +(>.6) (>.2) +(>,26) (>,?) 7>.226 FE_HEAFQ; C` prhb`bdcdoò of quf lueadhef fc tubh ahrrfat`gfetf fs of >.226

2>. ^`r` `rg`r c` sdiudfetf t`bc` sf mè tfedoh fe aufet` c`s ac`sdldaàdhefs;

A`cdldaàdñe

sfxh Gukfr

\`rñe

_H_@C

@. Bufe` B. Rfiuc`r

9 0>

4 =

0< 0=

A. Ofldadfetf

2

?

<

_H_@C 04 20 ?> Qd fetrf chs ?> `cugehs of odamh aursh, sf fcdif 0 `c `z`r, m`cc`r c` prhb`bdcdoò of quf; `) M`y` M`y` hbtf hbtfedo edoh h B fe c` fv`c fv`cuà uàdñe dñe ^(B) 7 e(B)/e(Τ) 0=/?> 7>,?6 FE_HEAFQ; C` prhb`bdcdoò of quf ue `cugeh `c `z`r m`y` hbtfedoh B fs >,?6 b) M`y` M`y` hbtfedoh hbtfedoh B s`bdfeo s`bdfeoh h quf fc `cugeh `cugeh fcfidoh fcfidoh fs fs v`rñe. v`rñe. ^(B/G) 7 ^(BɆG)/^(G) 7 (0>/?>)/(04/?>) 7 0>/04 7 >,62< FE_HEAFQ; C` prhb`bdcdoò of quf ue m`y` hbtfedoh B s`bdfeoh quf fc `cugeh fcfidoh fs v`rñe fs of >,?6.

9

ahetdfef 0= i`ccftdt`s i`ccftdt`s of s`cvòh y 0> of ìu`, sf fxtr`fe fxtr`fe 2 i`ccftdt`s i`ccftdt`s `c `z`r, 20. Of ue` c`t` quf ahetdfef suafsdv`gfetf y sde rfpftdadñe. A`cauc`r c` prhb`bdcdoò of quf c` prdgfr` i`ccftdt` fxtr`ço` sf` of s`cvòh y c` sfiueo` of ìu`. @; C` i`ccft` fs of s`cvòh B; C` i`ccft` fs of ìu` ^(@.B) 7 ^(@).^(B) ^(@).^(B) 7 0= /2= ∙ 0> /29 7 6 /20 7 >,21= FE_HEAFQ; C` prhb`bdcdoò of quf c` prdgfr` i`ccftdt` fxtr`ço` sf` of s`cvòh y c` sfiueo` of ìu` fs of >,21=. 22. Fc prhafsòhr, c` pcà` gòrf y c` gfghrd` tdfef ue 6%, 0>% y 2>% of prhb`bdcdoòfs of l`cc`r ètfs of uel`cc`s `ðh ètfs rfspfatdv`gfetf. fsahgphefetfs c` prhb`bdcdoò of ahgpr`r ue ahgputòhr quf prfsfet`rà of ue `ðh, fe¸Auàc chs trfs sfð`còhs8 @; Fc prhafsòhr l`cc` ètfs of ue `ðh. → ^(@) 7 >.>6 B; C` pcà` gòrf l`cc` ètfs of ue of ue `ðh. → ^(B) 7 >.0> A; C` gfghrd` l`cc` ètfs of ue `ðh. → ^(A) 7 >.2> F0 7 Chs trfs ahgphefetfs l`ccè ètfs of ue `ðh. ^ (F0) 7 ^(@). ^(B). ^(A) 7 (>.>6) (>.0>) (>.2>) 7 >.>>0 FE_HEAFQ; C` prhb`bdcdoò of ahgpr`r ue ahgputòhr quf prfsfet`rà l`cc`s ètfs of ue `ðh, fe chs trfs ahgphefetfs sfð`còhs fs of >.>>0. 21. Qfiûe ue delhrgf ofc tdfgph, sf prhehstda` p`r` g`ðè` ue` prhb`bdcdoò of ccuvd` of >.? y of

>.9 of quf mì` lrçh. Qd `gbhs suafshs she deofpfeodfetfs, ¸Auàc fs c` prhb`bdcdoò of quf g`ðè` Eh ccufv` ed t`gphah mì` lrçh8 @; G`ðè` ccufvf. → ^(@) 7 >.? y ^(@—) 7 0 ‑ ^(@)7 >.< B; G`ðè` màf lrdh. → ^(B) 7 >.9 y ^(B—) 7 0 ‑ ^(B)7 >.1 ^(@—∤B—) 7 ^(@—). ^(B—) 7 (>,,1) 7 >,0= FE_HEAFQ; C` prhb`bdcdoò of quf g`ðè` Eh ccufv` ed t`gphah mì` lrçh fs of >,0= 2?. Of àufro àufroh h ` ue` devfst devfstdi` diàdñ adñe e rf`cdz` rf`cdzò` o` fe ue` oftfrgd oftfrgdeò eò` ` aduoò aduoò àfra` àfra` of gukfrf gukfrfs s

g`yhrfs of 2> `ðhs sf m` ahgprhbòh quf fetrf htr`s ahs`s fc % tr`b`k` lufr` ofc mhi`r. Of c`s quf eh fstàe a`sò`s, fc 92 % tr`b`kè lufr` ofc mhi`r; @; _r`b`kè lufr` ofc Mhi`r. B0; Fs a`sò`. B2; Fs shctfr`. Gukfrfs g`yhrfs of 2> ^(Bd) ^(@/Bd) `ðhs B0 B2

>..12

>.?> >.92

`) ¸Puä phrafet`k phrafet`kf f of gukfrfs gukfrfs g`yhrfs g`yhrfs of 2> 2> `ðhs tr`b`k` tr`b`k` lufr` ofc ofc mhi`r mhi`r 2

^ ( @ ) 7

^ ( @ / B ) ^ ( Bd ) ∐ 7 d

d 0

^ (@)7 (>..?) +(>.12) (>.92) 7>.6>2? FE_HEAFQ; Fc phrafet`kf of gukfrfs g`yhrfs of 2> `ðhs tr`b`k` lufr` ofc mhi`r fs of 6>.2?%

b) Qd sf sfcfaadh sfcfaadhe` e` `c `z`r ue` gukfr gukfr g`yhr g`yhr of 2> `ðhs, ¸auàc ¸auàc fs c` prhb`bdc prhb`bdcdoò doò of quf eh fstä a`sò` ed tr`b`kf lufr`8

^ ( B2 ∤ @ )7 ^ ( @ ∖ B2 ) ⋆ ^ ( B2 ) 0

0

=

( >.2= ) ( >.12 )7>.>=4< FE_HEAFQ; C` prhb`bdcdoò of quf eh fstä a`sò` ed tr`b`kf lufr` fs of >.>=4<

26. Ve hrofeòhr pfrshe`c fstà ahet`gdeòh phr ue vdrus y tdfef a`riòhs ohs prhir`g`s ètdvdrus

quf àtûè deofpfeodfetfgfetf ueh ofc htrh. Fc prhir`g` p 0 oftfat` c` prfsfead` ofc vdrus ahe ue` prhb`bdcdoò of >.4 y fc prhir`g` p 2 oftfat` fc vdrus ahe ue` prhb`bdcdoò of >.=. ¸Auàc fs c` prhb`bdcdoò of quf fc vdrus eh sf` oftfatòh8 @; Fc prhir`g` ^0 oftfat` c` prfsfead` ofc vdrus. ^(@) 7 >.4 y ^(@—) 7 0 ‑ p(@) 7 >.0 B; Fc prhir`g` ^2 oftfat` c` prfsfead` ofc vdrus. ^(B) 7 >.= y ^(B—) 7 0 ‑ p(B) 7 >.2 ^ ( @ ' ∤ B ' )7 ^( @ ' ). ^ (B ' )7(>.0) (>.2)7>.>2 (>.2)7>.>2

2% of c` phbcàdñe tdfef a`bfcch a`st`ðh, 26% of c` phbcàdñe tdfef hkhs

a`st`ðhs y fc 06 % tdfef a`bfcchs y hkhs a`st`ðhs. Qf thg` `c `z`r ` 0 pfrshe`; @; _dfef a`bfcchs a`st`ðhs B; _dfef hkhs a`st`ðhs ^(@) 7 >,?>: ^(B) 7 >,26: ^(@ɆB) 7 >,06

`) Qd tdfef tdfef a`bfcch a`bfcch a`st`ðh, ¸auàc ¸auàc fs c` prhb`bdc prhb`bdcdoò doò of quf t`gbdäe t`gbdäe tfei` hkhs hkhs a`st`ðhs8 a`st`ðhs8 ^(B/@) 7 ^(@ɆB)/^(@) 7 >,06/>,?> 7 >,196

b) Qd tdfef tdfef hkhs a`st`ðhs a`st`ðhs ¸auàc ¸auàc fs c` prhb`bdcd prhb`bdcdoò oò of quf eh tfei` a`bfcch a`bfcchs s a`st`ðhs8 a`st`ðhs8 ^(@—/B) 7 0 ‑ ^(@/B) 7 0- Z>,06/>,26W 7 0 ‑ >,< 7 >,?

chs m` m`bd bdtè tètf tfs s of ue ue` ` oft oftfr frgd gdeò eò` ` phbc` phbcàd adñe ñe she she kubd kubdc` còhs ohs y htrh htrh 2> 2>% % she she 29. Fc 2>% of chs fstuodètfs. C` gûsda` acàsda` fstuodètfs. acàsda` cf iust` `c 96% of chs kubdcòhs, `c 6>% of chs fstuodètfs y `c 2>% ofc rfsth of c` phbcàdñe. phbcàdñe. A`cauc` A`cauc` c` prhb`bdcdoò prhb`bdcdoò of quf fcfido` fcfido` `c `z`r ue` pfrshe` ` c` quf cf iust` gûsda` acàsda` sf` kubdcò`. @7 Iust` fsauam` c` gûsda` acàsda`. B07 Fs Kubdcòh. B27 Fs fstuodètf. B17 Fc rfsth of c` phbcàdñe.

M bdtètfs B` 0 B2 B1

^.(>B2d) > >.>2 >.><

^ >.(9@6/Bd)

>.6> >.2>

2

^ ( @ ) 7

^ ( @ / B ) ^ ( Bd ) ∐ 7 d

d 0

^(@)7(>,96) (>,2)+(>,6) (>,2)+(>,2)(>,,19> (>,2)+(>,2)(>,,19>

^ ( @ / B 0 ) ( >,96 )( >,2 ) ^ ( B0 / @ )7 7 7 >,?>6 ^ ( @ ) >,19> 2=. Fe ue` feaufst` rf`cdzò` ` 26> acdfetfs fe ue afetrh ahgfrad`c, sf cfs prfiuetñ àfra` ofc

gheth of sus ahgpr`s y c` lhrg` fe c` au`c m`bçè rf`cdzòh su pìh. Qf feahetrñ quf fc ?=% acdfetfs acdf etfs mdadfrhe sus pìhs fe flfatdvh, flfatdvh, quf fc 12.=% of chs acdfetfs acdfetfs mdadfrhe ahgpr`s phr ue` 4

aètdoò gfehr ` $26>, t`gbdäe sf feahetrñ quf fc =>% chs acdfetfs quf pì`rhe ahe t`rkft` of aräodth mdadfrhe ahgpr`s phr gàs of $26>. @; ^ì`rhe fe flfatdvh B; Rf`cdzñ ahgpr`s gfehrfs ` 26> ^(@) 7 >,?= : ^(B) 7 >,12= : ^(B—/@—) 7 >,=>

`) Qd sf sfcfaadh sfcfaadhe` e` ` ue acdfetf acdfetf `c `z`r ¸auàc ¸auàc fs c` c` prhb`bdcdoò prhb`bdcdoò of quf quf m`y` pìòh pìòh ahe t`rkft` of aräodth8 ^(@—) 7 0 ‑ >,?= 7 >,62

b) Qd sf sfcfaad sfcfaadhe` he` ue acdfetf acdfetf `c `z`r ¸auàc ¸auàc fs c` prhb`bdc prhb`bdcdo` doò o of quf m`y` i`stòh i`stòh gàs gàs of $26> y m`y` pìòh ahe t`rkft` of aräodth8 ^( B ' ∤ @ ' )7^(B ' / @').^( @') (>,=>) (>,62)7>,?0<

a) Qd sf sfcfaad sfcfaadhe` he` `c `z`r ue acdfetf acdfetf quf pìñ fe flfat flfatdvh dvh ¸auàc ¸auàc fs c` prhb`bdcd prhb`bdcdoò oò of quf m`y` ahgpròh gfehs of $26>8 ^ ( @ ∤ B ) >,22? ^ ( B / @ )7 7 >,?,?= ^ ( @ ∫ B)7^( @)+^(B)−^( @ ∤ B) ^ ( @ ∤ B)7>,?=+>,12=−>,6=?7>,22? T t`gbdäe; ^( @ ∫ B)70−^( @ '∤ B ' )70−>,?0,6=?

24. Fc prdeadp`c acdfetf of ue` làbrda` tfxtdc rfirfsñ ue chtf of `rtçauchs ofbdoh ` oflfaths fe c`

fc`bhràdñe.. Fe c` làbrda` fc`bhràdñe làbrda` sf aufet` ahe trfs cçef`s of prhouaadñe prhouaadñe quf fc`bhrè fsf `rtçauch `rtçauch y sf phsffe o`ths of c` t`s` of chtfs oflfatuhshs fe aò` cçef` of prhouaadñe. Chs o`ths of c` prhphradñe of c` prhouaadñe tht`c fc`bhrò` phr aò` cçef` y su rfspfatdv` t`s` of chtfs oflfatuhshs sf gufstrè ` ahetdeuàdñe.

Cçef` 0 2 1

^rhphradñe ofc tht`c >.6 >.1 >.2

_`s` of chtfs oflfatuhshs 2% 6% .6 >.1 >.2

>.>2 >.>6 >.><

`) A`cauc` A`cauc` c` prhb`bdcd prhb`bdcdoò oò of quf ue chtf chtf fcfidoh fcfidoh `cf`thrd`gfet `cf`thrd`gfetf f sf` of c` cçef` cçef` 0 y fstä fstä oflfatuhsh. ^( B0∤ @)7^( @/B0) . ^(B0 )7>,>2∙>,67>,>0 0>

b) ¸Auà ¸Auàcc fs c` prhb prhb`b `bddcd cdo` oò o of sfc sfcfaad faadhe he`r `r ue cht htf f of c` cçe çef` f` 2 y quf quf rf rfsu suct ctf f oflfatuhsh8 ^( B2∤ @)7^( @/B2). ^(B2 )7>,>6∙>,17>,>06

a) ¸Auàc fs c` prhb`bdcd prhb`bdcdoò oò of quf ue chtf sfcfaadh sfcfaadheòh eòh `c `z`r `z`r sf` of c` cçef` cçef` 1 y fstä oflfatuhsh8 ^( B1∤ @)7^( @/ B1) .^(B1 )7>,>,27>,>02

o) A`cauc` A`cauc` c` prhb`bdcd prhb`bdcdoò oò of quf ue chtf chtf fcfidoh fcfidoh `c `z`r `z`r fstä oflfatuh oflfatuhsh. sh. 2

^ ( @ )7

^ ( B / @ ) ^ ( Bd ) ∐ 7 d

d 0

^( @)7(>,>2) (>,6)+(>,>6) (>,1)+(>,>,2)7>, >19

f) Qf fc`bhrñ fc`bhrñ ue cht chtf f quf rfsuct rfsuctñ ñ fst`r oflfatuh oflfatuhsh, sh, ¸auàc ¸auàc cçef` of prhouaad prhouaadñe ñe tdfef g`yhr prhb`bdcdoò of m`bfrch fc`bhròh8 ^ ( @ / B 0) ( >,>2 )( >,6 ) 7>,29> 7 ^ ( B0 / @ )7 ^ ( @ ) >,>19 ^ ( @ / B 2) ( >,>6 )( >,1 ) 7>,?>6 7 ^ ( B2 / @ )7 ^ ( @ ) >,>19 ^ ( @ / B1) ( >,>,2) ^ ( B1 / @ )7 7>,12? 7 ^ ( @ ) >,>19 1>. Fc 2>% of chs fgpcfòhs of ue` fgprfs` she deifedfrhs y htrh 2>% she fahehgdst`s. Fc 96%

of chs deifedfrhs haupè ue pufsth odrfatdvh y fc 6>% of chs fahehgdst`s t`gbdäe, gdfetr`s quf chs eh deifedfrhs y chs eh fahehgdst`s shc`gfetf fc 2>% haup` ue pufsth odrfatdvh. ¸Auàc fs c` prhb`bdcdoò of quf ue fgpcfòh odrfatdvh fcfidoh `c `z`r sf` deifedfrh8 B0; Fc fgpcfòh fs deifedfrh B2; Fc fgpcfòh fs Fahehgdst` B1; Fc fgpcfòh fs htrh prhlfsdhe`c @; Fc fgpcfòh haup` ue pufsth odrfatdvh ^(B0) 7 >,2>: ^(B 2) 7 >,2>: ^(B 1) 7 0- (>,2>+>,2>) 7 >,: ^(@/ B 0) 7 >,96: ^(@/ B2) 7 >,6>: ^(@/ B1) 7 >,2> ^( @)7∐ ^( @/Bd @/ Bd ). ^(Bd )7(>,96) (>,2>)+(>,6>) (>,2>)+(>,2>) (>,)7>,19

10. C` prhb`bdcdoò of quf m`y` ue àadofetf fe ue` làbrda` quf odsphef of `c`rg` fs >.0. C` prhb`bdcdoò prhb`bd cdoò of quf sufef fst` sç sf m` prhouadoh `ciûe deado deadofetf fetf fs of >.49 y c` prhb`bdcdoò prhb`bdcdoò

of quf sufef sd eh m` suafodoh edeiûe deadofetf fs >.>2. Fe fc supufsth of quf m`y` lueadheòh c` `c`rg`, ¸auàc fs c` prhb`bdcdoò of quf eh m`y` m`bdoh edeiûe deadofetf8 B0; Qf prhoukh ue deadofetf B2; Eh sf prhoukh ue deadofetf @; C` `c`rg` lueadhe` ^(B0) 7 >,0: ^(B2) 7 0 - >,0 7 >,4: ^(@/ B 0) 7 >,49: ^(@/ B 2) 7 >,>2 ^( @)7∐^( @/Bd ). ^(Bd )7(>,49) (>,0)+(>,>2) (>,4)7>,006

^ ( B2 / @ )7

∐ ^ ( @ / B2) . ^ ( B 2) ^ ( @ )

7

( >,>2 )( >,4 ) >,006

7 >,06% of c` phbcàdñe fsahc`r quf eh fstà

vàueòh ahetr`f irdpf. Ve` a`gp`ð` of vàueàdñe `caèz` ue` ahbfrtur` ofc 9>% of fst` 00

phbcàdñe. Qd of chs vàueòhs, phbcàdñe. vàueòhs, shch fc 0>% ahetr`f irdpf, ¸Auàc fs c` prhb`bdcdoò prhb`bdcdoò of quf ue fsahc`r ahetr`di` irdpf8 B0; Qf feaufetr` vàueòh B2; Eh sf feaufetr` vàueòh @; Ahetr`f irdpf ^(B0) 7 >,9>: ^(B2) 7 0 - >,9> 7 >,1>: ^(@/ B 2) 7 >,1>: ^(@/ B 0) 7 >,0> ^( @)7∐^( @/Bd ). ^(B d )7(>,0>) (>,9>)+(>,1>) (>,1>)7>,0<

11. Ve khvfe tdfef au`trh a`gds`s of chs sdiudfetfs ahchrfs; rhk` (R), bcèa` (B), efir` (E) y vfrof

(\), t`gbdäe phsff ohs pèt`chefs, irds (I) y `zuc (@). `) _ràf _ràf ue odìr` odìr`g` g` of àrbhc àrbhc of fstf fstf fxpfrd fxpfrdgfe gfeth. th. A`gds`s.

R

^èt`chefs.

I @

B

I @

E

I @

\

I @

b) ¸Of ¸Of au auàe àet` t`s s gèf gèfr` r`s s pufo pufofe fe ah ahgb gbde de`r `rsf sf chs chs pèt pèt`c `che hefs fs ah ahe e c`s c`s a` a`gd gds` s`s s h vdafvfrs`8 e(Τ)¸ ? ] 27=

Adfrt` ` ahgp`ðç` ahgp`ðç` lhrg`rà ue ahgdtä ahgdtä of pcèfàdñe pcèfàdñe ` c`rih pc`zh, ahe fc fea`rih of ofs`rrhcc`r ofs`rrhcc`r 1?. Adfrt ue pcè qudequfe`c fstr`täidah p`r` quf c` fgprfs` deirfsf `c gfraòh of ue eufvh prhouath. Fc prfsdofetf m` dofetdldaòh ` sdftf ifrfetfs a`pàfs ahgh aèodo`ths p`r` fc ahgdtä. ¸Of auàet`s gèfr`s sf pufof lhrg`r fc ahgdtä of trfs gdfgbrhs8 9! 9 A 17 716 1 ! ( 9 −1 )!

16. Fe ue` oftfrgdeò` aduoò, c`s `pcdaàdhefs of a`gbdh of zhedldaàdñe sdiufe ue prhafsh of

ohs ft`p`s; ue` rfvdsdñe phr c` ahgdsdñe of pcèfàdñe, y ue` ofadsdñe lde`c phr fc ahesfkh aduoòèh. Fe fc p`sh 0 c` ahgdsdñe of pcèfàdñe rfvds` c` pftdadñe of a`gbdh of zhedldaàdñe y fgdtf ue` rfahgfeoàdñe phsdtdv` h efi`tdv` àfra` ofc a`gbdh. Fe fc p`sh 2 fe ahesfkh aduo aduoò` òèh eh rfvd rfvds` s` c` rfah rfahgf gfeo eoà àdñ dñe e of c` ahgd ahgdsd sdñe ñe of pcèf pcèfà àdñ dñe e y vht` vht` `p `prh rhbàe bàeohc ohc` ` h rfam`zàeohc`. Fe `ciuehs a`shs fc vhth ofc ahesfkh aduoòèh aheahroñ ahe c` rfahgfeoàdñe of odam` ahgdsdñe. Fc ahestruathr of ue ahgpcfkh of vdvdfeo`s à`b` of prfsfet`r ue` shcdadtuo of a`gbdh of zhedldaàdñe. Ahesdofrf quf fc prhafs`gdfeth of c` shcdadtuo fs ue fxpfrdgfeth. `) ¸Auàeths ¸Auàeths pueths guästr`c guästr`cfs fs m`y p`r` p`r` fstf fxpfrdg fxpfrdgfeth8 feth8 Mì` ue` ue` cdst` cdst` of fcchs. fcchs. ^; Rfahgfeoàdñe phsdtdv`. 02

E; Rfahgfeoàdñe efi`tdv`. @; @prufb` c` rfahgfeoàdñe. R; Rfam`z` c` rfahgfeoàdñe. 27 ? e(Τ)¸ 2 ]   2

^@ @ , ^R , E@ , ER } Τ¸ { ^ b) _ràf _ràf ue odìr` odìr`g` g` of àrbhc àrbhc of fstf fstf fxpfrdg fxpfrdgfet feth. h.

@ ^

R

@ E R

1.6 >.? >.1

Kufz B >.< >.6 ->.0

A >.? >.1 >.1

Rfspufst`;

C`s prhb`bdcdoòfs ofc kufz @, y` quf c` sug` ofbf o`r 0. C`s prhb`bdcdoòfs ofc kufz B, y` quf eh fxdstf ue` prhb`bdcdoò gfehr quf >.

19. Fetrf chs => odrfatdvhs of ue` ahgp`ðç` m`y ?= a`sòhs, 16 she iròuòhs of fsaufc`s

supfrdhrfs y 22 of chs ?= a`sòhs t`gbdäe she iròuòhs. Qd sf fcdif uehs of fsths odrfatdvhs p`r` quf `sdst` ` ue` ahevfeadñe ¸auàc fs c` prhb`bdcdoò of quf c` pfrshe` fcfido` eh sf` a`sòh ed iròuòh8 @; Fc odrfatdvh a`sòh. B; Fc odrfatdvh iròuòh of fsaufc` supfrdhr. ^(@)¸ ?= / =>7>.< ^(B)¸ 16 / => 7>.?196 ^(B/@)¸ 22 / ?> 7>.?6=1 → ^(@ Ɇ B) 7 22/=> 7 >,296

∫ ¿ B) 7 ¸ 0−Z ^ ( @ ) + ^ ( B )− ^ ( @ ∤ B ) W7 0−( >.< + >.?− >.2 )7>.2 ^(@— ∤ ”B)¸ ^ ( @ 01

1=. Ve` fgprfs` l`rgàäutda` ccfvñ ` a`bh ue fstuodh p`r` fv`cu`r fc flfath of ue` gfodade` p`r`

`cd `cdvdh vdh of `cfrid `cfrid`s: `s: 26> pàdfe pàdfetfs tfs,, ahe sçethg sçethg`s `s quf deacuy deacuyfe fe hkhs hkhs drr drrdt` dtòhs ohs y tr`sthre tr`sthrehs hs fpdoärgdahs, rfadbdfrhe fc eufvh gfoda`gfeth. Chs rfsuctòhs ofc fstuodh she ahgh sdiuf; 4> of chs pàdfetfs tr`tòhs fxpfrdgfet`rhe `cdvdh fe chs hkhs, 016 sf aur`rhe of su `lfaadñe autàef` y ?6 fxpfrdgfet`rhe ` c` vfz `cdvdh fe chs hkhs drrdtòhs y fe c` pdfc. ¸Auàc fs c` prhb`bdcdoò of quf ue pàdfetf quf thg` fc gfoda`gfeth; @; Qdfetf `cdvdh fe chs hkhs. B; Qf aur` of c` `lfaadñe autàef`. ^ ( @ ) 74> / 26>7>.1< ^ ( B )7016 / 26>7 >.6< ^ ( @ Ɇ B ) 7?6 / 26> 7>.0= `)

fxpfrdgfetf `c `cdvdh fe fe phr ch gf gfehs ue ueh of of cch hs sçethg`s8 ^ ( @ ∫ B ) 7 ^ ( @ ) + ^ ( B )− ^ ( @ ∤ B )7>.1< + >.6.0=7>.9?

b)

shch fxpfrdgfetf `cdvdh fe chs hkhs8 ^ ( Qhch @ )7 ^ ( @ )− ^ ( @ ∤ B )7 >.1< −>.0= 7>.0=

a)

eh fx fxpfrdgfetf etf `c `cdvdh fe fe cch hs hk hkhs y eh eh sf sf au aurf of of su su `lf `lfa aadñe autà tàe ef`8

∫ ¿ B ) ¸ 0−>.9? 7 >.2< ^ ( @ — ∤ ” B ) 7 ^ ( @ o)

sf au aurf of of su su `l `lfaadñe au autàef` sd m` m` fx fxpfr pfrdgfet` tò oh `c `cdvdh fe fe cch hs hk hkhs8 ^ ( @ ∤ B ) >.0= ^ ( B / @ )7 7>.6 7 >. ^ ( @ )

14. Ve `p`r`th fcfatrñedah fcfatrñedah ahest` of ohs p`rtfs p`rtfs.. C` prhb`bdcdoò prhb`bdcdoò quf l`ccf c` prdgf prdgfr` r` fs >.2>, quf

l`ccfe c`s ohs p`rtfs fs >.06 y of quf l`ccf sñch c` sfiueo` p`rtfs >.?6. A`cauc`r c` prhb`bdcdoò quf ; @; C` prdgfr` p`rtf l`cc`. B; C` sfiueo` p`rtf l`cc` ^(@); >.2> ^ (Qhch B); >.?6 ^ (@ Ɇ B); >.06

`)

L`ccf sñch c` prdgfr` p`rtf

^ (Qhch B)¸ ^ ( @ )− ^ ( @ ∤ B )7>.2> −>.067 >.>6

b)

L`ccf c` c` pr prdgfr` p` p`rtf au auèoh sf sf s` s`bf qu quf l` l`ccñ c` c` sf sfiueo` 0?

^(@/B)¸

^ ( @ ∤ B ) ^ ( B )

7

>.06 >.?6 + >.06

7>.26

?>. Ve fes`gbcòhr of ahgputòhr`s us` p`rtfs quf prhvdfefe of 1 prhvffohrfs ^0, ^2 y ^1. Of

2>>> p`rtfs rfadbdo`s 0>>> prhvdfefe of ^0, > of ^2 y fc rfsth of ^1. Of fxpfrdfead`s p`sò`s fc fes`gbcòhr s`bf quf c`s p`rtfs oflfatuhs`s quf prhvdfefe of ^0, ^2 y ^1 she rfspfatdv`gfetf 1%, ?% y 6%. Qd sf fcdif ue` ahgputòhr` `c `z`r, y sd ahetdfef ue` p`rtf oflfatuhs`. @; _dfef p`rtfs oflfatuhs`s B0 ;^rhvdfef ofc prhvffohr ^0

B2 ;^rhvdfef ofc prhvffohr ^2 B1 ; ^rhvdfef ofc prhvffohr ^1

^(Bd)

^(@/Bd) >.>1

B0

0>>>/2>>>7>.6

>.>?

B2 >/2>>>7>.1 B 1 ?>>/2>>>7>.2

>.>6

`)¸Auàc fs c` prhb`bdcdoò of quf c` pdfz` sf` oflfatuhs`8 1

^ ( @ )7

^ ( @ / Bd ) ^ ( B ) ∐ − d 0

^ ( @ ) 7( >.>1 ) ( >.6 ) + ( >.>? ) ( >.1 ) + ( >.>6 ) ( >.2 )7>.>19 b)¸Auàc fs c` prhb`bdcdoò of m`y` sdoh prhvfçoh phr ^08 ^ ( @ / B 0) ( >.>1 )( >.6 ) ^ ( B0 / @ )7 7 7>.?>?6 >.>19 ^ ( @ ) a) ¸Auàc fs c` prhb`bdcdoò of m`y` sdoh prhvfçoh phr ^28 ^ ( @ / B 2) ( >.>? )( >.1 ) ^ ( B2 / @ )7 7 >.12?1 7 >.>19 ^ ( @ )

o)¸Auàc fs c` prhb`bdcdoò of m`y` sdoh prhvfçoh phr ^18 ^ ( @ / B1) ( >.>6)( >.2) ^ ( B1 / @ )7 7>.29>1 7 ^ ( @ ) >.>19

06

ahgputòhr`s tdfef gdc ahgputòhr`s ahgputòhr`s fe sthaj. sthaj. Ve` despfaadñe despfaadñe ` c` làbrda` m` ?0. Ve` làbrda` of ahgputòhr`s `rrhkòh `rrhk` oh c` del delhrg hrgàd àdñe ñe sdiud sdiudfetf fetf;; m`y > ahgput` ahgputòhr ohr`s `s eufv`s, eufv`s, 64> ahgput` ahgputòhr ohr`s `s ahe rfpufsths rfpufs ths fe bufe fstòh y 11> ahgputòhr`s ahgputòhr`s usò`s tdfefe rfpufsths fe g`c fstòh (fs ofadr, aò` ahgputòhr` tdfef c` prhb`bdcdoò of 0/0>>> of sfr sfcfaadheò`). Eufv`s (>) B0 Vsò`s (?>>) B2 Ahe rfpufsth fe bufe fstòh (64>) @ Ahe rfpufsth fe g`c fstòh (?0>) Eufv` y ahe rfpufsths fe bufe fstòh (62>) Eufv` y ahe rfpufsth fe g`c fstòh (=>) Vsò`s ahe rfpufsths fe g`c fstòh (11>) Vsò`s ahe rfpufsths fe bufe fstòh (9>) `) ¸Auàc fs c` prhb`bdcd prhb`bdcdoò oò of quf ue` ahgputòhr` ahgputòhr` sfcfaa sfcfaadheò` dheò` `c `z`r sf` eufv` eufv` y tfei` rfpufsths fe g`c fstòh8 





    

^ ( @ ' ∤ B2 )7

11> 0>>>

7 >,11

b) Qd ue` ahgputòhr` ahgputòhr` sfcfaa sfcfaadheò` dheò` `c `z`r tdfef rfpufsths rfpufsths fe bufe fstòh ¸Auàc fs prhb`bdcdoò of quf sf` eufv`8 ^ ( @ ∤ B 2 ) 9> / 0>>> 7>,00=< ^ ( B 2 / @ )7 7 64> / 0>>> ^ ( @ ) ?2. Fe trfs pcèt`s, @, B y A, l`brdaè fc 6> %, fc 1> % y fc 2> %, rfspfatdv`gfetf, ofc tht`c of chs

hbkfths of ue` fgprfs`. Chs phrafet`kfs of prhouaadñe oflfatuhs` of fst`s pcèt`s she, rfspfatdv`gfetf, fc 1 %, fc ? % y fc 6%. `) Qd sf sfcfaad sfcfaadhe` he` ue hbkfth `c `z`r, `z`r, ¸quä prhb`bdc prhb`bdcdoò doò tdfef tdfef of s`cdr oflfatuhs oflfatuhsh8 h8 1

^ ( @ ) 7

^ ( @ / Bd ) ^ ( Bd ) ∐ 7 d 0

^ ( @ )7( >,>1 )( >,6 )+( >,>? )( >,1 )+( >,>6 )( >,2 )7>,>19 b) Quphedfeoh quf fs oflfatuhsh, ¸auàc fs c` prhb`bdcdoò of quf sf m`y` prhouadoh fe c` pcèt` @8 ^ ( @ / B 0 ) ( >,>1 )( >,6 ) ^ ( B 0 / @ )7 7 >,?>6? 7 ^ ( @ ) >,>19 ?1. Ve oftfrgdeòh acub tdfef ue 96 % of sus gdfgbrhs quf she gukfrfs y ue 26 % quf she

mhgbrfs. Of fstf acub tdfef tfcälheh gñvdc ue 26 % of c`s gukfrfs y ue 6> % of chs mhgbrfs. @; _dfef tfcälheh B0; Gukfr B2; Mhgbrf `) A`cauc`r A`cauc`r fc phrafet`k phrafet`kf f of gdfgbrhs gdfgbrhs of fstf fstf acub acub quf eh tdfefe tdfefe tfcälheh tfcälheh gñvdc. gñvdc. 2

^ ( @ ) 7

^ ( @ / Bd ) ^ ( Bd ) ∐ 7 d 0

^ ( @ )7( >,26 )( >,96 )+( >,6> )( >,26 )7 >,1026 ^ ( @ ' )70 − >,10267 >,,6> >,26 2 ^ ( B / @ )7 ( / ^ ( @ ) ) ( ) 7 ( >,1026 )( ) 7>,?>

0<

??. Ohs `gdihs ahgp`rtfe pdsh. Fc prdgfrh prfp`r` c` ahgdo` fc ?> % of chs oç`s y fc rfsth of chs

oç`s ch màf fc sfiueoh. Fc phrafet`kf of vfafs quf sf cf qufg` `c prdgfrh fs fc 6 %, gdfetr`s quf fc ofc sfiueoh fs fc =%. @; Qf cf qufg` c` ahgdo` B0; ^rdgfrh fe prfp`r`r c` ahgdo` B2; Qfiueoh fe prfp`r`r c` ahgdo` `) A`ca A`cauc uc`r `r c` prhb prhb`b `bdc dcdo doò ò of quf quf ue oç oç`, `, fcfi fcfido doh h `c `z`r `z`r,, c` ah ahgd gdo` o` fstä fstä qufgò`. 2

^ ( @ / Bd ) ^ ( Bd ) ∐ 7

^ ( @ )7

d 0

^ ( @ )7( >,>6 )( >,?> )+( >,>= )( >, )7 >,>,>6 )( >,?> ) ^ ( B 0 / @ )7 7 7>,24?0 ^ ( @ ) >,> % of c`s pfcçauc`s she of àadñe

gdfetr`s quf fe fc sfiueoh ch she fc 9> %. Ve fspfatòhr fcdif `c `z`r ue adef sdiudfeoh ue gäthoh quf dgpcda` quf c` prhb`bdcdoò of fcfidr fc prdgfrh fs fc trdpcf quf c` of fcfidr fc sfiueoh. Ve` vfz ccfi` `c adef. @; ^fcçauc` of àadñe B0; ^rdgfr adef p`r` vfr c` pfcçauc` B2; Qfiueoh adef p`r` vfr c` pfcçauc` `) A`cauc`r A`cauc`r c` prhb`bdcdoò prhb`bdcdoò of quf c` pfcçauc` pfcçauc` quf vf` sf` of àadñe. àadñe. 2

^ ( @ )7

∐ ^ ( @ / B 0) ^ ( B 0)

D 7 0

^ ( @ )7(>,6> )( >,26 )+( >,9> )( >,96 )7 >,,6> )( >,26 ) ^ ( B 0 / @ )7 7 7>,0421 ^ ( @ ) >,% supfr` chs Q/. 4>, gdfetr`s quf c`s ahgpr`s rf`cdzò`s phr chs mhgbrfs sñch fc 1> % supfr` fs` aètdoò. @; Qupfrè chs 4> shcfs B0; ^hr gukfrfs B2; ^hr mhgbrfs `) Fcfidoh Fcfidoh ue tdajft tdajft of ahgpr` `c `z`r, ¸auàc ¸auàc fs c` prhb`bdcd prhb`bdcdoò oò of quf supfrf chs chs Q/. 4>8 2

^ ( @ ) 7

^ ( @ / ¸ B ) ^ ( B ) ¸ ∐ 7 0

0

d 0

^ ( @ ) 7( >.=> ) ( >.9> ) + ( >.1> ) ( >.1> )7>,, ¸auàc ¸auàc fs c` c` prhb`bdcdoò prhb`bdcdoò of quf quf c` ahgpr` m`y` sdoh mfam` phr ue` gukfr8 ^ ( @ ' / B 0 ) . ^ ( B 0) ( 0 −>,=> )( >,9> ) ^ ( B 0 / @ ' )7 7 7 >,? )7 0− ^ ( @ ) 0−>,

Ejercicios de Probabilidades 2.

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