20 b The prime numbers are 2,3,5,7,11,13,17,19.

Transcription

1 Topi : Probbility Short nswer tehnology- free The following my be of use in this test:! 0 0 Two rows of Psl s tringle re: ontiner holds irulr piees eh of the sme size. Written on eh is different number, one of the integers from to. One of the piees of pper is seleted t rndom. lulte the probbility the number on the seleted piee is: not multiple of b is prime number is either number greter thn or smller thn d is prime number given it is number greter thn e is even nd multiple of Nme: The multiples of re,,9,,,. Pr(not multiple of ) 0.7 b The prime numbers re,,,7,,,7,9. Pr( prime number) Pr( > or < ) 0. d Pr(prime > ) 0. e Pr(even multiple of ) 0. For events nd, Pr( ) 0., Pr( ) 0. nd Pr( Ç ) 0.. Show the given probbilities on probbility tble (Krnugh mp) nd omplete the entries in the tble. b lulte: i Pr( È ) ii Pr( ). For two events nd, Pr( ) nd Pr( ). If nd re mutully exlusive events, lulte Pr( È ). b If nd re independent events lulte Pr( È ). If Ì lulte Pr( È ) b i Pr( È ) Pr( ) + Pr( ) - Pr( Ç ) ii Pr( ) Pr( Ç ) Pr( ) Pr( È ) Pr( ) + Pr( ) - Pr( Ç ) Sine nd re mutully exlusive, Pr( Ç ) 0 \ Pr( È ) + b Sine nd re independent, Pr( Ç ) Pr( ) Pr( ) \ Pr( È ) + - Sine Ì, Pr( È ) Pr( ) John Wiley & Sons ustrli, Ltd

2 group of boys nd girls re rrnged in row. Wht is the totl number of possible rrngements? b In how mny of the rrngements re the boys together? Wht is the probbility tht the girls nd boys re in lternte positions? b 7 people n be rrnged in 7! wys. 7! 7! 00 so there re 00 totl possible rrngements. Tret the boys s one unit whih n internlly rrnge in! wys. The girls nd unit of boy rrnge in! wys. Hene, the totl number of rrngements is!! 7. For the girls nd boys to be in lternte positions, they must be st in the rrngement GGGG. There re!! suh rrngements.!! Pr(GGGG) 7!!! 7! How mny distint rrngements of the letters of the word TOMTO re possible if the letters re rrnged: i in row ii in irle? b lulte the probbility tht the two letters T re together when the letters of the word TOMTO re rrnged in row. In how mny wys n ommittee of people be hosen from group of men nd women without restrition b with t lest one mn on the ommittee? Jne is one of the women. lulte the probbility tht Jne is seleted for the ommittee. i There re letters, inluding identil T s nd identil O s. These letters rrnge in row in! 0 wys.!! ii The letters rrnge in irle in! 0 wys.!! b Group the T s together s one unit with no internl rrngements to onsider, sine the T s re identil. This gives + units to rrnge whih inlude two identil O s. b In row there re! 0 distint! rrngements. Pr(T s together) 0 0 Without ny restrition, people n be hosen from group of people in wys. The possible ommittees with t lest one mn re mn nd women, or men nd womn. These n be seleted in + + wys. (lterntively, - possibilities.) With Jne on the ommittee, more people must be seleted from the other people. Pr(Jne is on the ommittee) 0 0. John Wiley & Sons ustrli, Ltd

3 Multiple hoie The Venn digrm shows the results of survey of 00 people who dislosed whether they dded yoghurt (Y) or milk (M) to their brekfst grnol. Wht is the probbility tht rndomly hosen person from this group did not dd yogurt to their grnol? In kithen drwer there re 0 size btteries nd size btteries. Four of the btteries do not work nd three of the btteries do not work. One bttery is hosen t rndom from the drwer. The probbility the hosen bttery is size nd works is: 7 bg ontins green blls nd 7 yellow blls. Two blls re drwn without replement. The probbility tht both re green is: 0 n unbised oin is tossed three times. The probbility of t lest one Hed is: 7 Given Pr( ) 0., Pr( ') 0. nd Pr( ) 0., Pr( ) equls: 0 swimmer ompetes in two events in whih the hnes of her winning re 0. nd 0. respetively. The probbility she wins extly one of the events is: Numbers re formed using the digits 9,,7 nd without repetition. How mny numbers greter thn 00 is it possible to form? 9 0 John Wiley & Sons ustrli, Ltd

4 There re fition books nd non-fition books on shelf. If books re hosen t rndom, then the probbility of obtining n equl number of fition nd nonfition books is: In the expnsion of ( p+ q), the number of terms nd the term ontining respetively: terms, ( ) pq terms, ( ) pq terms, ( ) pq terms, ( ) pq 9 terms, ( ) pq p re, 9 multiple hoie test hs 0 questions, eh with possible nswers. student who hs not prepred for the test guesses the nswers to eh question. The probbility tht the student gets either or orret nswers is given by: 0 0 (0.) (0.) + (0.) (0.) 0 0 (0.) (0.) + (0.) (0.) 0 0 (0.) + (0.) 0 0 (0.) + (0.) (0.) (0.) + (0.) (0.) xtended response John Wiley & Sons ustrli, Ltd

5 Two friends, Jude nd Lee, regulrly ply Srbble nd bdminton with eh other. It is estimted tht in the long run Jude wins 70% of the gmes of Srbble nd 0% of the gmes of bdminton. On dy when Jude nd Lee ply Srbble in the morning nd bdminton in the fternoon, lulte the probbility tht: i Jude wins both gmes ii Jude loses both gmes iii Lee wins one of the gmes but not both. In the evening Jude deides to rry out simultion of the sitution using rndom number tble nd oin s follows: The digits 0,, re ssoited with Lee winning Srbble nd the digits,,,, 7,,9 re ssoited with Jude winning Srbble. Hed on the oin is ssoited with Jude winning bdminton nd Til with Lee winning bdminton. b xplin why this is n pproprite model for the results of the two gmes. The results of trils re shown. H H T T T T T H 0H 7H d H H T T 9H T T H T T Wht does the first result H men in the ontext of the simultion? Use the simultion results to estimte the probbility tht: i Jude wins Srbble ii Jude wins bdminton iii Jude wins both gmes. i Pr(Jude wins both gmes) ii Pr(Jude loses both gmes) iii ither Lee wins bdminton nd Jude Srbble, or Lee wins Srbble nd Jude wins bdminton. The probbility b Pr(0,, ) 0. nd Pr(,,,7,,9) 0.7, so these probbilities gree with the ft tht Lee wins 0% nd Jude 70% of the Srbble gmes. Pr(H) 0. Pr(T) grees with the ft tht Jude nd Lee eh win 0% of the gmes of bdminton. The simultion result H mens Lee won Srbble nd Jude won bdminton. d i In the trils, the digits,,,, 7,,9 our times. Hene n estimte of the probbility tht Jude wins Srbble is or 0.. ii In the trils, Hed ours 9 times. Hene n estimte of the probbility tht Jude wins tennis is 9 or 0.. iii In the trils, one of the digits,,,, 7,,9 ours with Hed times. n estimte of the probbility tht Jude wins both gmes is or 0.. John Wiley & Sons ustrli, Ltd

6 Two six sided die, one white nd the other blk, re rolled onto tble nd the number uppermost on eh die is reorded. isply the smple spe on lttie digrm. b Let be the event of obtining lrger number on the white die. Find Pr( ). Let be the event tht the sum of the uppermost numbers on the two die does not exeed. Wht is Pr( )? d Let be the event tht both of the uppermost numbers re odd numbers. Stte Pr( ). e lulte the following. f i Pr( Ç Ç ) ii Pr( È ') iii Pr( Ç ) etermine whether the events nd re independent nd give mthemtil explntion to justify your nswer. b d e n ( ) Pr( ) n( x ) n ( ) Pr( ) n( x ) n ( ) 9 Pr( ) n( x ) f i Pr( Ç Ç ) n ( Ç Ç )) n( x ) n ( È ') 7 ii Pr( È ') n( x ) n ( Ç Ç ) iii Pr( Ç ) n ( Ç ) If nd re independent events, then Pr( Ç ) Pr( ) Pr( ) n ( Ç ) LHS n( x ) RHS ¹ LHS The events nd re not independent. John Wiley & Sons ustrli, Ltd

7 Howrd n trvel to work either by riding his bike or by thing the bus. If the wether is forest s wet, there is 0% hne tht he will drive his r. However, if the wether is forest s fine, there is n 0% hne tht he will ride his bike. The wether bureu forests tht the hne of Mondy being wet is 0.. onstrut probbility tree digrm to show the given informtion nd define the symbols used. b lulte the probbility tht Mondy is fine nd Howrd drives his r to work. Wht is the probbility tht Howrd rides his bike to work on Mondy? d Given tht Howrd rode his bike to work, wht is the probbility tht Mondy ws fine? e The wether bureu s reords show tht if one dy is wet there is 0. hne the next dy will be wet; however, if one dy is fine, there is 0.7 hne tht the next dy will be fine. On Mondy, the wether is fine. lulte the probbility tht the Wednesdy of the sme week will lso be fine. Let W be the event of wet wether, F be the event of fine wether, be the event of riding the bike nd be the event of driving the r. b Pr( FÇ ) The probbility tht Mondy is fine nd Howrd drives his r is 0.. Pr( ) Pr( W Ç ) + Pr( FÇ ) d 0. The probbility tht Howrd rides his bike to work is 0.. Pr( FÇ ) Pr( F ) Pr( ) e Tuesdy ould either be wet or fine. Pr(Wednesdy is fine given Mondy is fine) John Wiley & Sons ustrli, Ltd 7

8 lol shire ounil hs submissions for 0 new housing developments nd new rod developments. udget restritions will only llow the ounil to pprove of the projet submissions. How mny different seletions of the projets re possible? b lulte the probbility tht the projets re for new housing developments nd new rods. Wht is the probbility tht t lest of the new rod developments re pproved? d The ounil omes to the deision to pprove new housing developments, H, H, H, H, nd new rod developments, R nd R. It must then onsider how to shedule the order in whih these developments will be undertken, one fter the other. i In how mny different wys n the ounil order the sheduling of these projets? ii Wht is the probbility tht the two rod projets R nd R re not sheduled with one immeditely fter the other? There re wys to hoose projets from the totl of projets submitted. 00 seletions b Four new housing developments nd new rods n be seleted in 0 wys. The probbility is 0 0. The possibilities re housing nd rod, or housing nd rod, or housing nd rod. The probbility of t lest new rod developments is: d i The projets n be ordered in! 7 wys. ii The probbility tht R nd R re seprted is the omplement of the probbility tht R nd R re together. Tret R nd R s one unit whih internlly rerrnges in!. There re now units to rrnge in! wys. Hene the totl number of rrngements with R nd R together is!! 0. Pr( R nd R re seprted) 0-7 John Wiley & Sons ustrli, Ltd