186 CHAPTER 6 Transformations 3. O x y 2 4 6 –2 –4 –2 2 4 6 8 M M The diagram shows a Cartesian plane. Point M' is the image of point M under a translation. Describe the translation. 4. The diagram is drawn on a 1-unit square grid. Draw the image of ABC under the translation –3 2 and label it as A'B'C'. A B C 5. A B C A B C The diagram is drawn on a square grid. A'B'C' is the image of ABC under a transformation. Describe the transformation. 6. The diagram is drawn on a square grid. Draw the image of triangle M under a UHÁHFWLRQLQWKHOLQHPQ. P Q M 7. The diagram shows a pentagon PQRST GUDZQRQDJULGRIHTXDOVTXDUHV%HJLQQLQJ from the line AB, draw a pentagon ABCDE which is congruent to pentagon PQRST A B P QR S T 8. Pentagon ABCDE and pentagon A'B'C'D'E', are drawn on a grid as shown. If A'B'C'D'E' is the image of ABCDE under an enlargement at centre P, mark and label the centre of enlargement. D A B C E A D E C B 9. The diagram shows two quadrilaterals, ABCD and A'B'C'D' drawn on a grid of HTXDO VTXDUHV A'B'C'D' is the image of ABCD under an enlargement with centre O 2QWKH GLDJUDP PDUN DQG ODEHO FHQWUHO B A C D D C A B ©Praxis Publishing_Focus On Maths
187 Transformations CHAPTER 6 10. The diagram shows a pentagon PQRST GUDZQ RQ D JULG RI HTXDO VTXDUHV 2Q WKH diagram, draw the image of PQRST under an enlargement with scale factor 1 2 at centre C P Q R S T C 11. –2 2 –2 2 y O x –4 4 M 7KH GLDJUDP VKRZV D &DUWHVLDQ SODQH State the coordinates of the image of point M under a rotation through 90° clockwise DERXWWKHSRLQW ± 12 7KHGLDJUDPLVGUDZQRQDVTXDUHJULG'UDZ the image of triangle KLM under a rotation WKURXJKffiFORFNZLVHDERXWFHQWUHO K L M O 13. R O 7KH GLDJUDP LV GUDZQ RQ D VTXDUH JULG 'UDZ WKH LPDJH RI WULDQJOH R under a URWDWLRQ WKURXJK ffi DQWLFORFNZLVH DERXW centre O 14. 2 4 –2 –4 –2 O x y D C B A A B C D 2 4 7KH GLDJUDP VKRZV D &DUWHVLDQ SODQH A'B'C'D' is the image of ABCD under a URWDWLRQ'HVFULEHWKHURWDWLRQ 15. In the diagram, A'B'C'D' is the image of ABCD under a rotation through 90° FORFNZLVH DERXW SRLQW N 2Q WKH GLDJUDP mark point N 2 4 –4 –2 O 2 4 x y –4 D C A B A B C D –2 16. 7KH GLDJUDP LV GUDZQ RQ D VTXDUH JULG A'B'C'D' is the image of ABCD under a URWDWLRQWKURXJKflDERXWSRLQWT2QWKH GLDJUDPODEHO (a) point T E SRLQW M', the image of point M under WKH VDPH URWDWLRQ A B M C D A B C D ©Praxis Publishing_Focus On Maths
7PROBABILITY Applications of this chapter 3UREDELOLW\ LV D PDWKHPDWLFDO À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©Praxis Publishing_Focus On Maths
Concept Map Learning Outcomes • 8QGHUVWDQGDQGXVHWKHFRQFHSWRIH[SHULPHQWDOSUREDELOLW\ • ([SORUHWKHH[SHULPHQWDOSUREDELOLW\RIDQHYHQWZKHQWKHQXPEHURIWULDOVDUHODUJH • 8QGHUVWDQGDQGXVHWKHFRQFHSWRIWKHRUHWLFDOSUREDELOLW\ 189 Gerolamo Cardano was a 16th-century Italian mathematician who used the game of throwing dice to develop the basic concepts of probability. In his book Liber de Ludo Alea (Book on Games of Chance) published in 1663, he introduced important concepts and laid the foundation for the development of probability theory. • Probability • Experiment • Trial • Outcome • Relative frequency • Experimental probability • Theoretical probability • Equally likely outcomes • Sample space Key Terms Maths History Introduction to Probability Probability Idea of probability Experimental probability Theoretical probability Complement of an event ©Praxis Publishing_Focus On Maths
CHAPTER 7 Probability 190 Flashback 1. &DOFXODWHHDFKRIWKHIROORZLQJ (a) 2 3 + 1 9 = (b) 5 6 – 4 5 = (c) 1 6 + 5 8 – 3 4 = (d) 1 – 3 7 = (e) 1 – 13 50 = 7KHUH DUH WKUHH ÁDYRXUV RI LFH FUHDP DYDLODEOH DW WKH VWDOOffl VWUDZEHUU\ FKRFRODWH DQG YDQLOOD -RKQ FDQ RQO\ RUGHU WZR VFRRSV RI LFH FUHDP DW D WLPH /LVW DOO WKH SRVVLEOH FRPELQDWLRQVRIRUGHUVWKDWKHFDQPDNH 1 If John can order three scoops of ice cream at a time, how many possible combinations of orders can he make? List down your answers. (Repetition is allowed) Critical Thinking 2. 6LPSOLI\HDFKRIWKHIROORZLQJ (a) 1 2 = 1 3 = (b) 2 3 = 3 8 = (c) 3 8 = 4 7 = (d) 7 10 = 4 9 = (e) 2 3 = 7 10 = 1 4 = Vanilla Chocolate Strawberry ©Praxis Publishing_Focus On Maths
Probability CHAPTER 7 191 7.1 Introduction to Probability A Idea of probability The probabilityRIDQHYHQWLVDPHDVXUHRIWKHOLNHOLKRRGWKDWWKHHYHQWZLOORFFXU3UREDELOLW\ FDQEHVKRZQXVLQJDWRSUREDELOLW\VFDOH Less likely to happen More likely to happen Impossible Unlikely Even chance Likely Certain 0 1 0 0.5 1 0% 50% 100% 1 2 8VH IUDFWLRQ 8VH GHFLPDO 8VH SHUFHQW 6RPHWKLQJZLWKDSUREDELOLW\RIZLOOQHYHUKDSSHQ)RUH[DPSOHWKHSUREDELOLW\RIDKRUVH EHLQJDEOHWRÁ\LV,QRWKHUZRUGVLWLVimpossible IRUDKRUVHWRÁ\ 6RPHWKLQJZLWKDSUREDELOLW\RIRUZLOOGHÀQLWHO\KDSSHQ)RUH[DPSOHWKHSUREDELOLW\RI DEDOOIDOOLQJGRZQDIWHUEHLQJWKURZQXSZDUGVLV7KXVLWLV certainIRUWKHHYHQWWRKDSSHQ 6RPHWKLQJWKDWLVDVOLNHO\WRKDSSHQDVLWLVQRWWRKDSSHQKDVDQHYHQFKDQFHRUDSUREDELOLW\ RIRU)RUH[DPSOHWKHSUREDELOLW\RIWRVVLQJDKHDGZLWKDIDLUFRLQLV7KHHYHQW KDVDQeven chanceRIRFFXUULQJ EXAMPLE 1 8VHWKHZRUGV¶,PSRVVLEOH·¶8QOLNHO\·¶(YHQFKDQFH·¶/LNHO\· DQG¶&HUWDLQ·WRGHVFULEHKRZOLNHO\WKHIROORZLQJ VLWXDWLRQV ZLOOKDSSHQ D $IDLUFRLQWKDWLVWRVVHGODQGV¶KHDGV·XS E $FDUGSLFNHGUDQGRPO\IURPDGHFNZRXOGEHWKH¶4XHHQ RI+HDUWV· F $SHUVRQXVHVKLVULJKWKDQGWRZULWH G 7KHVXQULVHVLQWKH(DVW H $QDQWLVKHDYLHUWKDQDQHOHSKDQW Solution: D (YHQFKDQFH E 8QOLNHO\ F /LNHO\ G &HUWDLQ H ,PSRVVLEOH ©Praxis Publishing_Focus On Maths
CHAPTER 7 Probability 192 B Experimental probability $QDFWLYLW\LQYROYLQJFKDQFHVXFKDVWRVVLQJDFRLQLVFDOOHGDQexperiment(DFKUHSHWLWLRQ RU REVHUYDWLRQRI DQ H[SHULPHQW LV Dtrial DQG HDFK UHVXOW LV DQoutcome$ VHW RI RQH RU PRUHRXWFRPHVLVDQevent([SHULPHQWVDUHSHUIRUPHGWRSURYLGHGDWDZKLFKFDQWKHQEH XVHGWRIRUHFDVWWKHRXWFRPHRIIXWXUHVLPLODUHYHQWV Objective:7RFRQGXFWDVLPSOHSUREDELOLW\H[SHULPHQW Instruction:'RWKLVDFWLYLW\LQJURXSVRIIRXU Material:&RLQ 1. 7RVVDFRLQWLPHV5HFRUGWKHRXWFRPHV ZKHWKHULWLVKHDGVRUWDLOV 2. &RPSOHWHWKHIROORZLQJWDEOHZLWKWKHRXWFRPHVUHFRUGHG Event Frequency of the outcome Frequency of the outcome Total number of trials *HWWLQJDKHDGV *HWWLQJDWDLOV 3. 5HSHDW6WHSDQG6WHSIRUWLPHVDQGWLPHV 4. The ratio IUHTXHQF\RIWKHRXWFRPH WRWDOQXPEHURIWULDOV LVFDOOHGUHODWLYHIUHTXHQF\:KDWLVWKHUHODWLRQVKLS EHWZHHQWKHUDWLRDQGWKHH[SHULPHQWDOSUREDELOLW\" 5. 3UHVHQW\RXUÀQGLQJVLQFODVV 1 Experimental probabilityLVRQHZD\RIHVWLPDWLQJWKHSUREDELOLW\RIDQHYHQW7KHH[SHULPHQWDO SUREDELOLW\RIDQHYHQWLVGHWHUPLQHGEDVHGRQWKHGDWDFROOHFWHGWKURXJKH[SHULPHQW 5HODWLYHIUHTXHQF\LVKRZRIWHQDSDUWLFXODUUHVXOWRFFXUVLQDUHSHDWHGH[SHULPHQW7KHUHODWLYH IUHTXHQF\RIDQHYHQWRFFXUULQJLVWKHH[SHULPHQWDOSUREDELOLW\RILWRFFXUULQJ Experimental probability = frequency of the outcome total number of trials A survey was conducted by a school among 50 students to determine their favourite lunch food. 37 students chose Nasi Goreng as their favourite. Based on this information, what is your prediction for the number of students who would prefer Nasi Goreng, out of a total of 600 students? Discuss with your classmates. $VVXPHWKDWWKHUHDUHQRVLJQL¿FDQWFKDQJes in the food preferences of students between the time of the survey and the time of the prediction.) INTERACTIVE ZONE ©Praxis Publishing_Focus On Maths
Probability CHAPTER 7 193 Objective:7RH[SORUHWKHH[SHULPHQWDOSUREDELOLW\RIDQHYHQWZKHQWKHQXPEHU RIWULDOVDUHODUJH Instruction:'RWKLVDFWLYLW\LQSDLUV 1. 2SHQWKHÀOHRIRolling a DiceXVLQJGeoGebra 2. 5ROOWKH GLFH E\ FOLFNLQJ RQWKH ¶Roll the dice once· ¶Roll the dice 10 times· RU ¶Roll the dice 100 times·EXWWRQV 3. &OLFNRQWKH¶Start all over again·EXWWRQLI\RXZDQWWRVWDUWDJDLQ 4. 0DNH D FRQFOXVLRQ DERXW WKH H[SHULPHQWDO SUREDELOLW\ RI DQ HYHQW ZKHQ WKH QXPEHU RI WULDOVDUHODUJHHQRXJK 5. 3UHVHQW\RXUÀQGLQJVLQFODVV 2 )URPWKHÀQGLQJVRI$FWLYLW\ZHIRXQGWKDWWKHH[SHULPHQWDOSUREDELOLW\RIREWDLQLQJHDFK QXPEHU RU DSSURDFKHV WR RQH YDOXH WKDW LV $OWKRXJK H[SHULPHQWDO SUREDELOLW\FDQÁXFWXDWHGXHWRFKDQFHEXWDVWKHQXPEHURIWULDOVLQFUHDVHVWKHH[SHULPHQWDO SUREDELOLW\WHQGVWRDSSURDFKDPRUHFRQVLVWHQWYDOXHWKDWUHSUHVHQWVWKHWUXHSUREDELOLW\RI WKHHYHQW EXAMPLE 2 7ZR SXSLOV HDFK VHOHFW VHHGV DW UDQGRP IURP D EDJ FRQWDLQLQJDYHU\ODUJHQXPEHURIUHGDQGEODFNVHHGV(DFK VHHGLVUHWXUQHGWRWKHEDJEHIRUHDQRWKHULVVHOHFWHG7KH WDEOHVKRZVWKHLUUHVXOWV Colour of seed Number of seeds Carl UHG 57 EODFN 43 Sherene UHG 47 EODFN 53 ©Praxis Publishing_Focus On Maths
CHAPTER 7 Probability 194 If an experiment or survey is conducted only a few times, the estimate of the relative frequency may not be accurate. To obtain an accurate estimate, the experiment must be repeated numerous times. D )LQGWKH UHODWLYHIUHTXHQFLHV RI UHG RU EODFN VHHGVIRU Carl’VUHVXOWV E )LQG WKH UHODWLYH IUHTXHQFLHV RI UHG RU EODFN VHHGV IRU 6KHUHQH’VUHVXOWV F &DOFXODWH WKH FRPELQHG UHODWLYH IUHTXHQFLHV RI UHG RU EODFNVHHGV Solution: D 5HODWLYHIUHTXHQF\RIUHGVHHGV 57 100 5HODWLYHIUHTXHQF\RIEODFNVHHGV 43 100 E 5HODWLYHIUHTXHQF\RIUHGVHHGV 47 100 5HODWLYHIUHTXHQF\RIEODFNVHHGV 53 100 F &RPELQHGUHODWLYHIUHTXHQF\RIUHGVHHGV 104 200 &RPELQHGUHODWLYHIUHTXHQF\RIEODFNVHHGV ffi 200 fl EXAMPLE 3 'HWHUPLQH WKH H[SHULPHQWDO SUREDELOLW\ RI HDFK RI WKH IROORZLQJHYHQWV D *HWWLQJQXPEHUZKHQDGLFHLVUROOHGLIQXPEHUWXUQHG XSflWLPHVLQUROOV E *HWWLQJWDLOVZKHQDFHQWFRLQLVWRVVHGLIKHDGVWXUQHG XSWLPHVLQflWRVVHV Solution: D ([SHULPHQWDOSUREDELOLW\ = )UHTXHQF\RIJHWWLQJQXPEHU 1XPEHURIWULDOV = fl 300 E ([SHULPHQWDOSUREDELOLW\ = )UHTXHQF\RIJHWWLQJWDLOV 1XPEHURIWULDOV = fl² fl )UHTXHQF\ RI JHWWLQJ WDLOV 1XPEHU RIWULDOV ²IUHTXHQF\ RI JHWWLQJ KHDGV ©Praxis Publishing_Focus On Maths
Probability CHAPTER 7 195 C Theoretical probability :KHQ D FRLQ LVWRVVHGWKHUH DUHWZR SRVVLEOH RXWFRPHV KHDGV RUWDLOV7R GHWHUPLQHWKH SUREDELOLW\RIJHWWLQJDKHDGV\RXFDQFRQGXFWDQH[SHULPHQWDQGUHFRUG\RXUUHVXOWVWRÀQG WKH H[SHULPHQWDO SUREDELOLW\ RU \RX FDQ FDOFXODWH WKH WKHRUHWLFDOO\ SUREDELOLW\ Theoretical probabilityLVXVHGWRÀQGWKHSUREDELOLW\RIDQHYHQWZKHQDOORXWFRPHVDUHHTXDOO\OLNHO\ (I) Equally likely outcomes 2XWFRPHV DUH equally likely WR RFFXU LI WKH\ KDYH WKH VDPH SUREDELOLW\ RI RFFXUULQJ 7KLV PHDQVWKDWWKHHYHQWLVIDLU:KHQZHWRVVDFRLQLWLVHTXDOO\SRVVLEOHWKDWZHZLOOJHWHLWKHU ¶KHDGV·RU¶WDLOV·,IZHWRVVDFRLQIRUDQLQÀQLWHDPRXQWRIWLPHZHZLOOÀQGWKDWKDOIRIWKH WLPHKHDGVZLOODSSHDUDQGWKHRWKHUKDOIWDLOVZLOODSSHDU7KXVZHFDQDVVXPHWKDWERWK RXWFRPHVDUHHTXDOO\OLNHO\ 2XWFRPHDUHnot equally likelyWRRFFXULIRQHRUPRUHRXWFRPHVDUHPRUHOLNHO\WRRFFXU WKDQRWKHUV7KLVPHDQVWKDWWKHHYHQWLVXQIDLU:KHQDPDWFKER[LVWKURZQDOOVL[IDFHVDUH QRWHTXDOO\OLNHO\WRFRPHXS,IDEDJFRQWDLQVEDOOVRIGLIIHUHQWVL]HVDQGDEDOOLVVHOHFWHG DWUDQGRPDOOWKHEDOOVDUHQRWHTXDOO\OLNHO\WREHVHOHFWHG EXAMPLE 4 $QDO\VHWKHIROORZLQJHYHQWVDQGGHWHUPLQHLIWKHLURXWFRPHV DUHHTXDOO\OLNHO\RXWFRPHV6WDWH\RXUUHDVRQV D 7RVVLQJDFRLQ E $PDUDWKRQZLWKUXQQHUV F 5ROOLQJDGLFH Solution: D <HV,WLVHTXDOO\OLNHO\WRJHWHLWKHUWKH¶KHDGV·RU¶WDLOV· IDFHXS E 1R7KHUXQQHUVDUHQRWHTXDOO\OLNHO\WRZLQWKHPDUDWKRQ ,WGHSHQGVRQKRZIDVWWKH\FDQUXQ F <HV,WLVHTXDOO\OLNHO\WRJHWRUE\UROOLQJ DGLFH (II) Sample space 7KHOLVWRIDOOSRVVLEOHRXWFRPHVRIDQH[SHULPHQWLVDsample space$VHWRIRQHRUPRUH RXWFRPHVLVDQevent)RULQVWDQFHWKHSRVVLEOHRXWFRPHVRIUROOLQJDGLFHDUHGRWV DQG6RWKHVDPSOHVSDFHRIUROOLQJDGLFHLVZULWWHQDVS ^`7KHVHW RIRXWFRPHVWKDWVDWLVÀHVDJLYHQFRQGLWLRQDQGLVDVXEVHWRIWKHVDPSOHVSDFHLVNQRZQDV HYHQW,IALVWKHHYHQWRIJHWWLQJPRUHWKDQLQDGLFHUROOLQJWKHQLWLVZULWWHQDVA ^` 6RPHH[SHULPHQWVKDYHWZRVWHSVWKDWJLYHDSDLURIUHVXOWVVXFKDVWRVVLQJFRLQVUROOLQJ GLFHVRUWRVVLQJDFRLQDQGUROOLQJDGLFH7KHVDPSOHVSDFHRIWKHVHWZRVWHSH[SHULPHQWV PD\EHGLVSOD\HGLQDtwo-way table or a tree diagram ©Praxis Publishing_Focus On Maths
CHAPTER 7 Probability 196 Steps of drawing tree diagram: Use branches to show the individual outcomes for the ¿UVWWULDO Relate each outcome IURP WKH ¿UVW WULDO ZLWK WKH outcome of the second trial. List the possible combinations of outcomes in order, that is, WKH RXWFRPH RI WKH ¿UVW WULDO followed by the outcome of the second trial. EXAMPLE 5 $ER[FRQWDLQVSLHFHVRIFDUGVQXPEHULQJWR$FDUG LVUHPRYHGUDQGRPO\IURPWKHER['HWHUPLQH D WKHVDPSOHVSDFHSRIWKHH[SHULPHQW E HYHQWADQGBDVIROORZ A $SULPHQXPEHULVFKRVHQ B $PXOWLSOHRILVFKRVHQ Solution: (a) S ^ fl ffi ` (b) A ^ ` B ^ fl` EXAMPLE 6 /LVWWKHVDPSOHVSDFHIRUHDFKRIWKHIROORZLQJH[SHULPHQWV 7KHQ UHSUHVHQW WKH VDPSOH VSDFH XVLQJ D WZRZD\ WDEOH DQGDWUHHGLDJUDP D 7RVVLQJFRLQVWRJHWKHU E 7RVVLQJDFRLQDQGUROOLQJDGLFH Solution: D /HWH UHSUHVHQWV KHDGV DQGT UHSUHVHQWV WDLOV 6DPSOH VSDFH ^HHHTTHTT ` 7ZRZD\ WDEOHffl 7UHH GLDJUDPffl Coin 1 Coin 2 Outcomes H T H T HH HT TH TT H T Coin 1 Coin 2 H T H HH TH T HT TT E 6DPSOH VSDFH ^ H H H H H (H T T T T T T ` If two coins are tossed simultaneously, what is the probability of getting exactly two heads? Discuss with your IULHQGVDQGSUHVHQW\RXU¿QGLQJWRWKHFODVV INTERACTIVE ZONE The outcomes in a sample space S are equally likely if each outcome has the same probability of occurring. ©Praxis Publishing_Focus On Maths
Probability CHAPTER 7 197 A pair of results can be written as an ordered pair. The ordered pair (H, 2) would correspond to obtaining a heads on the coin and a 2 on the dice. Tree diagrams are used to list all possible outcomes of two or more events that are not necessarily equally likely. 7ZRZD\WDEOHffl Coin Dice H T 1 (H (T 2 (H (T 3 (H (T 4 (H (T 5 (H (T (H (T 7UHHGLDJUDPffl Coin Dice Outcomes H 1 2 3 4 5 6 (H, 1) (H, 2) (H, 3) (H, 4) (H, 5) (H, 6) T 1 2 3 4 5 6 (T, 1) (T, 2) (T, 3) (T, 4) (T, 5) (T, 6) EXAMPLE 7 ,QDIRRWEDOOFRPSHWLWLRQ7HDP6N\KDVSOD\HGWZLFH7KH SRVVLEOHUHVXOWVRIWKHJDPHDUHZRQORVWRUWLHG'HWHUPLQH D WKH VDPSOH VSDFH RIWKHIRRWEDOO FRPSHWLWLRQ E HYHQWA7HDP 6N\ KDV ZRQ DW OHDVW RQFH Solution: D /Ht W UHSUHVHQWV ZRQ T UHSUHVHQWV WLHG DQG L UHSUHVHQW ORVW First game Second game Tree diagram Outcomes W W T L WW WT WL T W T L TW TT TL L W T L LW LT LL 6DPSOH VSDFH ^WW WT WL TW TT TL LW LTLL` (b) A = {WWWTWLTWLW` ©Praxis Publishing_Focus On Maths
CHAPTER 7 Probability 198 (III) Probability of an event ,IAUHSUHVHQWVDQHYHQWWKDWLVOLNHO\WRRFFXUDQGSUHSUHVHQWV WKHVDPSOHVSDFHRIDOOSRVVLEOHRXWFRPHVWKHQWKHSUREDELOLW\ RIHYHQWA RFFXUULQJLV P(A) = n(A) n(S) EXAMPLE 8 $ER[FRQWDLQVXQLWVRIEODFNSHQDQGXQLWVRIUHGSHQ ,IDSHQLVSLFNHGDWUDQGRPIURPWKHER[ÀQGWKHSUREDELOLW\ WKDWWKHSHQSLFNHGLVDUHGSHQ Solution: /HWA EHWKHHYHQWWKDWDUHGSHQLVSLFNHG n(A) = 15 n(S) = 25 + 15 = 40 7KHUHIRUHWKHSUREDELOLW\WKDWDUHGSHQLVSLFNHGLV 3 fl EXAMPLE 9 $IDLUGLFHLVUROOHG)LQGWKHSUREDELOLW\RIJHWWLQJ D DSULPHQXPEHU E DQXPEHUWKDWLVOHVVWKDQ Solution: S ^` +HQFH n(S D /HWA (YHQWRIJHWWLQJDSULPHQXPEHU 7KHQA ^` n(A) = 3 7KHUHIRUHWKHSUREDELOLW\RIJHWWLQJDSULPHQXPEHULV 1 2 P(A) = n(A) n(S) = 15 40 = 3 fl P(A) = n(A) n(S) = 3 = 1 2 ZKHUHn(A QXPEHURIRXWFRPHVIDYRXUDEOHWRHYHQWA n(S WRWDOQXPEHURISRVVLEOHRXWFRPHVRIWKHVDPSOHVSDFHS 7KLV LV FDOOHGWKHtheoretical probability RI HYHQWA DQG FDQ RQO\ EH DSSOLHGZKHQ DOOWKH RXWFRPHVRIWKHH[SHULPHQWDUHHTXDOO\OLNHO\ Note that 0 P(A) 1. (a) P(A) = 1 means that event A is certain to occur. (b) P(A) = 0 means that event A will not occur. ©Praxis Publishing_Focus On Maths
Probability CHAPTER 7 199 E /HWB (YHQWRIJHWWLQJDQXPEHUOHVVWKDQ 7KHQB ^` n(B) = 4 7KHUHIRUHWKHSUREDELOLW\RIJHWWLQJDQXPEHUOHVVWKDQLV 2 3 P(B) = n(B) n(S) = 4 = 2 3 Science Probability plays a vital role in everyday life, including weather forecast. In a weather forecast, all elements such as temperature, wind and precipitation contains information that accurately TXDQWL¿HV WKH LQKHUHQW XQFHUWDLQW\ $ SUREDELOLW\ IRUHFDVW LV an assessment of how likely an event can occur in terms of percentage and records the risks associated with weather. Maths LINK D Complement of an event &RPSOHPHQWRIHYHQWALVWKHVHWRIDOORXWFRPHVLQWKHVDPSOHVSDFHWKDWDUHQRWLQFOXGHG LQWKHRXWFRPHVRIHYHQWADQGGHQRWHGDVA'7KHSUREDELOLW\RIWKHFRPSOHPHQWRIHYHQWA LVJLYHQDVEHORZ P(A' ) = n(A' ) n(S) P(A') = 1 – P(A) n(A') = n(S) – n(A) Sample Space Event A Complement of A A A' EXAMPLE 10 $ ER[ FRQWDLQV JUHHQ FDSV EOXH FDSV DQG ZKLWH FDS 6LWL UDQGRPO\WDNHV RXW D FDSIURPWKH ER[ )LQGWKH SUREDELOLW\RI D JHWWLQJDJUHHQFDS E QRWJHWWLQJDJUHHQFDS Solution: /HWG1 DQGG2 EH JUHHQ FDSVB1 B2 DQGB3 EH EOXH FDSV DQGWEHZKLWHFDS 6DPSOHVSDFHS = {G1 G2 B1 B2 B3 W ` n(S ©Praxis Publishing_Focus On Maths
CHAPTER 7 Probability 200 D /HWA (YHQWRIJHWWLQJDJUHHQFDS = {G1 G2 ` n(A) = 2 (b) A' (YHQWRIQRWJHWWLQJDJUHHQFDS = {B1 B2 B3 W` n(A' ) = 4 EXAMPLE 11 1 1 2 3 1 1 1 2 1 2 3 1 1 2 3 7KHGLDJUDPVKRZVLGHQWLFDOFDUGVODEHOOHGDQG $ FDUG LV FKRVHQ DW UDQGRP 6WDWH WKH SUREDELOLW\ WKDW WKH FDUGFKRVHQLVnotODEHOOHG Solution: 6DPSOHVSDFHS ^` n(S) = 15 /HWA (YHQWRIREWDLQLQJDFDUGODEHOOHG ^` n(A fl 3 A) = fl 15 A' (YHQWRIQRWREWDLQLQJDFDUGODEHOOHG P(A' ²3 A) ² fl 15 = 7 15 P(A) = n(A) n(S) = 2 = 1 3 P(A' ) = n(A') n(S) = 4 = 2 3 P(A ² P(A) ² 1 3 = 2 3 A bag contains red, green and blue balls. A ball is chosen randomly from the bag. If P(a red ball) = 1 2 and P(not a green ball) = 2 3 , determine the probability that a blue ball is chosen. ©Praxis Publishing_Focus On Maths
Probability CHAPTER 7 201 In a box, there are 5 red balls and 7 black balls. What is the probability of getting a black ball? Critical Thinking 'HVFULEH KRZ OLNHO\ WKH IROORZLQJ VLWXDWLRQV ZLOOEHXVLQJWKHZRUGV¶,PSRVVLEOH·¶8QOLNHO\· ¶(YHQFKDQFH·¶/LNHO\·DQG¶&HUWDLQ· (a) *HWWLQJ D ¶WZR·IURP UROOLQJ D GLFH (b) :DONLQJ IURP %DQJNRN WR 1HZ <RUN LQ KRXUV (c) $ \HDU ROG NLG ZHDULQJ JODVVHV G &KRRVLQJDQRGGQXPEHUIURPDOLVWRI SULPH QXPEHUV UDQGRPO\ (e) $Q DGXOW JLUDIIHWDOOHU WKDQ FP I 3LFNLQJ RXW D JUHHQ PDUEOH DW UDQGRP IURP D EDJ FRQWDLQLQJ EOXH PDUEOHV \HOORZ PDUEOHV DQG fl JUHHQ PDUEOHV 7KHVSLQQHUVKRZQLVVSXQWLPHV7KH UHVXOWVDUHDVUHFRUGHGEHORZ Number 12345 Frequency 25 23 24 27 21 )LQGWKHUHODWLYHIUHTXHQF\RIVFRULQJ D E F G H *LYH\RXUDQVZHUVWRVLJQLÀFDQWÀJXUHV 'HWHUPLQH WKH H[SHULPHQWDO SUREDELOLW\ RI HDFKRIWKHIROORZLQJHYHQWV D *HWWLQJ D SDLU RI QXPEHU ZKHQ WZR GLFH DUH UROOHG LI QXPEHU WXUQHG XS LQ ERWK GLFH WLPHV LQ UROOV E :KLWHIDFH FKLSIDFLQJ XSZDUG ZKHQ D EODFNZKLWH FKLS LV WRVVHG LI WKH EODFN IDFHFKLSIDFLQJXSZDUGflWLPHVLQ WRVVHV 1 2 4 5 3 Result of Games Number of times Ann wins |||| |||| Number of times Badrul wins |||| |||| | Number of games |||| |||| |||| |||| 7KHVOLSRIUHVXOWVDERYHVKRZVWKHQXPEHU RI WLPHV ZRQ E\ WZR SOD\HUV LQ D VHYHUDO FKHVV JDPHV 'HWHUPLQH WKH H[SHULPHQWDO SUREDELOLW\ RI ZLQQLQJ RI HDFK SOD\HU 'XULQJDEDVNHWEDOOWUDLQLQJ&KRQJVFRUHV SRLQWVRXWRIffiDWWHPSWV'HWHUPLQHWKH H[SHULPHQWDOSUREDELOLW\RI&KRQJVFRULQJD SRLQW" 'HWHUPLQHLIWKHRXWFRPHVRIWKHIROORZLQJ HYHQWVDUHHTXDOO\OLNHO\6WDWH\RXUUHDVRQV IRU\RXUDQVZHUV D $Q DUFKHU LV HTXDOO\ OLNHO\ WR KLW WKH EXOO·V H\H RU PLVV LW E 6XGD LV HTXDOO\ OLNHO\WR SLFN RXW D UHG EHDQ DW UDQGRP IURP D FRQWDLQHU ZLWK UHG EHDQV DQG JUHHQ EHDQV PL[HG WRJHWKHU F 7UDNDUQLVHTXDOO\OLNHO\WRSLFNDFRIIHH ÁDYRXUHG VZHHW IURP D EDJ LI RI WKH EDJ FRQWDLQV QRQ FRIIHHÁDYRXUHG VZHHWV $ GD\ LV VHOHFWHG UDQGRPO\ IURP D ZHHN 'HWHUPLQH D WKH VDPSOH VSDFH RIWKH VHOHFWLRQ E HYHQW E VHOHFWLQJ GD\V VWDUWHG ZLWK letters S DQGT Practice 7.1 Basic Intermediate Advanced ©Praxis Publishing_Focus On Maths
202 CHAPTER 7 Probability C A D B E The diagram shows a disk spinner with ¿YHVHFWRUVRIWKHVDPHVL]H(DFKVHFWRU are labelled with letters A, B, C, D and E respectively. The spinner is spun twice. List the sample space for the experiment. $GLFHLVUROOHGWZLFH5HSUHVHQWWKHVDPSOH VSDFHIRUWKH H[SHULPHQW XVLQJ DWZRZD\ WDEOHDQGDWUHHGLDJUDP CAT 7KH GLDJUDP VKRZVWKUHH OHWWHU FDUGV LQ D ER[$ FDUG LV VHOHFWHG UDQGRPO\IURPWKH ER[DQGWKHOHWWHULVUHFRUGHG7KHVHOHFWHG FDUGLVSXWEDFNLQWKHER[7KHQWKHVHFRQG FDUGLVVHOHFWHGIURPWKHER[ D 5HSUHVHQW WKH VDPSOH VSDFH RI WKH H[SHULPHQW XVLQJ DWUHH GLDJUDP E /LVW WKH RXWFRPHV RI HDFK RI WKH IROORZLQJ HYHQWV (i) X (YHQW RI ERWK FDUGV VHOHFWHG KDYH WKH VDPH ODEHOV (ii) Y (YHQW RI RQH RIWKH FDUG VHOHFWHG LV ODEHOOHGT $OHWWHULVFKRVHQDWUDQGRPIURPWKHZRUG ¶BANANA·)LQGWKHSUREDELOLW\RIFKRRVLQJ (a) the letter N E DYRZHO F DFRQVRQDQW $ EDJ FRQWDLQV RQHGROODU QRWHV ÀYHGROODU QRWHV WHQGROODU QRWHV DQG À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ffifl`)LQGWKHSUREDELOLW\ WKDW WKH VHOHFWHG QXPEHU LV QRW D SULPH QXPEHU $QXPEHULVFKRVHQDWUDQGRPIURPWKHVHW {x : 5 x x LV DQ LQWHJHU`)LQGWKH SUREDELOLW\RIFKRRVLQJ D DQXPEHUWKDWLVGLYLVLEOHE\ E DQXPEHUWKDWLVQRWGLYLVLEOHE\ $EDJFRQWDLQVUHGEXWWRQVDQGxEODFN EXWWRQV$EXWWRQLVSLFNHGDWUDQGRPIURP WKHEDJ*LYHQWKDWWKHSUREDELOLW\RISLFNLQJ DUHGEXWWRQLV 2 3 ÀQG D WKHSUREDELOLW\RISLFNLQJDEODFNEXWWRQ E WKHYDOXHRIx $ GUDZHU FRQWDLQV UHG EURZQ DQG ZKLWH HQYHORSHV RIWKHP DUH UHG HQYHORSHV ,I)DWLPDKSLFNVDQHQYHORSHUDQGRPO\IURP WKHGUDZHUWKHSUREDELOLW\RIREWDLQLQJDUHG HQYHORSHLV 1 7KHSUREDELOLW\RIREWDLQLQJ DEURZQHQYHORSHLV 1 4 )LQG D WKH SUREDELOLW\ RI REWDLQLQJ D ZKLWH HQYHORSH E WKH WRWDO QXPEHU RI HQYHORSHV LQ WKH GUDZHU ©Praxis Publishing_Focus On Maths
Probability CHAPTER 7 203 Sample space The list of all possible outcomes. Event A set of one or more outcomes. Summary Probability Experimental probability Experimental probability = frequency of the outcome total number of trials Theoretical probability If A = event that is likely to occur and S = sample space, then P(A) = n(A) n(S) Complement of an event Complement of event A denoted as A'. P(A') = n(A') n(S) or P(A') = 1 – P(A) Section A 1. A bag contains 100 pens of brand P and 66 pens of brand Q. A pen is chosen randomly from the box. Find the probability that the pen chosen is brand Q. A 11 25 C 1 66 B 33 83 D 1 166 2. A basket contains 8 bean dumplings, 6 meat dumplings and 4 chicken dumplings. A dumpling was taken randomly from the basket. Let A is an event of getting bean dumpling and S is the sample space. Which of the following is true? A n(A) = 9 B n(S) = 19 C P(A) = 1 3 D A' is the event of getting other than bean dumpling. 3. There are several cards printed with number 16 to 40 in a box. A card is chosen randomly from the box. Find the probability of choosing a card that is not a prime number. A 1 3 C 18 25 B 17 24 D 19 25 4. A basket contains 4 blue balls. The probability of choosing yellow and red balls are 1 2 and 1 3 respectively. Find the total number of balls in the basket. A 12 C 20 B 16 D 24 5. Ben and Eddy are the last two candidates who apply for a post in a company. It is found that the probability that Ben is accepted is 3 8 and the probability that Eddy is accepted is 4 7 . If the probability of Ben or Eddy is accepted is 181 280 , what is the probability that both of them are accepted? A 3 10 C 3 40 B 3 14 D 19 70 6. Two pupil representatives are chosen from 8 pupils in Grade 8 and 7 pupils in Grade 9 to attend a national conference. Find the probability that both pupils chosen are Grade 8? A 3 56 C 4 15 B 11 56 D 1 210 ©Praxis Publishing_Focus On Maths
204 CHAPTER 7 Probability 7. In a bookshelf containing 80 books, there are 25 mathematics books, 35 history books, and the remaining are art books. Dahlan randomly chooses two books, one at a time. If he selects a history or an art book, he returns the book before choosing the second one. However, if he selects a mathematics book, he keeps the book and then chooses the second book. What is the probability of him choosing an art book followed by a mathematics book? A 5 64 B 35 256 C 25 316 D 175 1264 8. A box contains 14 envelopes such that 4 envelopes are red, 7 envelopes are blue and the rest are green. Two white envelopes are added into the box and three blue envelopes are taken out. An envelope is chosen at random from the box, the probability that the chosen envelope is white or blue is A 6 13 B 4 13 C 3 13 D 2 13 9. At a school, 350 students ride the bus to school, 30 students are sent by their parents with car and the rest walk to school. If a students is chosen randomly, the probability of choosing a student who rides the bus to school is 7 8 . Find the probability of choosing a student who does not walk to school. A 3 4 B 1 8 C 19 20 D 37 45 10. Pupils Prefect Not prefect Class A 8 12 Class B 20 18 The table above shows the number of prefects and non prefects in Class A and Class B. If two pupils are randomly chosen, ¿QG WKH SUREDELOLW\ WKDW ERWK SXSLOV FKRVHQ come from the same class. A 47 87 B 37 87 C 27 87 D 10 87 11. The table below shows the number of Science and Accounting teachers in a district. Department Science Accounting Numbers of teachers 54 36 After some time, 6 Science teachers transferred to another district, 2 Accounting teachers retired and 3 new Science teachers began working in the district. If 2 teachers are randomly selected from the district, what is the probability that they are from different departments? A 16 35 C 18 35 B 17 35 D 324 595 ©Praxis Publishing_Focus On Maths
205 Probability CHAPTER 7 Section B 1. In a survey on the favourite colour of a group of girls, the following results are obtained. Colour Blue Green Red Yellow Number of girls 45 36 24 15 A girl is selected at random from the group. Find the probability that the girl’s favourite colour is (a) blue, (b) green, (c) red, (d) yellow. 2. Frequency 10 2 6 4 8 Time (minutes) 14 16 25 18 17 Ann keeps a record of the time she takes to jog around a garden. The table above shows the time, to the nearest minute, she takes in each trial. Find the probability that the time taken by Ann is (a) 18 minutes, (b) not more than 18 minutes. 3. A survey shows that the probability of a secondary school teacher involved in any charity activities is 0.62. If the survey LQYROYHG WHDFKHUV ¿QG WKH H[SHFWHG number of teachers who are involved in charity activities. 4. E V E N T 7KH GLDJUDP VKRZV ¿YH FDUGV 7KH FDUGV are placed in a bag. Michael draws a card randomly from the bag and records its outcome. The card is then placed back into the bag before a second card is drawn at random. The process is repeated 200 times and the results are shown in the table below. Card EVNT Number of occurrence 82 40 36 42 %DVHGRQWKHUHVXOW¿QGWKHSUREDELOLW\WKDW (a) a ҊEҋFDUG is drawn, (b) a ҊVҋFDUG is drawn 5. Students Apple Orange Girl 8 16 Boy 12 4 The table shows a group of students who preferred apple or orange. (a) If a student is chosen at random from the JLUOV¿QGWKHSUREDELOLW\WKDWWKHVWXGHQW preferred apple. (b) If a student is chosen at random from the JURXS¿QGWKHSUREDELOLW\RIFKRRVLQJ (i) a boy who preferred orange, (ii) a student who preferred orange. 6. The probability that Kassim plays soccer this evening is 5 8 . What is the probability that Kassim does not play soccer this evening? 7. A box contains 20 bulbs. 6 of the bulbs are faulty. A bulb is selected at random from the box. Find the probability that the bulb selected is not faulty. 8. 2 7 9 Y Z Box J Box K The diagram shows three number cards in Box J and two letter cards in Box K. A card is selected at random from Box J and another card is selected at random from Box K. By listing the sample of all the possible outcomes RIWKHHYHQW¿QGWKHSUREDELOLW\WKDW (a) a card with an odd number and the card labelled Z are selected, (b) a card with a prime number or the card labelled Y is selected. 9. In a box, there are two red cups, two blue cups and two green cups. If two cups are VHOHFWHG DW UDQGRP IURP WKH ER[ ¿QG WKH probability that D WKH ¿UVW FXS VHOHFWHG LV UHG DQG WKH second cup selected is blue, (b) the two cups selected are of the same colour. ©Praxis Publishing_Focus On Maths
206 CHAPTER 7 Probability 10. A fair dice is rolled and a fair coin is tossed. Find the probability that (a) a heads or an odd number is obtained, (b) a heads and an odd number are obtained. 11. Based on records, the probability of rainfall on any given day in November at Town P is 1 8 . Determine the probability of (a) rainfall occurring on two consecutive days, (b) rainfall happening on one day but not on the following day. 12. 1 4 3 2 The spinner is spun twice. Find the probability that (a) both numbers obtained are even, (b) an odd number and an even number are obtained. 13. Madam Chin selects a monitor and an assistant monitor from class 9A which consists of 12 male pupils and 14 female pupils. (a) Determine whether this event is a dependent or independent event. Give \RXUMXVWL¿FDWLRQ (b) Find the probability that both the class monitor and assistant monitor are male. 14. G I V I N G A L E C T U R E Fifteen cards labelled with letters as shown above are put into a bag. Two cards are randomly drawn from the bag, one by one without replacement. Calculate the probability that D WKH ¿UVW FDUG GUDZQ LV OHWWHU * DQG WKH second card drawn is letter I. E WKH ¿UVW FDUG GUDZQ LV D YRZHO DQG WKH second card drawn is consonant. (c) both cards drawn are the same letter. (d) at least one of the cards drawn is a vowel. 15. Given that sample space, S = {17, 18, 19, …, 36}, event H = {20, 24, 28, 32, 36} and event K = {18, 24, 30, 36}. Find (a) P(H), (b) P(H and K). 16. The probability that Kamal passes his English, Mathematics and Science tests are 0.65, 0.82 and 0.75 respectively. Find the probability that Kamal (a) passes his English test only, (b) passes two of the three tests, (c) fails all the three tests. Give your answers correct to 3 decimal places. ©Praxis Publishing_Focus On Maths
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JBRB221243 ISBN 978-981-17293-4-8 FOCUS-ON TEXTBOOK MATHS 9 FOCUS-ON MATHS is a complete mathematics programme specially written in line with the latest Indonesian Mathematics syllabus (Phase D) for Grade 7 to Grade 9 students. The topic coverage in each grade is arranged to address all the learning achievements (Capaian Pembelajaran) as prescribed by the Indonesian Ministry of Education. The series adopts the Singapore Maths method which is a world-class maths teaching approach. This comprehensive series builds on the foundations laid in primary mathematics and prepares learners for embarking on higher-level mathematics. With 21st Century Skills and Higher Order Thinking Skills infused in the contents; this series challenges students with engaging problem-solving tasks in real-word contexts, enabling them to become independent maths learners and build foundations for future success. Focus-on Maths comprises: • Textbook • Workbook • Teacher’s Guide • Teaching Aids ©Praxis Publishing_Focus On Maths