John plays Peter at tennis. The first to win two sets wins the match. Illustrate the sample space using a tree diagram.
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1 TEE DIMS ND INOMIL OILITIES (Chpter 20) 405 Exmple 2 ohn plys eter t tennis. The first to win two sets wins the mth. Illustrte the smple spe using tree digrm. If mens ohn wins the set nd mens eter wins the set then the tree digrm is: We ould write the smple spe in set nottion s S = f,,,,, g. 1st set 2nd set 3rd set 2 Use tree digrm to illustrte the smple spe for the following: d e The genders of 4-hild fmily. g ontins red nd white mrles nd g ontins lue nd yellow mrles. g is seleted nd one mrle is tken from it. Hts, nd C eh ontin pink nd purple tikets. ht is seleted nd then two tikets re tken from it. Two tems, X nd Y, ply footll. The first tem to kik 3 gols wins the mth. ody nd etri ply tennis. The first to win two sets in row or totl of 3 sets, wins the mth. 3 g ontins five mrles. One is lue, one is red, nd three re green. Two mrles re seleted from the g without replement. Use tree digrm to show ll possile outomes. OILITIES FOM TEE DIMS Tree digrms n e used to illustrte smple spes providing there re not too mny different lterntives. One the smple spe hs een illustrted, the tree digrm n e used to determine proilities. INDEENDENT EVENTS Suppose Chrles hs proility 4 5 of pssing hysis nd 7 10 of pssing Chemistry. We ssume tht these events re independent, whih mens the proility of Chrles pssing hysis is not relted in ny wy to him pssing Chemistry. tree digrm showing ll possile outomes is shown longside with eing pss nd F eing fil. t_ hysis F q_p_ Dq_p_ q_p_ Dq_p_ Chemistry F F outomes proilities F F FF t_ q_p_ Ip_ t_ Dq_p_ Wp_ q_p_ t_p_ Dq_p_ Dt_p_
2 406 TEE DIMS ND INOMIL OILITIES (Chpter 20) The proilities for pssing nd filing eh sujet re mrked long the rnhes. When using tree digrms to solve proility questions: ² the proility for eh rnh is lulted y multiplying the proilities long tht rnh ² if two or more rnhes meet the desription of the ompound event, the proility of eh rnh is found nd then they re dded. Exmple 3 essi hs proility 4 5 of getting n in Mthemtis nd proility 2 5 n for English. Disply this informtion on tree digrm. Wht is the proility essi gets one in the two sujets? of getting is the event of getting n. N is the event of not getting n. Mthemtis t_ N English N N outomes N N NN proilities t_ Qw_ Sw_t_ essi n get one either y getting n in Mthemtis ut not in English, or y getting n in English ut not in Mthemtis. 4 So, the totl proility is = These events re highlighted on the tree digrm. EXECISE spinner hs proility 1 3 of finishing on lue nd proility 2 3 of finishing on green. Xin spins the spinner two times. Disply this informtion on tree digrm. Find the proility tht the spinner finishes on lue one nd on green one. 2 ndrew hs proility 1 5 of winning the 100 metre sprint nd proility 1 3 the 200 metre re. Drw tree digrm showing ll of ndrew s hnes. Wht is the proility ndrew wins oth res? Wht is the proility tht ndrew wins extly one of the two res? d Wht is the proility ndrew loses oth res? e Find the sum of the proilities in, nd d. Explin why this nswer is to e expeted. of winning
3 DEENDENT EVENTS TEE DIMS ND INOMIL OILITIES (Chpter 20) 407 Suppose ht ontins 5 red nd 3 lue tikets. One tiket is rndomly hosen, its olour is noted, nd it is then thrown wy. seond tiket is rndomly seleted. Wht is the hne tht it is red? If the first tiket ws red, (seond is red) = 4 7 If the first tiket ws lue, (seond is red) = reds remining 7 to hoose from 5 reds remining 7 to hoose from So, the event of the seond tiket eing red depends on wht olour the first tiket ws. In suh se we sy we hve dependent events. Two or more events re dependent if they re not independent. Dependent events re events where the ourrene of one of the events does ffet the ourrene of the other event. If nd re dependent events then ( then ) = ( given tht hs ourred). typil exmple of dependent events is when we smple two ojets without replement. This mens tht the first ojet is not repled efore the seond is seleted. It therefore nnot e seleted twie. Exmple 4 ox ontins 4 red nd 2 yellow tikets. Two tikets re rndomly seleted one fter the other from the ox, without replement. Disply this informtion on tree digrm. Wht is the proility tht oth re red? Wht is the proility tht one is red nd the other is yellow? Let e the event tht red tiket is drwn nd Y e the event tht yellow tiket is drwn. Note tht the outome of the seond event depends on the first. 1st seletion ( nd then ) = ( given tht hs ourred) y_ Y t_ 2nd seletion y_ Y y_ t_ Y = = 2 5 (one is nd the other is Y ) = ( nd then Y or Y nd then ) fhighlighted rnhesg = = or 15
4 408 TEE DIMS ND INOMIL OILITIES (Chpter 20) EXECISE ox ontins 7 red nd 3 green lls. Two lls re rndomly seleted from the ox one fter the other. The first is not repled in the ox efore the seond is seleted. Determine the proility tht: 3 melie hs g ontining two different vrieties of pples. They re pproximtely the sme size nd shpe, ut one vriety is red nd the other is green. There re 4 red pples nd 6 green ones. She selets one pple t rndom, ets it, nd then tkes nother, lso t rndom. Determine the proility tht: d e oth pples were red oth pples were green the first ws red nd the seond ws green the first ws green nd the seond ws red she te one red nd one green pple. 4 Mrjut hs rton ontining 10 ns of soup. 4 ns re tomto nd the rest re pumpkin. She selets 2 ns t rndom without looking t the lels. oth re red the first is green nd the seond is red green nd red re otined. 2 ht ontins the nmes of the 7 plyers in tennis squd inluding the ptin nd the vie ptin. tem of 3 plyers is hosen t rndom y drwing the nmes from the ht. Disply this informtion on tree digrm. Distinguish etween the ptin, the vie ptin, nd the other plyers. Find the proility tht the tem: i does not ontin the ptin ii ontins neither the ptin nor the vie ptin iii ontins either the ptin or the vie ptin, ut not oth. Let T represent tomto nd represent pumpkin. Drw tree digrm to illustrte this smpling proess. Wht is the proility tht oth ns were tomto soup? Wht is the proility tht one n ws tomto nd the other n ws pumpkin soup? Seleting items simultneously mens there is no replement. LE SMLE SCES Sometimes the numer of possile outomes is suffiiently lrge tht it is wste of time to drw tree digrm. We n still use the sme priniples we used with the tree digrms to lulte the proilities.
5 Exmple 5 TEE DIMS ND INOMIL OILITIES (Chpter 20) 409 ht ontins 20 tikets with the numers 1, 2, 3,..., 19 nd 20 printed on them. If 3 tikets re drwn from the ht without replement, determine the proility tht ll re prime numers. There re 20 numers of whih 8 re primes: f2, 3, 5, 7, 11, 13, 17, 19g ) (3 primes) = (1st drwn is prime nd 2nd is prime nd 3rd is prime) = ¼ 0: primes out of 20 numers 7 primes out of 19 numers fter suessful first drw 6 primes out of 18 numers fter two suessful drws EXECISE lottery hs 100 tikets whih re pled in rrel. Two tikets re drwn t rndom from the rrel to deide 2 prizes. If ohn hs 2 tikets in the lottery, determine his proility of winning: first prize first nd seond prize seond prize ut not first prize. d none of the prizes. 2 in ontins 12 identilly shped hooltes of whih 8 re strwerry rems. If 3 hooltes re seleted t rndom from the in, determine the proility tht: they re ll strwerry rems none of them re strwerry rems. 3 g ontins two white nd five red mrles. Three mrles re seleted simultneously. Determine the proility tht: ll re red only two re red t lest two re red. SELECTION WITH ND WITHOUT ELCEMENT You my hve notied tht when we work with tree digrms, the proilities of independent nd dependent events re lulted using the sme method. The following exmple ompres two suh events. Consider ox ontining 3 red, 2 lue nd 1 green mrle. Suppose we wish to smple two mrles: ² with replement of the first efore the seond is drwn ² without replement of the first efore the seond is drwn. Exmine how the tree digrms differ:
6 410 TEE DIMS ND INOMIL OILITIES (Chpter 20) With replement Without replement This rnh represents lue with the first drw nd ed with the seond drw. This is written s. 1st 2nd 1st 2nd n t hve Notie tht: ² with replement (two reds) = = 1 4 ² without replement (two reds) = = 1 5 We n thus see why replement is importnt. Exmple 6 with replement without replement. For the exmple ove of the ox ontining 3 red, 2 lue nd 1 green mrle, find the proility of getting red nd lue: ( red nd lue) ( red nd lue) = ( or ) ftiked onesg = ( or ) frossed onesg = = = 1 3 = = = 2 5 EXECISE 20.4 Use tree digrms to help nswer the following questions: 1 r ontins 4 lue nd 2 red mrles. r ontins 1 lue nd 5 red mrles. jr is rndomly seleted nd one mrle is tken from it. Determine the proility tht the mrle is lue. 2 Two mrles re drwn in suession from ox ontining 2 purple nd 5 green mrles. Determine the proility tht the two mrles re different olours if: the first is repled the first is not repled. 3 5 tikets numered 1, 2, 3, 4 nd 5 re pled in g. Two re tken from the g without replement. Determine the proility tht: oth re odd oth re even one is odd nd the other is even.
7 TEE DIMS ND INOMIL OILITIES (Chpter 20) r ontins 3 red nd 2 green tikets. r ontins 3 red nd 7 green tikets. die hs 4 fes mrked nd 2 fes mrked. The die is rolled nd the result is used to selet either jr or jr. When jr hs een seleted, two tikets re rndomly seleted from it without replement. Determine the proility tht: oth re green they re different in olour. Exmple 7 g ontins 5 red nd 3 lue mrles. Two mrles re drwn simultneously from the g. Determine the proility tht t lest one is red. drw 1 Ti_ Ei_ drw 2 u_ Eu_ Tu_ Wu_ (t lest one red) = ( or or ) = = = = lterntively, (t lest one red) =1 (no reds) fomplementry eventsg =1 nd so on. 5 g ontins four red nd two lue mrles. Three mrles re seleted simultneously. Determine the proility tht: ll re red only two re red t lest two re red. 6 ox ontins 3 red nd 4 green mrles. ox ontins 5 red nd 2 green mrles. One mrle is rndomly seleted from ox nd its olour noted. If it is red, 2 reds re dded to ox. If it is green, 2 greens re dded to ox. C mrle is then seleted from ox. Find the proility tht the mrle seleted from ox is green. In mny situtions there re only two possile outomes. For exmple: ² tennis plyer either wins or loses gme ² you either th us or you don t ² you either mke free throw or you miss. INOMIL OILITIES inomil experiments re onerned with the repetition of severl independent trils where there re only two possile outomes, suess nd filure. Eh tril hs the sme proility of suess.
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