#VLJUBCOIFSIBLLTBLMESWF\":%*/:\":*/-\"3*OBBJUUJSTBZMZBTBOOIÐLÐNMFSJOF HËSFLJUBCOEÐ[FOJ NFUOJ TPSVWFõFLJMMFSJLTNFOEFPMTBIJ¿CJSõFLJMEFBMOQZBZNMBOB- NB[ GPUPLPQJZBEBCBõLBCJSUFLOJLMF¿PóBMUMBNB[ :BZO4PSVNMVTV $BO5&,÷/&- :BZO&EJUÌSÑ %J[HJ–(SBGJL5BTBSN &TSB:·,4&-)BLBO\"ó$\" *4#//P :BZOD4FSUJGJLB/P \"ZEO:BZOMBS%J[HJ#JSJNJ #BTN:FSJ ÷MFUJöJN &SUFN#BTN:BZO-UEõUJr \":%*/:\":*/-\"3* JOGP!BZEJOZBZJOMBSJDPNUS 5FMr 'BLT 0533 051 86 17 aydinyayinlari aydinyayinlari * %¸O¾P.DSDáñ11.SINIF KARMA TEST - 2 0MBTML Karma Testler www.aydinyayinlari.com.tr 7. MODÜL #JS[BSBSLBBSLBZBLF[BUMZPS Modülün sonunda –3 –2 –1 0 1 2 3 11. SINIF #VÑÀBUöUBOJLJTJOJO CJSJOJOHFMNFPMBTMô LBÀUS :VLBSEBLJTBZEPôSVTVOEBOTFÀJMFOUBNTB- \" 1 # 1 $ 1 % 1 & 1 ZOONVUMBLEFôFSDFGBSLOOUFOCÑZÑLPMNB 6 36 72 128 216 PMBTMôLBÀUS Alt bölümlerin \" 1 # 1 $ 2 % 3 & 2 EDĜOñNODUñQñL©HULU 3 2 37 7 *UPSCBEBTBS MBDJWFSU **UPSCBEBTBS MBDJ- tüm alt bölümleri WFSUCJMZFWBSES4F¿JMFOCJSUPSCBEBOCJSCJMZF¿FLJ- MJZPS OLASILIK ¦FLJMFOCJMZFOJOTBSPMNBPMBTMôLBÀUS AB L©HUHQNDUPDWHVWOHU \" 1 # 1 $ 2 % 3 & 4 \"UPSCBTOEBTJZBIWFCFZB[ #UPSCBTOEBTJ- 2 3 34 5 ZBIWFCFZB[UPQWBSES5PSCBMBSOIFSCJSJOEFO õFSUPQ¿FLJMJQEJóFSUPSCBZBBUMZPS ³ Basit Olayların Olasılıkları t 2 )FSCJSUPSCBEBOÀFLJMFOUPQVOBZOSFOLUFPM- \\HUDOñU ³ Koşullu Olasılık t 6 EVôV CJMJOEJôJOF HÌSF ZFS EFôJöUJSNFEFO TPO- 0MBTML <D]ñOñ6RUXODUñ SB\"WF#UPSCBMBSOEBLJUPQMBSOSFOLMFSJOFHÌ- SFEBôMNOEBEFôJöJNPMNBNBPMBTMôLBÀUS ³ Bağımlı ve Bağımsız Olaylar t 12 L[WFFSLFLZBOZBOBTSBMBOEôOEBIFSIBO- \" 2 # 1 $ 1 % 1 & HJJLJFSLFôJOZBOZBOBPMNBNBPMBTMôLBÀUS 3 3 24 \" 1 # 1 $ 1 % 1 & 1 ³ Bileşik Olayların Olasılığı t 18 42 35 28 24 15 ³ Deneysel ve Teorik Olasılık t 25 6ñQñIð©LðĜOH\\LĜ ôFLJMEFLJBóB¿EJZBHSBNOEBCJSQBSBOOBSUBSEB 11. SINIF .0%·- 0-\"4*-*, www.aydinyayinlari.com.tr YAZILI SORULARILF[BUMNBTTPOVDVZB[WFUVSBHFMNFEVSVNMBS WFSJMNJõUJS #JSUPSCBEBCFZB[ NBWJCJMZFWBSES#VUPSCB- #JSNBEFOJQBSB:OOLF[IBWBZBBUMNBTEFOF- EBO¿FLJMFOCJMZFHFSJLPONBNBLÐ[FSFBSUBSEBJLJ : ZJOEFÑTUZÑ[FFTOB[LF[ZB[HFMEJôJCJMJOEJôJ- ³ Karma Testler t 27 ,0õ6--60-\"4*-*, CJMZF¿FLJMJZPS : OFHÌSF LF[ZB[HFMNFPMBTMôLBÀUS ÷MJöLJMJ,B[BONMBS D6 C \"#$%ZBNVL YYYTTTT + YYYYTT +:YYYYYT + YYYYYY A 6! + 6! + 6! T+ 1 ³ Yazılı Soruları t 3011.7.1 : ,PõVMMVPMBTMóB¿LMBZBSBLQSPCMFNMFS¿Ë[FS ¦FLJMFO[ACBJS]JO//D[CJ DCJ]MZFOJO CFZB[ JLJODJ CJMZFOJO 2NXO\\D]ñOñVñQDYODUñQGD NBWJPMNBPMBTMôLBÀUS 3! .3! 4! .2! 5! ©ñNDELOHFHNVRUXODUñL©HULU %XE¸O¾PGHNL¸UQHN TANIM ÖRNEK 3 VRUXODUñQ©¸]¾POHULQH | |AB =DN + 15 + 6 + 1=PMVS DNñOOñWDKWDX\\JXODPDVñQGDQ XODĜDELOLUVLQL] #JSJODJCFZB[ JLJODJNBWJPMEVôVOEBOTSBTCFMMJEJS#V- ³ Yeni Nesil Sorular t 31&ËSOFLV[BZOEB\"WF#JLJPMBZPMTVO#PMB- | |12 OBHÌSBF 4D·C3 ==2DNPMVS ÷TUFOF:OPMBTML= : 6 1 ZOOHFS¿FLMFõNFTJIBMJOEF\"PMBZOOHFS¿FL- 76 7 YYYYYT = PMVS &ÌSOFLV[BZOOJLJPMBZ\"WF#PMTVO 42 7 = T 42 MFõNFPMBTMóOB \"PMBZOO#PMBZOBCBôM P (A) = 1 , P (B) = 1 , P (A , B) = 7 ise \"#$% ZBNVôVOVOJÀ ACÌ=MH{F0T,J1O,E2F, 3T,F4À}JMFLOÐNCFJTSJOJOFMFNBOMBSLVMMBOMB- T OPLUBOO [AB] LFOBSOBSB PLMFBMOEFVF[EBJLMFMCôJMOFDOFL[CUÐDN]пCBTBNBLMTBZMBSCJ- |LPöVMMVPMBTMôEFOJSWF1 \" # JMFHËTUFSJMJS 23 12 : T | P^A + B h |a) P ( A # PMBTMôLBÀUS LFOBSOB PMBO V[BLMôOEBO EBIB LÑÀÑL PMNB T 1 # áPMNBLÐ[FSF 1 \" B = |C P ( A #h PMBTMôLBÀUS SFSLBSUBZB[MQCJSUPSCBZBLPOVZPS PMBTMôLBÀUS P^ B h õFLMJOEFEJS \" 1 # 5 $ 51PSCBE%B O 7SBTUHF&MF T2FÀJMFO CJS TBZ#OVOOBÀHJGÌUSPF M-ZB[UVSBHFMNFPMBTMôLBÀUS |D P ( B A PMBTMôLBÀUS 9 &FõPMVNMVËSOFLV[BZWFT # áJTF 36 E6VôVCJMJO1E2JôJOFHÌS9F SBLBNMBSOO\"G B S1LMPMN#B 1 $ &3Ë SOF%L V[1BZOEB&\" WF5#PMBZMBSWFSJMJZPS | s^A + B h PMBTMôLBÀUS 8 4 8 1 2 P^ h 18WF1 \"b# = 3 1 \"h = , 1 \" # = PMVS a) P(A b B) = P(A) + P(B) - P(A a B) B = s^ B h 7 = 1 + 1 - P^ A + B h 23 4 12 2 3 &WSFOTFMLÑNFÀJGUÑÀCBTBNBLMTBZMBSES C A 4B5 3D = 60 28 |E D PMEVCôVOBHÌSF 1 \" # EFôFSJLBÀUS 1 P^ A + B h = 3 1 = PMVS 1. ? 2. ? 1. ? 2. ? 12 4 0 ÖRNEK 1 ÷TUFOFOSBLBNMBSGBSLMPMBOES 1 11 2 P(A') = ise P(A) = &ËSOFLV[BZOOJLJPMBZ\"WF#PMTVO P^ A \\ B h = P^ A + B h = 4 3 = UÑS 4 22 P^ B h 14 4 3 1 + 3 3 2 = 30 P(A b B) = P(A) + P(B) - P(A a B) P (A›) = 1 ve P (A + B) = 3 C 3 02 53 1 65 4 P(A a B) = - = PMVS AB 30 1 6 4 12 JTUFOFOPMBTML= = PMVS |PMEVôVOBHÌSF P ( B \" PMBTMôLBÀUS 1 60 2 111 P^ A + B h 12 1 <HQL1HVLO6RUXODU 4 4 12 öLJUPSCBEBË[EFõCJM0ZFMBMFTSMWLBSES#JSJODJUPSCBEB P(A\\B) = == P(A'aB) P^ B h 14 P(A) = 1 - P(A') 3 15 P(AaB') $OW%¸O¾P7HVWOHUL <(1m1(6m/6258/$5 P(A) = 1- = P(AaB) Her alt bölümün 66 VRQXQGDRE¸O¾POHLOJLOL 1 WHVWOHU\\HUDOñU TEST - 3 NBWJ CFZB[CJMZF JLJODJUPSCBEBNBWJ CFZB[ 3 ,PöVMMV0MBTML P^ A + B' h 4 3 CJMZFWBSES#JSJODJUPSCBEBOCJSCJMZF¿FLJMJQJLJODJ EJLEËSUHFO 1 ôFLJMEFLJ ZFMEFóJS- P(A \\ B') = == UPSCBZBBUMZPS÷LJODJUPSCBEBOBMOBOCJSCJMZF- IFEFG P^ B + A h 5 18 P^ B' h 28 OJONBWJPMNBPMBTMôOBôBÀEJZBHSBNZÌOUF- ôFLJMEFLJ CJ¿JNJOEFLJ UBIUBT Fõ P (B \\ A) = = = PMVS 2 NFOJOJOLPMMBSGBSL- P^ A h 5 25 3 LBSFMFSEFOPMVõNVõUVS 6 öLJ[BSCJSMJLUFBUMZPS 1EFO ZF LBEBS OVNBSBMBONõ LBSUUBO JLJTJ 6 M SFOLMFSEF CPZBO- P^ B + A h BZ4OBO1EB¿FLJMJZPS Nõ WF LPMMBSB CJSFS 0RG¾O¾QJHQHOLQGH\\RUXP = = PMVS \\DSPDDQDOL]HWPHYE ;BSMBSOÑTUZÑ[FZMFSJOFHFMFOTBZMBDS OU1P #QM=B\"N =- 12 NJJMFCVMVOV[ OVNBSB WFSJMNJõUJS EHFHULOHUL¸O©HQNXUJXOX OOPMEVôVCJMJOEJôJOFHÌSF CVTBZMBSOJLJTJ- P^ A h ,2BSUMBSO Ñ[FSJOEF ZB[M TBZMBSO UPQMBNOO 3Ð[HBS CFMMJ CJS I- 10 m B A OJOEFUFLTBZPMNBPMBTMôLBÀUS ÀJGUTBZPMEVôVCJMJOEJôJOFHÌSF IFSJLJLBSUUBLJ 4 [O BMUOB EÐõUÐ- 9B óÐOEF ZFMEFóJSNF- ÖRNEK 2 \" 2 # 1 $ 2 % Ö3RN EK 4 2 TBZOOÀJGUPMNBPMBTMôLBÀUS 12 m File 12 m & 2B 9 35 #JS5TOGUBLJËó3SFODJMFSJOV\"N B3UFNBUJL# 7GJ[$JL WF1 % 8 & 7 5 5 M :VLBSEB FOJ N WF CPZV N PMBO E#JLVEËISFUHEFFOGUBIUBTOBBSU5BSEBBUöAZ4BQBOCJSB3U- OJEVSNBLUBES)FS 34 9 EVSEVóVOEB LPMMB- #JSÀJGU[BSBUMEôOEB CJSJODJ[BSHFMNJöJTFJLJ[BS- VIFSJLJEFSTUFOEFLBMNõU8S 16 2 17 EBLJÑTUUFLJTBZMBSOUPQMBNOOEFOLÑÀÑLPMNB CJ¿JNJOEFCJSUFOJTLPSUVWFSJMNJõUJS DOOJLJBUöUBEBIFEFGJWVSEVôVCJMJOEJôJOFHÌ- SOEBOCJSJ\"OPLUB- 3BTUHFMFTFÀJMFOCJSÌôSFODJOJOGJ[JLUFOLBMEôCJMJO- 3 EJôJOFHÌSF NBUFNBUJLUFOHFÀNJöPMNBPMBTMôOF- 3 9 B r \" OPLUBTOEBLJ UFOJT¿J LFOEJ BMBOOSFO FBOóBSM[LC JSJOEFUBSBMTCOÌOMHIFJ[ZBJTWVOSBNHVFöMNPFMNLUBFEPJSMB-:FMEFóJSNFOJHÐOJ¿JO- PMBTMôLBÀUS 5M NFSLF[JOEF CVMVOBO # OPLUBTOEBLJTUFMOôJT¿LJBZÀFUS EFEFGBEVSVZPS EJS 6M CJSTFSWJTBUõZBQBDBLUS 9 UÑNEVSVNMBSPMVS÷T- 1 - BUöUBWVSBNBN BPMB#TVMOôB HÌSF \"OPLUBTOEBEVSBOLPMMBSOBSEöL UFOFOEVSVNMBS WF EJS MF 4OGUBLJ ÌôSFODJ TBZT Y 2 5 3 6 28 r #OPLUBTOEBLJUFOJT¿J LFOEJ¿FWSFTJOF16N1W6F 2 2 OVNB4SBM5PMNBPMBTMôLBÀUS ·+·= 2 1 #JS TOGUBLJ ËóSFODJMFSJO GJ[JL V LJNZB PMTVO.WF'NBUFNBUJLWFGJ- NEFOEBIBB[V[BLMLUBEÐõFOUPQ1M-BS24LB·S2õ4- = 1 - 3 · 3 =1- = 9 # 1 $ 2 % 1 & 5 [JLöLUFJO[BLSBCMBJOSMJÌLUôFSFBOUDMJMZFSPJOS LÑ- 5 9 5 9 45 MBZBCJMNFLUFEJS \" 59 6 94 18 #VOBHÌSF PMBTML 6 = 3 WPFMVS JIFSJLJEFSTUFOEFLBMNõUS 20X 10X 30X N;FTBJSPMBMTSVOO ÑTU ZÑ[FZJOF HFMFO TBZMBSO UPQMBN- 36 3O0xO=3PMEPVMVôSV CJMJOEJôJOFHÌSF [BSMBSEBOCJSJOJO 4OGUBOSBTUHFMFTFÀJMFO CJSÌôSFODJOJO LJNZB- 40xHFMN4FPMBTMôLBÀUS #VOB HÌSF TFSWJT BUö ZBQBO \" OPLUBTOEBLJ VRUXODUD\\HUYHULOPLĜWLU 2 EBOLBMEôCJMJOEJôJOFHÌSF CVÌôSFODJOJOGJ[JL- UF1OJTÀJOJO28UPQVLBSöBMBOBEÑöÑSEÑôÑCJMJOEJôJ- 1 1 5 7 UFOEFLBMNöPMNBPMBTMôLBÀUS OF2HÌSF Q45VBOBMNBPMBTMôLBÀUS 74 9 18 11 3 \" 3 # 2 $ 1 % 1 & 1 \" 1- π # 1- 2π $ 1 - π 83 # 1 $ 1 6 % 1 & 1a) 4 C 3 D 1 4 3 3 2 3 4 15 15 5 \" 7 65 4 25 82 4 & 1- π 2 $\\UñFDPRG¾OVRQXQGD % 1- π 3 .JMMJQJZBOHPCJMFUJÐ[FSJOEFLJOVNBSBMBSSBLBNM- Okul WDPDPñ\\HQLQHVLOVRUXODUGDQ ES Ev ROXĜDQWHVWOHUEXOXQXU #JSUPSCBOOJ¿JOEFBZOCÐZÐLMÐLUFTBS LSN[ AB CJMZFWBSES5PSCBEBOBZOBOEBCJMZF¿FLJMJZPS ÷MLSBLBNUVUUVôVCJMJOFOCJSCJMFUJOTPOJLJSB- LBUMCJSBQBSUNBOO\"WF#HJSJõLBQMBSWBSES ¦FLJMFOCJMZFMFSJOBZOSFOLMJPMEVôVCJMJOEJôJOF LBNOOEBUVUNBPMBTMôLBÀUS \"LBQTOEBOHJSEJLUFOTPOSBBTBOTËSUFLOVNBSB- HÌSF JLJTJOJOEFTBSPMNBPMBTMôLBÀUS MLBUMBSB #LBQTOEBOHJSEJLUFOTPOSBBTBOTËS¿JGU \" 1 # 1 $ 1 % 1 & 1 OVNBSBMLBUMBSBHJUNFLUFEJS \" 7 # 3 $ 5 % 1 & 3 3 6 10 90 100 8 4 8 2 8 r \"WF#LBQMBSOEBHJSJõõJGSFTJUÐS r )FSJLJLBQEBLJUVõMBSOUVõVCP[VLUVSWF 1 .VSBUZPMVOCBõOEBLJFWJOEFO PLZËOÐOEF HF¿UJ- óJZPMEBOUFLSBSHF¿NFNFLÐ[FSFZPMB¿LZPS.V- 3 PMBTMLMBUVõVOVBMHMBZQ 2 PMBTMLMBUVõV ö¿JOEFBZOCÐZÐLMÐLUFNBWJ CFZB[WFTJZBI LJõJMJLCJSTOGUBGVUCPMPZOBZBOMBSLJõJ WPMFZ- SBUhO TPMVOB EËONF PMBTMó 1 TBóOB EËONF UPQCVMVOBOCJSUPSCBEBOSBTUHFMFпUPQ¿FLJMJZPS CPMPZOBZBOMBSLJõJWFIFSJLJTJOJEFPZOBZBOMBS 3 3 LJõJEJS BMHMBNBNBLUBES \"ZSDB \" LBQTOEBO HJSJMEJ- PMBTMó 1 WFEÐ[HJUNFPMBTMóEB 1 ES óJOEFLJ BTBOTËSÐO EF UVõV BZO LVSBMB HËSF 26 CP[VLUVS r \"LBQTOEBOHJSFO\"INFU#FZ LBUUB #LBQ- TOEBOHJSFO0SIBO#FZEFLBUUBPUVSVZPSMBS #V ÑÀ UPQUBO ZBMO[ CJSJOJO NBWJ PMEVôV CJMJO- #V TOGUBO SBTUHFMF TFÀJMFO CJS LJöJOJO GVUCPM #VOB HÌSF .VSBUhO PLVMB HJUNF PMBTMô LBÀ- ÷LJTJOJOEFJMLEFOFNFEFEBJSFMFSJOFVMBöNBPMB- EJôJOF HÌSF IFS SFOLUFO CJS UPQ ÀFLJMNJö PMNB TMôLBÀUS PMBTMôLBÀUS PZOBEô CJMJOEJôJOF HÌSF CV LJöJOJO WPMFZCPM US EBPZOBNBPMBTMôLBÀUS \" 1 # 1 $ 1 % 1 & 1 \" 1 # 1 $ 1 % 1 & 1 36 18 12 6 3 3 9 27 81 243 \" 5 B 3 $ 5 % 7 & 5 \" 2 # 1 $ 2 % 1 & 2 18 11 9 3 6 11 6 5 2 3 B B 31 E C C C E C 9 B D E E
www.aydinyayinlari.com.tr 11. SINIF 11. SINIF 7. MODÜL OLASILIK ³ Basit Olayların Olasılıkları t 2 ³ Koşullu Olasılık t 6 ³ Bağımlı ve Bağımsız Olaylar t 12 ³ Bileşik Olayların Olasılığı t 18 ³ Deneysel ve Teorik Olasılık t 25 ³ Karma Testler t 27 ³ Yazılı Soruları t 30 ³ Yeni Nesil Sorular t 31 1. ? 2. ? 1 1. ? 2. ?
KONUYA HAZIRLIK #VTBZGBEBLJJÀFSJLi0MBTMLuLPOVTVZMBJMHJMJTOGUBHÌSEÑôÑOÑ[LPOVMBSOUFLSBSBNBDZMBIB[SMBONöUS #\"4÷50-\":-\"3*/0-\"4*-*,-\"3* ÷MJöLJMJ,B[BONMBS 10.1.2.1 : ±SOFLV[BZ EFOFZ ¿LU CJSPMBZOUÐNMFZFOJ LFTJOPMBZ JNLºOT[PMBZ BZSLPMBZWFBZSLPMNBZBOPMBZLBW- SBNMBSOB¿LMBS 10.1.2.2 : 0MBTMLLBWSBNJMFJMHJMJVZHVMBNBMBSZBQBS 7$1,0%m/*m ÖRNEK 3 &= {x : 5 # x # 30 , x ` Z}PMNBLÐ[FSF #JSEFOFZEFFMEFFEJMFOTPOV¿MBSOIFSCJSJOFP EFOFZF BJU ¿LU EFOJS #JS EFOFZJO CÐUÐO ¿LU- &LÑNFTJOEFOSBTUHFMFTFÀJMFDFLCJSTBZOOBTBMTB- MBSOOLÐNFTJOFPEFOFZJOÌSOFLV[BZEFOJS ZWFZBÀJGUTBZPMNBPMBZOOFMFNBOTBZTLBÀUS ±SOFLV[BZHFOFMMJLMF&JMFHËTUFSJMJS 0MBZMBSBZSLPMBZMBSES ÖRNEK 1 8 + 13 =EJS [[ #JS NBEFOÆ QBSBOO BSLB BSLBZB JLJ LF[ BUMNBT EF- \"TBMTBZÀJGUTBZ OFZJOEFÀLUMBSWFÌSOFLV[BZCVMVOV[ ¦Ì[ÑN T 7$1,0%m/*m Y (Y, T), (Y, Y) \"PMBZ &FõPMBTMËSOFLV[BZOEBCJSPMBZPM- Y NBL Ð[FSF \" PMBZOO PMNB PMBTMó 1 \" PM- NBLÐ[FSF T P^ A h = s^ A h EJS T (T, T), (T, Y) s^ E h Y 0 #1 \" # 1 1 \" +1 \"h = 1 #VEFOFZJO¿LUMBS : 5 : : 5 5 5 : ±SOFLV[BZ, A a B = qJTF1 \"b# =1 \" +1 # A a#áqJTF &= { : 5 : : 5 5 5 : }EJS 1 \"b# =1 \" +1 # -1 \"a # ÖRNEK 2 ÖRNEK 4 #JS[BSWFCJSQBSBOOCJSMJLUFBUMNBTEFOFZJOJOÌS- LJõJMJLCJSTOGOÐFSLFLUJS&SLFLMFSJOTWF OFLV[BZOCVMVOV[ L[MBSOTJNBUFNBUJLEFSTJOEFOCBõBSMES #V TOGUBO SBTUHFMF TFÀJMFO CJS ÌôSFODJOJO L[ WFZB 123456 NBUFNBUJLUFOCBöBST[PMNBTPMBTMôLBÀUS Y (Y, 1) (Y, 2) (Y, 3) (Y, 4) (Y, 5) (Y, 6) T (T, 1) (T, 2) (T, 3) (T, 4) (T, 5) (T, 6) TANIM Erkek #BöBSM #BöBST[ ,[ 16 8 #JSËSOFLV[BZOJLJBZSPMBZOOLFTJõJNJCPõ 12 6 LÐNFJTFCVJLJPMBZBBZSLPMBZEFOJS 12 + 6 + 8 26 13 \"JMF#BZSLPMBZMBSJTF == A a B = qEJS 42 42 21 E = {(Y, 1), (Y, 2), (Y, 3), (Y, 4), (Y, 5), (Y, 6), (T, 1), (T, 2), 2 13 (T, 3), (T, 4), (T, 5), (T, 6)} 21 21
KONUYA HAZIRLIK #VTBZGBEBLJJÀFSJLi0MBTMLuLPOVTVZMBJMHJMJTOGUBHÌSEÑôÑOÑ[LPOVMBSOUFLSBSBNBDZMBIB[SMBONöUS ÖRNEK 5 ÖRNEK 7 #JS&ÌSOFLV[BZO\"WF#PMBZMBSJÀJO #JSBUZBSõOEB WFZBõMBSOEBпBUWBSESZB- õOEBLJBUOZBSõLB[BONBPMBTMóZBõOEBLJBUOLB- 1 \"h = 5 , P (A , B) = 1 ve P (A + B) = 1 [BONB PMBTMóOO ZBSTOB ZBõOEBLJ BUO LB[BONB 62 12 PMBTMóOOJTFпLBUOBFõJUUJS PMEVôVOBHÌSF #PMBZOOPMBTMôLBÀUS #VZBSöZBöOEBLJBUOLB[BONBPMBTMôLBÀUS P(A b B) = P(A) + P(B) – P(A a B) 2 3 4 0MBTML Y Y Y 1 = 1 + P^ B h- 1 Y+Y+Y= Y= 1 26 12 1 P^ B h = 5 x= 12 10 ÖRNEK 6 ÖRNEK 8 #JSBQBSUNBOOHJSJõJOF WFSBLBNMBSOEBO ,\"-&.-ö,LFMJNFTJOJOIBSGMFSJJMFZB[MBCJMFDFLIBSG- PMVõBOCJSõJGSFEÐ[FOFóJLVSVMVZPS MJBOMBNMWFZBBOMBNT[LFMJNFMFSCJSFSLºóEBZB[MBSBL CJSLVUVZBLPOVZPS #V LVUVEBO CJS L»ôU ÀFLJMEJôJOEF , OJO IFNFO TB- ôOEBLJ IBSGJO - WF IFNFO TPMVOEBLJ IBSGJO \" PMNB PMBTMôLBÀUS 8! FWSFOTFMLÑNFPMTVO 2! .2! AKL &.-÷,jJTUFOFOduruNPMVS 6! 6! 4 1 0MBTML= = = = PMVS 8! 8.7.6! 8.7 14 2! .2! 2.2 r )FSUVõBZBMO[CJSTFGFSCBTMBCJMJS ÖRNEK 9 B ,ËõFMFSJ õFLJMEFLJ ¿FN- CFS Ð[FSJOEF CVMVOBO r ôJGSFIBOFMJCFMJSMFONJõPMVQ CVпSBLBNOUPQ- A MBN¿JGUJTFLBQB¿MS C OPLUBEBO пРPMBO CJS H пHFOSBTUHFMFTF¿JMJZPS r ôJGSFZBOMõHJSJMJSTFBMBSN¿BMBS #VOBHÌSF LBQZBÀNBZBÀBMöBOCJSLJöJOJOBMBSN G #V ÑÀHFOJO CJS LÌöF- ÀBMESNBPMBTMôZÑ[EFLBÀUS F TJOJO&WFZB)OPLUBT D PMNBPMBTMôLBÀUS 55:WFZB¦¦5PMVSTBBMBSNÀBMBS E TTT + Ç ÇT 3 33 5ÑN PMBTML EVSVNMBSOEBO & WF ) OO PMNBELMBSO f p + f pf p ÀLBSBMN 3 21 10 1 6 = == fp 6 6 20 2 fp fp 3 59 33 1 - = 1 - = PMVS 8 14 14 fp 3 5 11 9 3 10 14 14 12
TEST - 1 #BTJU0MBZMBSO0MBTMLMBS #VTBZGBEBLJJÀFSJLi0MBTMLuLPOVTVZMBJMHJMJTOGUBHÌSEÑôÑOÑ[LPOVMBSOUFLSBSBNBDZMBIB[SMBONöUS \"WF#BZOËSOFLV[BZEBJLJPMBZPMNBLÐ[FSF L[ FSLFLËóSFODJOÐGVTDÐ[EBOMBSOCJSUPSCB- P^ A , B h = 23 , 1 \"h = 1 ZBLPZVZPSMBS 30 3 5PSCBEBOSBTUHFMFOÑGVTDÑ[EBOÀFLJMEJôJOEF WF1 \"a# h = 7 TJOJOFSLFLÌôSFODJZF JOJOL[ÌôSFODJZFBJU 10 OÑGVTDÑ[EBOPMNBPMBTMôLBÀUS PMEVôVOBHÌSF 1 # LBÀUS \" 5 # 5 $ 1 % 15 & 4 \" 2 # 3 $ 1 % 3 & 4 112 56 2 28 7 5 7 2 5 5 #JSCJSJZMFBSLBEBõMLZBQBOBSLBEBõOTJBWVLBU- US#VLJõJCJSTSBEBPUVSBDBLMBSES #JSNBEFOÅQBSBBSUBSEBLF[IBWBZBBUMZPS \"WVLBUPMBOMBSOZBOZBOBPUVSNBPMBTMôLBÀ- US #VQBSBOOLF[ZB[HFMNFPMBTMôLBÀUS \" 5 # 3 $ 1 % 1 & 1 \" 1 # 2 $ 1 $ 2 & 1 8 8 3 4 8 6 7 43 2 LJõJMJLCJSTOGOJFSLFLUJS&SLFLMFSJOTJL[- 0óV[hVO CJS BUõUB IFEFGJ WVSNB PMBTMó 1 :B- MBSOJNÐ[JLEFSTJOJTF¿NJõUJS 3 WV[hVOJTF 1 UÐSöLJTJEFIFEFGFCJSFSBUõZBQZPS- 4 MBS 4OGUBO SBTUHFMF TFÀJMFO CJS ÌôSFODJOJO L[ WF- #VOB HÌSF BUöMBS TPOVDVOEB TBEFDF CJSJOJO ZBNÑ[JLEFSTJOJTFÀFOCJSÌôSFODJPMNBPMBTM- IFEFGJWVSNVöPMNBPMBTMôLBÀUS ôLBÀUS \" 27 # 11 $ 1 % 7 & 1 \" 1 # 1 $ 1 % 5 & 1 12 4 3 12 2 40 20 2 40 8 L[WFFSLFóJOCVMVOEVóVCJSHSVQUBOLJõJMJLCJS SBLBNMBSJMFZB[MBCJMFDFLUÐNCB- PZVOFLJCJTF¿JMFDFLUJS TBNBLMTBZMBSCJSFSLBSUBZB[MBSBLCJSUPSCBZBLP- OVZPS #VFLJQUFFOB[L[CVMVONBPMBTMôLBÀUS 5PSCBEBO SBTUHFMF ÀFLJMFO CJS LBSUUBLJ TBZOO SBLBNMBSGBSLMWFJMFCÌMÑOFCJMFOCJSTBZPM- NBPMBTMôLBÀUS \" 9 # 3 $ 1 % 1 & 1 \" 1 # 7 $ 3 % 1 & 9 10 4 25 10 25 100 16 5 25 A B B A 4 D B D D
#BTJU0MBZMBSO0MBTMLMBS TEST - 2 #VTBZGBEBLJJÀFSJLi0MBTMLuLPOVTVZMBJMHJMJTOGUBHÌSEÑôÑOÑ[LPOVMBSOUFLSBSBNBDZMBIB[SMBONöUS GBSLM SFOLUF õJõFZF BJU LBQBLMBS JMF CV õJõFMFS ö¿JOEFBZOCÐZÐLMÐLUFCFZB[ TBSCJMZFCVMV- SBTUHFMFLBQBUMZPS OBO CJS UPSCBEBO SBTUHFMF CJS CJMZF ¿FLJMJQ J¿JOEF CFZB[ WF TBS CJMZF CVMVOBO JLJODJ CJS UPSCBZB 5ÑNöJöFMFSJOLBQBLMBSOOLFOEJSFOHJOEFPMNB BUMZPS PMBTMôLBÀUS #VOBHÌSF JLJODJUPSCBEBOSBTUHFMFTFÀJMFOCJS CJMZFOJOTBSPMNBPMBTMôLBÀUS \" 1 # 1 $ 1 % 1 & 1 120 100 80 35 5 \" 2 # 3 $ 7 % 3 & 1 3 5 15 7 5 öLJ[BSCJSMJLUFBUMZPS ;BSMBSOÑTUZÑ[FZJOFHFMFOTBZMBSOUPQMBNOO Y Z [EFOTBEFDFCJSJOJOLB[BOBDBóZBSõUB ZBS- PMNBPMBTMôLBÀUS õYJOLB[BONBPMBTMó ZOJOLBUES \" 1 # 5 $ 2 % 5 & 1 [OJOZBSöLB[BONBPMBTMôEBYJOLBUPMEV- 3 18 9 36 9 ôVOBHÌSF YJOZBSöLB[BONBPMBTMôLBÀUS \" 2 B 8 $ 16 % 16 & 16 9 81 27 81 243 %Ð[MFNEFIFSIBOHJÐEPóSVTBMPMNBZBO\" # $ % &OPLUBMBSWFSJMJZPS ,ÌöFMFSJCVOPLUBMBSPMNBLÑ[FSF ÀJ[JMFCJMFDFL \" #WF$BUõQPMJHPOMBSOEBõFLJMEFLJHJCJOVNBSB- ÀPLHFOMFSEFO SBTUHFMF CJSJ TFÀJMEJôJOEF CVOVO CJSÑÀHFOPMNBPMBTMôLBÀUS MBOESMNõCFõIFEFGUBIUBTWBSES \" 5 # 1 $ 3 % 1 & 1 12 3 8 2 84 8 ôFLJMEFLJ\"#$%EJLEËSUHFOJCJSJNLBSFEFOPMVõ- A B 4 5 NBLUBES DC C AB öLJPL¿VCJSFSBUõZBQQIFEFGMFSEFOCJSJOJWVSVZPSMBS õFLJMEFLJ EÌSUHFOMFSEFO SBTUHFMF TFÀJMFO CJS #VOBHÌSF IFSIBOHJCJSQPMJHPOEBLJUÑNIFEFG- EÌSUHFOJOLBSFPMNBPMBTMôLBÀUS MFSJOWVSVMNBPMBTMôLBÀUS \" 1 # 1 $ 2 % 5 & 13 \" 1 # 13 $ 3 % 21 & 5 3 5 12 30 5 25 5 25 A E A B 5 C A B
11. SINIF .0%·- 0-\"4*-*, www.aydinyayinlari.com.tr ,0õ6--60-\"4*-*, ÷MJöLJMJ,B[BONMBS 11.7.1 : ,PõVMMVPMBTMóB¿LMBZBSBLQSPCMFNMFS¿Ë[FS TANIM ÖRNEK 3 &ËSOFLV[BZOEB\"WF#JLJPMBZPMTVO#PMB- &ÌSOFLV[BZOOJLJPMBZ\"WF#PMTVO ZOOHFS¿FLMFõNFTJIBMJOEF\"PMBZOOHFS¿FL- P (A) = 1 , P (B) = 1 , P (A , B) = 7 MFõNFPMBTMóOB \"PMBZOO#PMBZOBCBôM ise |LPöVMMVPMBTMôEFOJSWF1 \" # JMFHËTUFSJMJS 23 12 | P^A + B h |a) P ( A # PMBTMôLBÀUS 1 # áPMNBLÐ[FSF 1 \" B = P^ B h |C P ( A #h PMBTMôLBÀUS õFLMJOEFEJS |D P ( B A PMBTMôLBÀUS &FõPMVNMVËSOFLV[BZWFT # áJTF | s^A + B h a) P(A b B) = P(A) + P(B) - P(A a B) 1 \" # = s^ B h PMVS 7 = 1 + 1 - P^ A + B h 12 2 3 P^ A + B h = 3 1 = PMVS 12 4 ÖRNEK 1 1 &ËSOFLV[BZOOJLJPMBZ\"WF#PMTVO P^ A |B h = P^ A + B h = 4 3 P (A›) = 1 ve P (A + B) = 3 = UÑS 65 P^ B h 14 |PMEVôVOBHÌSF P ( B \" PMBTMôLBÀUS C 3 AB 111 P(A'aB) 4 4 12 P(A) = 1 - P(A') P(AaB') 15 P(AaB) P(A) = 1- = 1 66 P^ A + B' h 4 3 3 P(A | B') = == P^ B' h 28 P^ B + A h 5 18 P (B | A) = = = PMVS 3 P^ A h 5 25 1 6 P^ B + A h 4 1 D 1 #]\" = = = PMVS P^ A h 12 2 ÖRNEK 2 ÖRNEK 4 #JSÀJGU[BSBUMEôOEB CJSJODJ[BSHFMNJöJTFJLJ[BS- #JSTOGUBLJËóSFODJMFSJOVNBUFNBUJL GJ[JLWF EBLJÑTUUFLJTBZMBSOUPQMBNOOEFOLÑÀÑLPMNB VIFSJLJEFSTUFOEFLBMNõUS PMBTMôLBÀUS 3BTUHFMF TFÀJMFO CJS ÌôSFODJOJO GJ[JLUFO LBMEô CJ- UÑNEVSVNMBSPMVS÷T- MJOEJôJOF HÌSF NBUFNBUJLUFO HFÀNJö PMNB PMBTMô UFOFOEVSVNMBS WF EJS LBÀUS 21 MF 4OGUBLJ ÌôSFODJ TBZT Y #VOBHÌSF PMBTML = PMVS 20X 10X 30X PMTVO.WF'NBUFNBUJLWFGJ- [JLUFO LBMBO ÌôSFODJMFSJO LÑ- 63 NFTJPMTVO 30x 3 = PMVS 40x 4 18 1 6 331 3 a) C D 482 4 25 3
www.aydinyayinlari.com.tr 0-\"4*-*, .0%·- 11. SINIF ÖRNEK 5 ÖRNEK 8 A = {B C D E F}LÐNFTJOJOBMULÐNFMFSJCJSFSLBSUBZB- \"INFUQBSBõÐUJMF\"#$%CËMHFTJOFBUMBZõZBQZPS [MQ CJSLVUVZBLPOVMVZPS#VLVUVEBOSBTUH FMFCJSLBSU A EB ¿FLJMJZPS KL ¦FLJMFO LBSUUBLJ LÑNFOJO FMFNBOM CJS LÑNF PMEV- ôVCJMJOEJôJOFHÌSF CVLÑNFEFBOOCVMVONBPMB- D MC TMôLBÀUS %&$FõLFOBSпHFOCËMHFTJZFõJMBMBOWFпHFOF, - 5 .OPLUBMBSOEBUFóFUPMBO¿FNCFSJTFJOJõQJTUJEJS FMFNBOMBMULÑNFTBZTf p = 10 PMVSBOOCVMVOEVôV \"INFUhJOZFöJMBMBOBJOEJôJCJMJOEJôJOFHÌSF JOJöQJT- UJOFJONFPMBTMôLBÀUS 3 FMFNBOMBMULÑNFTBZT {a, –, –} 4 63 f p = 6 PMVS÷TUFOFOPMBTML = PMVS 2 10 5 ÖRNEK 6 E r 3 30° 30° { } LÐNFTJOJO FMFNBOMBSOEBO SBTUHF- 60° MFJLJTJTF¿JMJZPS rM 4FÀJMFO TBZMBSO UPQMBNOO ÀJGU PMEVôV CJMJOEJôJOF HÌSF IFSJLJTJOJOEFUFLTBZPMNBPMBTMôLBÀUS r3 DC Dairenin Alanı 22 π .r π .r 4 == 4FÀJMFOTBZMBSOJLJTJEFUFLJTFf p = 6 GBSLMEVSVNPMVS 2 EDC Ü çgeninin Alanı ^ 2r 3 h 3 2 3 3 .r 3 4 4FÀJMFOTBZMBSOJLJTJEFÀJGUJTF f p = 3 GBSLMEVSVNPMVS π = PMVS 2 33 6 62 ÷TUFOFOPMBTML = = PMVS 6+3 9 3 ÖRNEK 7 ÖRNEK 9 #JS UBWMB [BS BUMEôOEB HFMFO TBZOO UFL TBZ PM \"UPSCBTOEBTBS MBDJWFSU #UPSCBTOEBTBS MB- EVôVCJMJOJZPSTB CVTBZOOBTBMTBZPMNBNBT PMB- DJWFSU UPQ WBSES 3BTUHFMF TF¿JMFO CJS UPSCBEBO CJS UPQ TMôLBÀUS ¿FLJMJZPS 1 ¦FLJMFOUPQVOTBSPMEVôVCJMJOEJôJOFHÌSF#UPSCB- 5FLTBZMBS BTBMPMNBZBOUFLTBZEJS0MBTML TOEBOÀFLJMNJöPMNBPMBTMôLBÀUS 3 3 PMVS #UPSCBTOEBOTBSUPQÀFLNFPMBTMô WF\"UPSCBTO- 10 4 EBOTBSUPQÀFLNFPMBTMô EVS÷TUFOFOPMBTMk 9 3 B den sarı top = 10 A dan sarı top + B den sarı top 4+ 3 9 10 3 10 27 = = PMVS 67 67 90 3 2 1 7 π 27 5 3 3 33 67
11. SINIF .0%·- 0-\"4*-*, www.aydinyayinlari.com.tr ÖRNEK 10 ÖRNEK 13 \"UPSCBTOEBCFZB[ NBWJ #UPSCBTOEBCFZB[ #JSÀJGU[BSIBWBZBBUMEôOEBÑTUZÑ[FHFMFOTBZMB- NBWJUPQWBSES SOUPQMBNOOFOB[PMEVôVCJMJOJZPSTB CVJLJTBZ- OOBZOPMNBPMBTMôLBÀUS 5PSCBMBSEBOIFSIBOHJCJSJOEFOSBTUHFMFBMOBOJLJUP- QVOJLJTJOJOEFNBWJPMEVôVCJMJOEJôJOFHÌSF\"UPSCB- (1, 6) (6, 1) (4, 4) TOEBOBMONöPMNBPMBTMôLBÀUS (2, 5) (5, 2) (4, 5) (5, 4) (2, 6) (6, 2) (4, 6) (6, 4) 2 (3, 4) (4, 3) (5, 5) fp (3, 5) (5, 3) (5, 6) (6, 5) 21 UÑNEVSVNMBS \"EBONBWJÀFLNFPMBTMô = 46 31 fp ÷TUFOFOPMBTML= = PMVS 2 21 7 #EFONBWJÀFLNFPMBTMô 4 fp 2 62 = = PMVS 7 21 7 fp 2 11 6 67 ÷TUFOFOPMBTML= = = PMVS 1+2 19 19 6 7 42 ÖRNEK 14 ÖRNEK 11 öLJ UPSCBEBO CJSJODJTJOEF NBWJ ZFõJM JLJODJTJOEF NBWJ ZFõJM CJMZF WBSES 5PSCBMBSEBO CJSJ SBTUHFMF BM- FSLFLWFL[EBO FSLFLWFL[OHË[MFSJNB- OQ J¿JOEFOCJSCJMZF¿FLJMJZPS WJEJS ¦FLJMFO CJMZFOJO NBWJ PMEVôV CJMJOJZPSTB JLJODJ UPS- #VHSVQUBOSBTUHFMFTFÀJMFOCJSJFSLFLJTFNBWJHÌ[- CBEBOÀFLJMNJöPMNBPMBTMôLBÀUS MÑPMNBPMBTMôLBÀUS 4 II. torbadan mavi 10 14 == I. torbadan mavi + II. torbadan mavi 54 39 + 7 10 Mavi gözlü erkek sayısı 3 = PMVS Erkek sayısı 10 ÖRNEK 12 ÖRNEK 15 #JSNBEFOJQBSBOOLF[IBWBZBBUMEôCJSEFOFZ- .\"/5*, TÌ[DÑôÑOÑO IBSGMFSJZMF PMVöUVSVMBCJMFDFL EFÑTUZÑ[FFOB[LF[UVSBHFMEJôJCJMJOEJôJOFHÌSF BOMBNM ZB EB BOMBNT[ TÌ[DÑLMFSEFO SBTUHFMF TFÀJ- LF[UVSBHFMNFPMBTMôLBÀUS MFOCJSJOJO\"IBSGJJMFCBöMBEôCJMJOJZPSTB 5IBSGJJMF CJUNFPMBTMôLBÀUS TTTTY T T T YY + T T T T Y + T T T T T 5! &WSFOTFMLÑNF A = 5! T = 4! 4! 5 5 ÷TUFOFOEVSVN A = = = PMVS 4! 1 5! 5! 10 + 5 + 1 16 + +1 0MBTML= = 3! .2! 4! 5! 5 7 3 5 8 1 14 1 19 10 16 7 39 5
,PöVMMV0MBTML TEST - 3 öLJ[BSCJSMJLUFBUMZPS EFO ZF LBEBS OVNBSBMBONõ LBSUUBO JLJTJ ;BSMBSOÑTUZÑ[FZMFSJOFHFMFOTBZMBSOUPQMBN- BZOBOEB¿FLJMJZPS OOPMEVôVCJMJOEJôJOFHÌSF CVTBZMBSOJLJTJ- OJOEFUFLTBZPMNBPMBTMôLBÀUS ,BSUMBSO Ñ[FSJOEF ZB[M TBZMBSO UPQMBNOO ÀJGUTBZPMEVôVCJMJOEJôJOFHÌSF IFSJLJLBSUUBLJ \" 2 # 1 $ 2 % 3 & 2 TBZOOÀJGUPMNBPMBTMôLBÀUS 9 3 55 3 \" 3 # 7 $ 1 % 8 & 7 8 16 2 17 34 #JS TOGUBLJ ËóSFODJMFSJO GJ[JL V LJNZB öLJ[BSCJSMJLUFBUMZPS WFJIFSJLJEFSTUFOEFLBMNõUS 4OGUBOSBTUHFMFTFÀJMFO CJSÌôSFODJOJOLJNZB- ;BSMBSO ÑTU ZÑ[FZJOF HFMFO TBZMBSO UPQMBN- EBOLBMEôCJMJOEJôJOFHÌSF CVÌôSFODJOJOGJ[JL- UFOEFLBMNöPMNBPMBTMôLBÀUS OOPMEVôVCJMJOEJôJOFHÌSF [BSMBSEBOCJSJOJO HFMNFPMBTMôLBÀUS \" 1 # 1 $ 1 % 1 & 1 \" 3 # 2 $ 1 % 1 & 1 8 7 65 4 4 3 23 4 #JSUPSCBOOJ¿JOEFBZOCÐZÐLMÐLUFTBS LSN[ .JMMJQJZBOHPCJMFUJÐ[FSJOEFLJOVNBSBMBSSBLBNM- CJMZFWBSES5PSCBEBOBZOBOEBCJMZF¿FLJMJZPS ES ÷MLSBLBNUVUUVôVCJMJOFOCJSCJMFUJOTPOJLJSB- ¦FLJMFOCJMZFMFSJOBZOSFOLMJPMEVôVCJMJOEJôJOF LBNOOEBUVUNBPMBTMôLBÀUS HÌSF JLJTJOJOEFTBSPMNBPMBTMôLBÀUS \" 1 # 1 $ 1 % 1 & 1 \" 7 # 3 $ 5 % 1 & 3 3 6 10 90 100 8 4 8 2 8 ö¿JOEFBZOCÐZÐLMÐLUFNBWJ CFZB[WFTJZBI LJõJMJLCJSTOGUBGVUCPMPZOBZBOMBSLJõJ WPMFZ- UPQCVMVOBOCJSUPSCBEBOSBTUHFMFпUPQ¿FLJMJZPS CPMPZOBZBOMBSLJõJWFIFSJLJTJOJEFPZOBZBOMBS LJõJEJS #V ÑÀ UPQUBO ZBMO[ CJSJOJO NBWJ PMEVôV CJMJO- #V TOGUBO SBTUHFMF TFÀJMFO CJS LJöJOJO GVUCPM EJôJOF HÌSF IFS SFOLUFO CJS UPQ ÀFLJMNJö PMNB PMBTMôLBÀUS PZOBEô CJMJOEJôJOF HÌSF CV LJöJOJO WPMFZCPM EBPZOBNBPMBTMôLBÀUS \" 5 B 3 $ 5 % 7 & 5 \" 2 # 1 $ 2 % 1 & 2 18 11 9 3 6 11 6 52 3 C C E C 9 B D E E
TEST - 4 ,PöVMMV0MBTML &ËSOFLV[BZOEBJLJPMBZ\"WF#EJS öLJ[BSBZOBOEBBUMZPS 1 \"h = 2 WF1 \"a# = 1 ·TUZÑ[FHFMFOTBZMBSOUPQMBNOOBTBMTBZPM- 34 EVôVCJMJOEJôJOFHÌSF CVUPQMBNOEFOCÑZÑL PMNBPMBTMôLBÀUS |PMEVôVOBHÌSF 1 # \" LBÀUS \" 1 # 1 $ 3 % 4 & 6 \" 1 # 2 $ 1 % 1 & 5 4 3 45 7 7 15 5 3 7 &ËSOFLV[BZOEBJLJPMBZ\"WF#EJS P^ A h = 1 , P^ B h = 3 WFP^ A , B h = 5 %ËSUNBEFOJQBSBBZOBOEBBUMZPS 24 6 &OB[JLJTJOJOUVSBHFMEJôJCJMJOEJôJOFHÌSF ÑÀÑ- |PMEVôVOBHÌSF 1 \" # LBÀUS OÑOUVSB CJSJOJOZB[HFMNFPMBTMôLBÀUS \" 1 # 2 $ 3 % 4 & 5 \" 1 # 4 $ 1 % 2 & 3 4 5 45 9 11 11 2 3 4 SBLBNMBSOO UÐNÐ LVMMBOMBSBL BMU )FSLFTJO FO B[ CJS EJM CJMEJóJ CJS UVSJTU LBGJMFTJOJO CBTBNBLMTBZMBSZB[MZPS 'SBOT[DB JöOHJMJ[DFCJMNFLUFEJS #VTBZMBSOJÀJOEFOTFÀJMFOCJSTBZOO JMFCÌ- MÑOEÑôÑCJMJOEJôJOFHÌSF CVTBZOOJMFCBöMB- #V LBGJMFEFO SBTUHFMF TFÀJMFO CJS LJöJOJO 'SBO- T[DBCJMEJôJCJMJOEJôJOFHÌSF ÷OHJMJ[DFCJMNFNF NBPMBTMôLBÀUS PMBTMôLBÀUS \" 1 # 1 $ 2 % 5 & 11 \" 1 # 2 $ 1 % 3 & 2 3 2 36 12 4 5 25 3 A = {-3, -2, -1, 0, 1, 2, 3} LÐNFTJOJOFMFNBOMBSOEBOSBTUHFMFJLJUBOFTJTF¿JMJ- .\"53ö4LFMJNFTJOJOIBSGMFSJZFSEFóJõUJSJMFSFLZB[- ZPS MBCJMFOBMUIBSGMJLFMJNFMFSEFOCJSJSBTUHFMFTF¿JMJZPS 4FÀJMFO JLJ FMFNBOO ÀBSQNOO OFHBUJG PMEV- #VLFMJNFOJO5IBSGJJMFCBöMBEôCJMJOEJôJOFHÌ- ôVCJMJOEJôJOFHÌSF ÀBSQNMBSOOÀJGUTBZPMNB PMBTMôLBÀUS SF TFTMJIBSGMFSJOZBOZBOBHFMNFEJôJCJSLFMJNF PMNBPMBTMôLBÀUS \" 1 # 4 $ 5 % 2 & 7 \" 1 # 1 $ 2 % 3 & 4 3 9 93 9 10 5 55 5 C E A C B B C D
,PöVMMV0MBTML TEST - 5 öLJUPSCBEBOCJSJODJTJOEFTBS ZFõJM JLJODJTJOEF ôFLJMEFLJPUPQBSLUBCJSCJSJOFFõJUV[BLMLUBпBSBCB TBS ZFõJMCJMZFWBSES5PSCBMBSOIFSIBOHJCJSJO- QBSLFUNJõUJS EFOSBTUHFMFCJSCJMZF¿FLJMJZPS ¦FLJMFO CJMZFOJO TBS PMEVôV CJMJOEJôJOF HÌSF 22 JLJODJUPSCBEBOÀFLJMNJöPMNBPMBTMôLBÀUS 44 \" 1 # 1 $ 5 % 3 & 7 4 2 8 4 8 22 24 2 A = { } \"SBCBMBSBNFUSFWFZBNFUSFEFOEBIBZBLOCJS DJTJNZBLMBõUóOEBBSBCBMBSOBMBSNMBS¿BMZPS LÐNFTJOJOUÐNBMULÐNFMFSJCJSFSLºóEBZB[MQCJS 0UPQBSLB HJSFO CJS LFEJOJO öFLJMEFLJ ÑÀHFOTFM UPSCBZBBUMZPS CÌMHFEF HF[EJôJ CJMJOEJôJOF HÌSF BSBCBMBSO BMBSNMBSOOIFSIBOHJCJSJTJOJÀBMESNBPMBTMô 5PSCBEBOCJSL»ôUTFÀJMEJôJOEFCVL»ôUUBFMF- LBÀUS NBOM CJS LÑNFOJO PMEVôV CJMJOEJôJOF HÌSF CV LÑNFEFJOCVMVOVQ OJOCVMVONBNBPMBTM- \" 3 π # 2 3 π $ 3 π ôLBÀUS 9 9 12 \" 1 # 2 $ 3 % 4 & 5 % 3 π & 3 π 7 7 7 7 7 6 24 ôFLJMEFLJOPLUBEBOUBOFTJTF¿JMJZPS ABCDE #JS UPSCBEB TBS NBWJ WF ZFõJM CJMZF WBSES d1 5PSCBEBOBSUBSEBHFSJLPOVMNBNBLÐ[FSFпCJMZF ¿FLJMJZPS F GH K MN ¦FLJMFOJMLCJMZFOJOTBSPMEVôVCJMJOEJôJOFHÌ- d2 4FÀJMFO CV OPLUBOO CJS ÑÀHFOJO LÌöFMFSJ PM- SF ÑÀ CJMZFOJO EF GBSLM SFOLUF PMNB PMBTMô EVôVCJMJOEJôJOFHÌSF ÑÀHFOJOJLJLÌöFTJOJOE1 LBÀUS EPôSVTVÑ[FSJOEFPMNBPMBTMôLBÀUS \" 1 # 3 $ 2 % 5 & 3 7 14 7 14 7 \" 1 # 2 $ 1 % 4 & 2 10 9 3 9 3 0, 0, 1, 1, 2, 2, 2, 3, 3 EFOBLBEBSOVNBSBMBOESMNõZVNVSUBCJS UPSCBZBLPOVMVZPS:VNVSUBMBSEBOJLJTJLSMZPS SBLBNMBSOOUÐNÐLVMMBOMBSBLTBZMBSZB[MZPS ,SMBO ZVNVSUBMBSO OVNBSBMBSOO UPQMBNOO OCBöBHFMNFEJôJCJMJOEJôJOFHÌSF ZB[MBOTB- ÀJGUTBZPMEVôVCJMJOEJôJOFHÌSF JLJOVNBSBOO ZOOTPOVOBHFMNFPMBTMôLBÀUS EBUFLTBZPMNBPMBTMôLBÀUS \" 1 # 1 $ 1 % 2 & 3 \" 2 # 1 $ 1 % 2 & 3 4 3 23 4 7 3 23 5 C B D A 11 E B C
11. SINIF .0%·- 0-\"4*-*, www.aydinyayinlari.com.tr #\"ó*.-*7&#\"ó*.4*;0-\":-\"3 ÷MJöLJMJ,B[BONMBS 11.7.1.2 : #BóNMWFCBóNT[PMBZMBSB¿LMBZBSBLHFS¿FLMFõNFPMBTMLMBSOIFTBQMBS 7$1,0%m/*m ÖRNEK 2 &ËSOFLV[BZOEBJLJPMBZ\"WF#PMTVO#PMBZ- )JMFTJ[CJSUBWMB[BSJMFEÐ[HÐOCJSNBEFOÅQBSBCJSMJLUF OOHFS¿FLMFõNFPMBTMó\"PMBZOOHFS¿FLMFõ- BUMZPS NF PMBTMóO FULJMFNJZPSTB \" PMBZ # PMBZO- ;BSOÀJGUTBZWFQBSBOOZB[HFMNFPMBTMôLBÀUS EBOCBôNT[ESEFOJS ;BSO ÀJGUHFMNFPMBTMô¦ QBSBOOZB[HFMNFPMBTMô: \"WF#PMBZMBSCBóNT[JTF PMTVOP^ Ç h = 1 WFP^ Y h = 1 EJS0MBZMBSCBôNT[PM- | 1 \" # =1 \" PMVS 22 EVôVOEBO1 ¦a Y) = P(Ç)1 : | P^A + B h 1 \" # = P^ B h PMEVóVOEBO 11 1 = · = PMVS P^ A + B h = P^ A hPMVS P^ B h 22 4 #VOBHËSF \"WF#CBóNT[PMBZMBSJTF ÖRNEK 3 1 \"a# =1 \" 1 # PMVS #JS LVUVEB CVMVOBO CFZB[ WF NBWJ QJOQPO UP \"PMBZOOHFS¿FLMFõNFTJ#PMBZOOHFS¿FLMFõ- QVOEBOSBTUHFMFÑBMOSTBFOB[JLJTJOJONBWJPM- NBPMBTMôLBÀUS NFTJOJ FULJMJZPSTB \" WF # PMBZMBSOB CBôNM 45 PMBZMBSEFOJS ÷LJTJNBWJ CJSJCFZB[JTF.f p f p = 40 PMVS #VEVSVNEBPa B Ak= P^ A + B h PMVQCVSB- 12 5 P^ A h ·ÀÑEFNBWJJTFf p = 10 PMVS 3 dan 9 ²SOFLV[BZf p = 84 PMVS | 1 \"a# =1 \" 1 # \" PMVS 3 40 10 50 25 ÖRNEK 1 ÷TUFOFOPMBTML + = = PMVS 84 84 84 42 \"WF#CBóNT[PMBZMBSES P^ A h = 2 , P^ B h = 3 PMEVôVOBHÌSF ÖRNEK 4 58 #JS UPSCBEB CFZB[ NBWJ CJMZF WBSES #V UPSCBEBO a) 1 \"a# HFSJCSBLMNBNBLLPõVMVJMFBSUBSEBJLJCJMZF¿FLJMJZPS C 1 \"b# ¦FLJMFO CJSJODJ CJMZFOJO NBWJ JLJODJ CJMZFOJO CFZB[ PMBTMLMBSLBÀUS PMNBPMBTMôLBÀUS a) P ( A a B ) =1 \" 1 # ¦FLJMFOUPQZFSJOFLPONBEôOEBOPMBZMBSCBôNMPMBZMBS- 23 3 ES P(M a# 1 . 1 #=. =·= 5 8 20 3 5 15 = · = PMVS C 1 \"b B) = P( A) + P ( B ) - P ( A a B ) 23 3 5 8 7 56 =+- = 5 8 20 8 a) 35 12 1 25 15 C 20 8 4 42 56
www.aydinyayinlari.com.tr 0-\"4*-*, .0%·- 11. SINIF ÖRNEK 5 ÖRNEK 7 #JSUPSCBEB TBSWFLSN[CJMZFWBSES#VUPSCBEBO \"UPSCBTOEBCFZB[ ZFõJM #UPSCBTOEBCFZB[ BSUBSEBJLJCJMZFBMOZPS ZFõJMUPQWBSES #VCJMZFMFSEFO \"EBOCJSUPQÀFLJMJQ#ZFBUMELUBOTPOSB#EFOÀF- B #JSJODJOJOTBS JLJODJOJOLSN[PMNB LJMFOCJSUPQVOZFöJMPMNBPMBTMôLBÀUS C öLJTJOJOEFLSN[PMNB D öLJTJOJOEFBZOSFOLUFPMNB EVSVN E #JSJOJOTBS CJSJOJOLSN[PMNB 2 6 12 PMBTMLMBSOCVMVOV[ 7 ·= . B 0MBZMBSCBôNMES#VOBHÌSF 11 77 . P(S a, =1 4 1 ,=4 43 2 \"UPSCBTOEBOCFZB[#UPSCBTOEBOZFöJM ÀFLNFPMBTMô ÀFLNFPMBTMô =·= 76 7 EVSVN 32 1 5 7 35 C 1 ,a, = · = 7 ·= 76 7 . 11 77 D P^ S + S h + P^ K + K h = 4 · 3 + 3 · 2 = 3 . 76 76 7 \"UPSCBTOEBOZFöJM #UPSCBTOEBOZFöJM E 4SBCFMMJPMNBEôOEBO ÀFLNFPMBTMô ÀFLNFPMBTMô 43 34 4 12 35 47 P(S a, 1 ,a S) = · + · = PMVS ÷TUFOFOPMBTML + = PMVS 76 76 7 77 77 77 ÖRNEK 6 ÖRNEK 8 \" WF # UPSCBMBSOO IFS JLJTJOEF EF CFZB[ LSN- UBOFTJ ÀBUMBL PMBO ZVNVSUB BSBTOEBO BSU BSEB [UPQWBSES TFÀJMFOZVNVSUBEBOCJSJOJOTBôMBN JLJTJOJOÀBUMBL PMNBPMBTMôLBÀUS \"EBOCJSUPQÀFLJMJQ#ZFTPOSBEB#EFOCJSUPQÀF- LJMJQ\"ZBBUMEôOEBSFOLCBLNOEBOJMLEVSVNVFM- EVSVN EFFUNFPMBTMôLBÀUS :VNVSUBMBS ¦ ÀBUMBL 4 TBôMBN PMNBL Ñ[FSF ¦ ¦ 4 IBSG- EVSVN \" UPSCBTOEBO CFZB[ CJMZF ÀFLJMJQ # UPSCBTOB MFSJOJO GBSLM TSBMBNBMBS QFSNÑUBTZPO LBEBS GBSLM öFLJM- EFTFÀJMFCJMJS BUMNöTB #UPSCBTOEBOEBCFZB[CJMZFÀFLJMNFMJLJJMLEV- 3! SVNFMEFFEJMTJO#VEVSVNEBCVPMBZB*PMBZEJZFMJN ¦¦4TSBMBNBMBS = 3 GBSLMöFLJMEFZB[MBCJMJS 5FLSBS- P^ I h = 3 · 42 2! = MQFSNÑUBTZPO 9 10 15 4 36 1 .. P(Ç a Ç a S) = · · = \"UPSCBTOEBOCFZB[#UPSCBTOEBOCFZB[ 10 9 8 10 13 ÀFLNFPMBTMô ÀFLNFPMBTMô ÷TUFOFOPMBTML ·3 = PMVS 10 10 EVSVN\"UPSCBTOEBOLSN[ÀFLJMNJöTF #UPSCBTOEBO EBLSN[ÀFLJMNFLJJMLEVSVNFMEFFEJMTJO#VPMBZB**PMB- ZEFSTFL P^ II h = 6 · 77 = 9 10 15 .. \"UPSCBTOEBOLSN[#UPSCBTOEBOLSN[ ÀFLNFPMBTMô ÀFLNFPMBTMô P^ I h + P^ II h = 2 + 7 = 9 3 = 15 15 15 5 2134 3 13 47 3 a) C D d) 7777 5 77 10
11. SINIF .0%·- 0-\"4*-*, www.aydinyayinlari.com.tr ÖRNEK 9 ÖRNEK 11 #JSUPSCBEBCVMVOBOCFZB[WFNBWJCJMZFEFOBSU #JS UPSCBEB TJZBI NBWJ ZFõJM CJMZF CVMVONBLUB- BSEBTFÀJMFOCJMZFEFOFOB[CJSJOJONBWJPMNBPMB- ES ¥FLJMFO CJMZF ZFSJOF LPONBL Ð[FSF BSU BSEB JLJ CJM- TMôLBÀUS ZF¿FLJMJZPS ¦FLJMFOJLJCJMZFOJOEFTJZBIPMNBPMBTMôLBÀUS \"SU BSEB ÀFLJMEJôJOEFO ÀFLJMNF TSBTOB HÌSF #. .# WF .. EVSVNMBSOEBO CJSJ HFSÀFLMFöJS #V EVSVNEB JTUFOFO P^ S + S h = 4 · 4 = 16 PMVS PMBTML 9 9 81 P ( B a M) + P ( M a B ) + P ( M a M ) [[ 64 46 43 2 · + · + · = PMVS 10 9 10 9 10 9 3 ÖRNEK 10 ÖRNEK 12 öLJBUDEBOCJSJODJOJOIFEFGJWVSNBPMBTMó 1 JLJODJOJO 4 5\"5÷- IFEFGJWVSNBPMBTMó 3 UJS EV 5 1 2 8 3 #÷4÷,-&5 #JSFS BUö ZBQMNBT TPOVDVOEB IFEFGJO ZBMO[ CJSJ )&%÷:&¦&,÷ 7 4 #6;%0-\"#* UBSBGOEBOWVSVMNBPMBTMôLBÀUS 6 5 #JSZBSöNBOOLVSBMMBSöVöFLJMEFCFMJSMFONJöUJS #JSJODJ BUDOO IFEFGJ WVSNB PMBTMô \" JLJODJ BUDOO IF- EFGJWVSNBPMBTMô#PMTVO r :BSõNBDEBSUUBIUBTOBBSUBSEBJLJBUõZBQBS P^ A h = 1 WFP^ B h = 3 UJS)FEFGJZBMO[CJSJOJOWVSNB- r &óFSWVSEVóVJLJTBZOOUPQMBN¿JGUTBZJTFIFEJ- 45 ZFQBOPTVOEBLJIFEJZFMFSEFOCJSJOJOõóZBOBSWF THFSFLJS#VOBHÌSF JTUFOFOPMBTML PIFEJZFZJLB[BOS 1 2 3 3 1 9 11 r 7VSEVóVTBZMBSOUPQMBNUFLTBZJTFFMFOJS P (A + B' ) + P (A' + B) = · + · = + = r :BQMBOBUõ¿J[HJZFEFOLHFMJSTFWFZBBZOTBZZ 4 5 4 5 10 20 20 WVSVSTBBUõUFLSBSFEJMJS PMVS #VOBHÌSF ZBSöNBZBLBUMBOCJSZBSöNBDOOFWLB- ÖRNEK 13 [BONBPMBTMôLBÀUS #JS[BSBSUBSEBJLJLF[BUMZPS TBZMBSOEBO JLJTJ WFZB TBZMBSOEBO JLJ- ÷MLBUöUBÀJGUWFJLJODJBUöUBBTBMTBZHFMNFPMBTM- TJWVSVMNBMES ôLBÀUS ÷LJTJEFÀJGU+ ÷LJTJEFUFL 31 ¦JGUHFMNFPMBTMô= = 43 4 33 · + ·= 62 87 8 77 31 1 &WJOöôOOZBONBPMBTMô UJS0MBZMBSCBôNT[PMEV- \"TBMTBZHFMNFPMBTMô= = 62 11 1 ÷TUFOFOPMBTML= · = 22 4 5 31 3 ôVOEBO · = CVMVOVS 7 5 35 2 3 14 16 11 1 81 20 4 3 35
#BôNMWF#BôNT[0MBZMBS TEST - 6 #JSUPSCBEBCVMVOBOUÐLFONF[LBMFNEFOJCP- #JSTOGUBL[ FSLFLËóSFODJWBSES [VLUVS5PSCBEBOSBTUHFMFLBMFN¿FLJMJZPS 4OGUBOSBTUHFMFTFÀJMFOLJöJEFOCJSJOJOL[EJ- ôFSJOJOFSLFLPMNBPMBTMôLBÀUS ¦FLJMFOLBMFNMFSEFOZBMO[ CJSJOJOCP[VLPMNB PMBTMôLBÀUS \" 1 # 2 $ 7 % 1 & 8 \" 50 # 48 $ 45 % 7 & 1 3 5 15 2 15 91 91 91 15 3 #JSUPSCBEB FõJUTBZEBTJZBIWFCFZB[CJMZFMFSWBS- #JSNBEFOÅQBSBBSUBSEBLF[IBWBZBBUMZPS ES #VOBHÌSF QBSBOOFOB[CJSLF[ZB[HFMNFPMB- TMôLBÀUS \"SUBSEBÀFLJMFOJLJCJMZFOJOEFTJZBIPMNBPMB- TMô 7 PMEVôVOBHÌSF JMLEVSVNEBUPSCBEB \" 4 # 5 $ 8 % 17 & 31 30 5 18 13 18 32 UPQMBNLBÀCJMZFWBSES \" # $ % & #JSUPSCBOOJ¿JOEFBZOCÐZÐLMÐLUFTBS LSN[ \"JMF#PMBZMBSBZOÌSOFLV[BZEBJLJCBôNT[ CJMZFWBSES5PSCBEBOBZOBOEBCJMZF¿FLJMJZPS PMBZPMNBLÑ[FSF ¦FLJMFOCJMZFMFSJOBZOSFOLMJPMEVôVCJMJOEJôJOF P^ A h = 2 ve P^ A + B h = 1 HÌSF JLJTJOJOEFTBSPMNBPMBTMôLBÀUS 55 PMEVôVOBHÌSF 1 # LBÀUS \" 7 # 3 $ 5 % 1 & 3 \" 2 # 8 $ 2 % 1 & 1 8 4 82 8 25 25 15 2 3 5FL BUõUB CJS IFEFGJ WVSNB PMBTMLMBS 3 WF 4 NBEFOÅQBSBCJSMJLUFIBWBZBBUMZPS 59 #VOMBSEBO JLJTJOJO UVSB EÌSEÑOÑO ZB[ HFMNF PMBOJLJBUDCVIFEFGFCJSFSBUõZBQZPSMBS PMBTMôLBÀUS #VOBHÌSF CVIFEFGJOWVSVMNVöPMNBPMBTMô \" 1 # 15 $ 1 % 3 & 1 4 64 8 32 64 LBÀUS \" 8 # 3 $ 2 % 7 & 8 45 5 3 9 9 C B E D 15 E E D B
TEST - 7 #BôNMWF#BôNT[0MBZMBS #JSUVSJTULBGJMFTJOEFöOHJMJ[ \"MNBOWBSES #JSNBEFOÆQBSBLF[BUMEôOEBFOÀPLLF[ ,BGJMFEFOSBTUHFMFTFÀJMFOLJöJEFOCJSJOJO÷OHJ- ZB[HFMNFPMBTMôLBÀUS MJ[EJôFSJOJO\"MNBOPMNBPMBTMôLBÀUS \" 8 # 7 $ 6 % 5 & 3 \" 1 # 5 $ 3 % 7 & 1 11 11 11 11 11 4 16 8 16 2 #JSUPSCBEBBZOCÐZÐLMÐLUFFMNBT ZBLVU LV- #JSUPSCBEBBZOCÐZÐLMÐLUFJODJ FMNBTWFZB- WBSTUBõWBSES LVUUBõWBSES 5PSCBEBO SBTUHFMF ÀFLJMFO UBöO ÑÀÑOÑO EF GBSLMDJOTUFOPMNBPMBTMôLBÀUS 5PSCBEBO BSU BSEB ÀFLJMFO UBöO ÑÀÑOÑO EF \" 1 # 5 $ 1 % 3 & 3 BZODJOTUFOPMNBPMBTMôLBÀUS & 5 2 12 3 10 11 \" 3 # 1 $ 1 % 5 42 11 12 14 84 #JSUPSCBEBBZOCÐZÐLMÐLUFTBS NBWJCJMZFWBS- #JSUVSJTULBGJMFTJOEF\"MNBO 'SBOT[UVSJTUWBS- ES ES 5PSCBEBOSBTUHFMFÀFLJMFOCJMZFOJOJLJTJOJOEF ,BGJMFEFOSBTUHFMFTFÀJMFOJLJLJöJOJOCJSJOJO\"M- BZOSFOLUFPMNBPMBTMôLBÀUS NBO EJôFSJOJO'SBOT[PMNBPMBTMôLBÀUS \" 8 # 7 $ 2 % 5 & 4 \" 1 # 2 $ 8 % 13 & 2 9 9 39 9 15 5 15 5 3 LJõJMJL CJS TQPSDV HSVCVOEBO SBTUHFMF TF¿JMFDFL L[ FSLFLËóSFODJLJNMJLMFSJOJCJSLVUVZBLPZV- JLJLJõJOJOJLJTJOJOEFZÐ[ÐDÐPMNBPMBTMó 5 PM- ZPSMBS 42 ,VUVEBOSBTUHFMFLJNMJLÀFLJMEJôJOEFJLJTJOJO EVóVOBHËSF HSVQUBLBÀZÑ[ÑDÑWBSES FSLFL ÌôSFODJZF CJSJOJO L[ ÌôSFODJZF BJU PMNB PMBTMôLBÀUS \" # $ % & \" 5 # 5 $ 15 % 4 & 6 112 56 28 7 7 C E E C 16 B D C C
#BôNMWF#BôNT[0MBZMBS TEST - 8 #JS UPSCBEB EFO F LBEBS OVNBSBMBOESMNõ #JSUPSCBEBTJZBI NBWJCJMZFWBSES#VUPSCB- CJMZFWBSES5PSCBEBOBSUBSEBCJMZF¿FLJMJZPS EBO BSU BSEB CJMZF ¿FLJMJQ TPO CJMZF ¿FLJMEJLUFO TPOSBCJSQBSBIBWBZBBUMZPS ¦FLJMFOCJMZFMFSJOCÑZÑLUFOLÑÀÑôFEPôSVB[B- MBOTSBEBÀFLJMNJöPMNBPMBTMôLBÀUS ¦FLJMFOCJMZFMFSJOBZOSFOLWFQBSBOOZB[HFM- NFPMBTMôLBÀUS \" 1 # 1 $ 1 % 1 & 2 24 12 6 3 3 \" 1 # 2 C 1 % 1 & 7 10 15 65 30 :B[ HFMNF PMBTMó UVSB HFMNF PMBTMóOO LBU #JSTOGUBËóSFODJWBSES)FSËóSFODJLFOEJPLVM PMBOCJS[BSJLJLF[BUMZPS OVNBSBTOCJSLºóEBZB[Q CJSUPSCBZBLPZVZPS ÷LJBUöOEBUVSBHFMNFPMBTMôLBÀUS )FSÌôSFODJCJSFSLBSUÀFLUJôJOEFIFSÌôSFODJOJO \" 1 # 1 $ 1 % 1 & 9 LFOEJOVNBSBTOÀFLNFPMBTMôLBÀUS 16 32 16 8 2 \" 1 # 1 $ 1 % 1 & 1 9 18 9! 18! 36! ôFLJMEFLJ EËONF EPMBCO IFS CËMNFTJOF EFO ZFLBEBSOVNBSBMBSWFSJMNJõUJS ö¿JOEF TBS NBWJ WF TJZBI UPQMBSO CVMVOEVóV CJS 1 12 UPSCBEBO¿FLJMFOCJSUPQVONBWJPMNBPMBTMó 1 2 WFTBSPMNBPMBTMó 2 EVS 3 9 11 3 5PSCBEBUBOFTBSUPQCVMVOEVôVOBHÌSF LBÀ 10 4 UBOFTJZBIUPQWBSES \" # $ % & 9 5 8 6 7 A öLJUPSCBEBOCJSJODJTJOEFTBS NBWJ JLJODJTJOEF %ËONF EPMBC ¿BMõUSBO HËSFWMJ EËONF EPMBC ¿B- MõUSELUBOTPOSBBSUBSEBEFGBEËONFEPMBCEVS- TBS NBWJCJMZFWBSES#JSJODJUPSCBEBOCJSCJMZF EVSVZPS )FS EVSEVSNB FTOBTOEB CBõMBOH¿ OPL- ¿FLJMJQJLJODJUPSCBZBBUMZPS UBTPMBO\"OPLUBTIJ[BTOEBCJSCËMNFEVSVZPS ÷LJODJ UPSCBEBO CJS CJMZF ÀFLJMEJôJOEF ÀFLJMFO #VJöMFNJOTPOVDVOEB\"CÌMNFTJIJ[BTOEBEV- CJMZFOJOTBSPMNBPMBTMôLBÀUS SBOÑÀCÌMNFOJOOVNBSBMBSOOLÑÀÑLUFOCÑZÑ- ôFEPôSVBSUBOTSBEBPMNBTPMBTMôLBÀUS \" 11 # 3 $ 17 % 2 & 1 50 10 50 5 2 \" 11 # 55 $ 1 % 1 & 229 144 432 3 2 532 A B D C 17 E D B
11. SINIF .0%·- 0-\"4*-*, www.aydinyayinlari.com.tr #÷-&õ÷,0-\":-\"3*/0-\"4*-*ó* ÷MJöLJMJ,B[BONMBS 11.7.1.3 : #JMFõJLPMBZB¿LMBZBSBLHFS¿FLMFõNFPMBTMóOIFTBQMBS TANIM ÖRNEK 3 #JSEFOFZEFOFMEFFEJMFOWFUFLCJSTPOV¿WFSFO ôFLJMEFLJUPSCBEBTBS NBWJWFCFZB[UPQWBSES PMBZMBSBCBTJUPMBZMBSEFOJS#JS[BSBUMEóOEB HFMNFTJPMBZCBTJUPMBZES 4 öLJ WFZB EBIB ¿PL PMBZO CJSMJLUF WFZB BSU BSEB 6M HFS¿FLMFõNFTJOF CJMFöJL PMBZ EFOJS #JS [BSO 2B BSUBSEBJLJLF[BUMEóOEBÐTUZÐ[FHFMFOTBZMB- SOUPQMBNMBSOO¿JGUPMNBTPMBZCJMFõJLPMBZES 5PSCBEBOBZOBOEBBMOBOJLJUPQUBOJLJTJOJOEFTBS WFZBJLJTJOJOEFNBWJPMNBPMBTMôLBÀUS ÖRNEK 1 \"SU BSEB LF[ BUMBO CJS NBEFOÆ QBSBOO JLJODJ BUö- UB ZB[ HFMNF PMBTMôO BôBÀ EJZBHSBN LVMMBOBSBL CVMVOV[ 1 :B[ 4 2 Tura fp 21 :B[ 1 :B[ ÷LJTJOJOEFTBSPMNBPMBTMô = 1 2 Tura 12 11 2 fp 2 Para 6 1 fp 2 1 25 2 ÷LJTJOJOEFNBWJPMNBPMBTMô = PMVS Tura 12 22 fp 1 2 2 15 7 11 11 1 0MBZMBSCBôNT[PMEVôVOEBO + = PMVS · + · = PMVS 11 22 22 22 22 2 ÖRNEK 2 ÖRNEK 4 #JS[BSJMFCJSNBEFOÅQBSBCJSMJLUFBUMZPS \"MJNBTBUFOJTJ \"ZõFTBUSBO¿UVSOVWBTOBLBUMBDBLMBSES 1BSBOOZB[WFZB[BSOUFLTBZHFMNFPMBTMôLBÀUS \"MJhOJONBTBUFOJTJUVSOVWBTOLB[BONBPMBTMó 3 , Ay- 1 4 1BSBOOZB[HFMNFPMBTMô1 : = EJS õFhOJOTBUSBO¿UVSOVWBTOLB[ BONBPMBTMó 2 UJS 2 5 1 ,BUMELMBS CV UVSOVWBMBSEBO Ali WFZB \"ZöFOJO ka- ;BSOUFLTBZHFMNFPMBTMô1 5 = EJS [BONBPMBTMôLBÀUS 2 P(A b B) = P(A) + P(B) - P(A a B) 1 :WFZB5 = P(Y b T) = P(Y) + P(T) - P(Y a T) 3 2 3 2 17 1 1 11 13 + -·= = + - · = 1 - = PMVS 4 5 4 5 20 2 2 22 44 1 3 18 7 17 22 20 2 4
www.aydinyayinlari.com.tr 0-\"4*-*, .0%·- 11. SINIF ÖRNEK 5 ÖRNEK 8 öLJ[BSCJSMJLUFBUMZPS ¶¿NBEFOJQBSBBZOBOEBIBWBZBBUMZPS ;BSMBSO ÑTU ZÑ[ÑOF HFMFO SBLBNMBSO UPQMBNOO 1BSBOOÑTUZÑ[ÑOFFOB[CJSLF[UVSBHFMNFPMBTM- WFZBPMNBPMBTMôLBÀUS ôLBÀUS (1, 4) , (4, 1) 1 - P(YYY) 17 PMEVôVOEBOP^ 5 h = 4 1 = PMVS 1 - = PMVS 36 9 88 (1, 5) (5, 1) (2,4 ) (4, 2) PMEVôVOEBOP^ 6 h = 5 PMVS 36 P(5 b 6) = P(5) + P(6) 15 1 = + = PMVS 9 36 4 ÖRNEK 6 ÖRNEK 9 #JSNBEFOÅQBSBWFCJS¿JGU[BSBUMZPS #JSBSBCBZBSõOEB\" # $QMBLBMпBSBCBWBSES:B- 1BSBOO UVSB WF [BSMBSO ÑTU ZÑ[ÑOF HFMFO TBZMBSO Sõ\"WF#OJOLB[BONBPMBTMLMBSFõJUWF$OJOLB[BO- UPQMBNOOPMNBTPMBTMôLBÀUS NBPMBTMóOOLBUES #VOB HÌSF CV ZBSö \" WFZB $ OJO LB[BONB PMBTM- ôLBÀUS 1BSBOOUVSBHFMNFPMBZ\"PMTVOP^ A h = 1 EJS A BC 2 4x 4x x PMBTMLMBSPMTVO ;BSMBSOÑTUZÑ[ÑOFHFMFOTBZMBSOUPQMBNOOPMNB 1 PMBZ#PMTVO Y+Y+Y=PMEVôVOEBO x = PMVS 9 (3, 6) (6, 3) P^ A h = 4 , P^ B h = 4 , P^ C h = 1 99 9 (4, 5) (5, 4) P^ A , C h = 4 + 1 = 5 PMVS 99 9 P^ B h = 4 1 = PMVS 36 9 11 1 P(A a B) =1 \" 1 # = · = PMVS 2 9 18 ÖRNEK 7 ÖRNEK 10 Y Z [CJSËSOFLV[BZPMVõUVSBOBZSLPMBZES #JSUPSCBEBEFOFLBEBSOVNBSBMBOESMNõCJM- YWFZBZPMBZOOPMBTMó 4 EJS ZFWBSES 7 #JSUPQÀFLJMEJôJOEF ÀFLJMFOUPQVOUFLTBZOVNBSB- #VOBHÌSF [PMBZOOPMBTMôLBÀUS MWFZBEFOLÑÀÑLOVNBSBMPMNBPMBTMôLBÀUS 4 5FLOVNBSBMBS5LÑNFTJ EFOLÑÀÑLMFS,LÑNFTJPM- P(X b: 1 9 1 : TVO 7 T = {1, 3, 5, 7, 9, 11, 13, 15} P(X b Y b Z) = 1 ,= {1, 2, 3, 4, 5, 6}PMVS P(X) + P(Y) + P(Z) = 1 4 + P^ Z h = 1 T b,= {1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15} 7 P^ Z h = 3 PMVS ÷TUFOFOPMBTML s^ T , K h 11 = PMVS 7 s^ E h 15 1 1 3 19 7 5 11 89 15 4 18 7
11. SINIF .0%·- 0-\"4*-*, www.aydinyayinlari.com.tr ÖRNEK 11 ÖRNEK 13 \"SUBSEBLF[BUMBOCJSQBSBOOPMBTMóIFTBQMBOSLFO #JSPUPCÐTUFLJZPMDVMBSOTJFSLFLUJS0UPCÐTUFLJCB- õFLJMEFLJHJCJBóB¿EJZBHSBNLVMMBOMBDBLUS ZBOMBSO 2 JWFFSLFLMFSJOZBSTJMLEVSBLUBJOFDFLUJS :\";* :\";* 5 TURA TURA #VOBHÌSF SBTUHFMFTFÀJMFOCJSZPMDVOVOJMLEVSBLUB JOFDFLPMNBWFZBFSLFLPMNBPMBTMôLBÀUS :\";* PARA TURA 0UPCÑTUFLJöJPMTVO EB ÷OFDFL 10 32 %JZBHSBNEB J¿JOEF ZB[ WFZB UVSB ZB[BO LVUV LVMMB- ÷ONFZFDFL 10 48 OMNõUS 52 13 #VOBHÌSF BSUBSEBLF[BUMBOCJSQBSBJÀJOÀJ[JMFO = PMVS EJZBHSBNEBLJ LVUV TBZT BSU BSEB LF[ BUMBO CJS QBSBJÀJOÀJ[JMFOEJZBHSBNEBLJLVUVTBZTOEBOLBÀ 100 25 GB[MBES ,VUV TBZMBS öFLMJOEF OJO LVWWFUMFSJ ÖRNEK 14 PMBSBLBSUBS LF[BUMSTB + 22 + 23+ 24+ 25BEFU B1 C LF[BUMSTB+ 22+ 23 + 24 + 25 + 26 + 27 + 28BEFU 1 31 LVUVPMVöVS 3 'BSLMBS6 + 27 + 28PMVS 3D G 26 (1 + 2 + 22) ==LVUVPMVS 1 1 1 3 3 31 3 A 1 1 H 3 2 E1 3 1 2K F ÖRNEK 12 ôFLJMEFLJLBSBZPMVOEB\"OPLUBTOEBOZPMB¿LBOCJSBSB¿ ZPMBZSNMBSOOUÐNÐOEFSBTUHFMFCJSZPMTF¿NJõWFHF¿- {B C D E F}LÐNFTJOJOBMULÐNFMFSJOEFOJLJUBOFTJSBTU- UJóJZPMEBOHFSJEËONFNJõUJS HFMFTF¿JMJZPS #VOBHÌSF CVLJöJOJO&WFZB)OPLUBTOEBOHFÀNJö PMNBPMBTMôLBÀUS #VBMULÑNFMFSEFOCJSJOJOJLJFMFNBOMWFEJôFSJOJOÑÀ FMFNBOMPMNBPMBTMôLBÀUS 25 =BMULÑNFTJWBSES 11 1 &EFOHFÀNFPMBTMô · = 55 f p.f p 32 6 2 3 10.10 25 11 1 = = PMVS )EFOHFÀNFPMBTMô · = 16.31 124 33 9 32 11 5 fp P(E b) + = PMVS 2 6 9 18 448 25 13 5 124 25 18
#JMFöJL0MBZMBSO0MBTMô TEST - 9 ö¿JOEFBZOCÐZÐLMÐLUFTJZBI CFZB[WFZFõJMUPQMBS A = {x : 1 # x # 50 , x `/} CVMVOBO CJS UPSCBEBO ¿FLJMFO CJS UPQVO ZFõJM PMNB LÑNFTJOEFOSBTUHFMFTFÀJMFOCJSTBZOOWFZB PMBTMó 1 CFZB[PMNBPMBTMó 7 EJS JMFCÌMÑOFCJMNFPMBTMôLBÀUS 4 12 5PSCBEB ZFöJM UPQ CVMVOEVôVOB HÌSF UPSCB- \" 2 # 12 $ 3 % 17 & 24 EBLJTJZBIUPQMBSOTBZTLBÀUS 5 25 5 25 25 \" # $ % & #JSLVUVEBÐ[FSMFSJOEFB C DIBSGMFSJOJOCVMVOEV- óVLBSUWBSES,VUVEBOCJSLBSU¿FLJMJQ CJSNBEF- OJQBSBBUMZPS )BWB EFOJ[WFLBSBZPMVJMFTFZBIBUFUNFZJEÐõÐ- ¦FLJMFOLBSUOCWFQBSBOOZB[HFMNFTJPMBT- OFOLJõJWBSES MôLBÀUS )FQTJOJOBZOZPMMBTFZBIBUFUNFPMBTMôLBÀ- \" 1 # 1 $ 1 % 1 & 2 US 12 8 6 3 3 \" 1 # 1 $ 3 % 1 & 4 81 64 64 27 27 ôFLJMEFLJFõJUCËMNFMJIFEFGUBIUBTOBZBQMBOIFS ôFLJMEFLJBóB¿EJZBHSBNOEBCJSËóSFODJOJOBSUBS- BUõ UBIUB Ð[FSJOEFLJ TBZMBSB BZSMBO CËMNFMFSEFO EBTPSV¿Ë[EÐóÐOEFLJEPóSVWFZBOMõEVSVNMBS CJSJOFLFTJOMJLMFJTBCFUFUNFLUFEJS WFSJMNJõUJS 9 0 D 8 1 D 7 2 : 6 D 3 D 4 : 5 : D D : : D : : #VOB HÌSF BSU BSEB ZBQMBO JLJ BUöUB WVSVMBO #VOB HÌSF CV ÌôSFODJOJO EPôSV WF ZBOMö CÌMNFMFSEFLJTBZMBSOUPQMBNOOPMNBPMBT- MôLBÀUS ZBQNBPMBTMôLBÀUS \" 1 # 1 $ 1 % 2 & 2 \" 1 # 1 $ 1 % 3 & 1 10 9 5 9 5 16 8 48 2 C D A 21 B C D
TEST - 10 #JMFöJL0MBZMBSO0MBTMô #JSTOGUBLJËóSFODJMFSJOVNBUFNBUJLUFO LJõJMJLCJSTOGOVFSLFLUJS&SLFLMFSJOJ L[- TJLJNZBEBOWFVIFSJLJTJOEFOLBMNõUS MBSOJTPMBLUS 3BTUHFMFTFÀJMFOCJSÌôSFODJOJOLJNZBEBOLBMBO 4OGUBOSBTUHFMFTFÀJMFOCJSÌôSFODJOJOL[WFZB WFZBIFSJLJEFSTUFOEFHFÀFOCJSÌôSFODJPMNB TPMBLPMNBPMBTMôLBÀUS PMBTMôZÑ[EFLBÀUS \" 2 # 14 $ 3 % 4 & 5 \" # $ % & 5 25 5 5 6 :BSÀBQDNPMBOCJSEBJSFOJOÑ[FSJOEFOTFÀJ- MFO CJS OPLUBOO EBJSFOJO NFSLF[JOF DN WFZB DNEFOEBIBV[BLPMNBPMBTMôLBÀUS ôFLJMEFLJ EÐ[FOFLUF ÐTUUFO CSBLMBO CJMZFOJO п- \" 1 # 1 $ 1 % 1 & 3 HFO FOHFMMFSJO TBóOEBO ZB EB TPMVOEBO HF¿NF 8 6 42 4 PMBTMLMBSFõJUUJS * ** *** *7 #JSUPSCBEBBZOCÐZÐLMÐLUFTBS ZFõJMUPQWBS- #VOBHÌSF ÑTUUFOCSBLMBOCJSCJMZFOJO**WFZB ES ***OVNBSBMCPöMVLUBOEÑöNFPMBTMôLBÀUS 5PSCBEBO SBTUHFMF ÀFLJMFO JLJ CJMZFOJO JLJTJOJO \" 1 # 1 $ 1 % 1 & 3 8 7 64 4 EF TBS SFOL WFZB JLJTJOJO EF GBSLM SFOL PMNB PMBTMôLBÀUS \" 3 # 5 $ 6 % 7 & 8 7 7 78 9 #JSNBEFOJQBSBBSUBSEBLF[IBWBZBBUMZPS A = { } #VOBHÌSF QBSBOOFOB[CJSLF[UVSBHFMNFWF- LÐNFTJOJO FMFNBOMBS JMF п CBTBNBLM SBLBNMBS ZBJLJLF[ZB[HFMNFPMBTMôLBÀUS GBSLMTBZMBSZB[MZPS \" 1 # 5 $ 3 % 7 & 15 #VTBZMBSEBOCJSJTFÀJMEJôJOEFUFLTBZWFZB 2 8 48 16 ÑOLBUPMNBPMBTMôLBÀUS \" 13 # 23 $ 35 % 45 & 51 30 30 44 52 64 C E C E 22 B E B
#JMFöJL0MBZMBSO0MBTMô TEST - 11 \"õBóEBLJUBCMPEBCJSËóSFODJOJOCFõHÐOEF¿Ë[EÐ- #JSTOGUBL[WFFSLFLËóSFODJWBSES óÐTPSVTBZTWFCVTPSVMBSOLB¿OOZBOMõPMEV- #VTOGUBOBSUBSEBTFÀJMFOJLJLJöJEFOCJSJODJOJO óVWFSJMNJõUJS L[ JLJODJOJOFSLFLPMNBPMBTMôLBÀUS 1UFTJ 4BM ¦BSö 1FSö $VNB \" 20 # 3 $ 2 % 14 & 5 25 35 40 50 5PQMBN 77 11 7 77 TPSV 10 5 10 15 TBZT 30 :BOMö 5 TPSV TBZT #VOBHÌSF CVTPSVMBSBSBTOEBOTFÀJMFOCJSTP- SVOVOTBMHÑOÑÀÌ[ÑMFOCJSTPSVWFZBZBOMöCJS TPSVPMNBPMBTMôLBÀUS #JSUPSCBEBBZOCÐZÐLMÐLUFNBWJWFZFõJMCJMZF \" 1 # 4 $ 14 % 1 & 4 WBSES 15 15 45 3 9 5PSCBEBOSBTUHFMFCJMZFÀFLJMEJôJOEFCJMZFMF- SJOGBSLMSFOLUFPMNBPMBTMôLBÀUS \" 1 # 1 $ 2 % 3 & 4 3 2 3 4 5 4BEFDFTBSWFNPSCJMZFMFSJOCVMVOEVóVCJSUPSCB- EBLJTBSCJMZFTBZTNPSCJMZFTBZTOOLBUES 5PSCBEBO SBTUHFMF ÀFLJMFO JLJ CJMZFOJO GBSLM SFOLUFPMNBPMBTMô 3 PMEVôVOBHÌSF UPSCB- 7 EBLBÀTBSCJMZFWBSES \" # $ % & #JSLFOBSDNPMBO\"#$%LBSFTJWFJ¿UFóFU¿FN- \" OPLUBTOEBO MBCJSFOUF HJSFO GBSF HF¿UJóJ ZPMEBO CFSJWFSJMNJõUJS DC HFSJEËONFZFDFLõFLJMEFMBCJSFOUUFIBSFLFUFEJZPS BC O 4 D AB AE F \"#$%LBSFTJJÀJOEFBMOBOIFSIBOHJCJSOPLUB- OOÀFNCFSJONFSLF[JOFPMBOV[BLMôOODN WFZBDNEFOLÑÀÑLPMNBPMBTMôLBÀUS #VOBHÌSF $WFZB%OPLUBTOEBOÀLNBPMBTM- \" π – 2 # π $ π 4 4 6 ôLBÀUS \" 1 # 1 $ 1 % 1 & 2 % π & π 8 6 43 3 8 12 D A D 23 A B B
TEST - 12 #JMFöJL0MBZMBSO0MBTMô #JSTPSVOVO \"LJõJTJUBSBGOEBO¿Ë[ÐMFCJMNFPMBT- #JS[BSOпZÐ[ÐOEF JLJZÐ[ÐOEFWFEJóFSZÐ- Mó 1 , B LJõJTJUBSBGOEBO¿Ë[ÐMFCJMNFPMBTMó 1 [ÐOEFZB[NBLUBES 24 ;BSJLJLF[BUMEôOEB ÑTUZÑ[FCJSLF[WFCJS WF$LJõJTJUBSBGOEBO¿Ë[ÐMFCJMNFPMBTMó 2 UÐS 3 LF[HFMNFPMBTMôLBÀUS #VOBHÌSF CVTPSVOVOFOB[CJSJUBSBGOEBOÀÌ- \" 1 # 1 $ 1 % 1 & 2 6 2 43 3 [ÑMFCJMNFPMBTMôLBÀUS \" 1 # 1 $ 5 % 3 & 7 16 2 8 4 8 öLJNBEFOJQBSBWFCJS[BSCJSMJLUFBUMZPS öLJ[BSBSUBSEBBUMZPS 1BSBMBSEBOFOB[CJSJOJOUVSBWF[BSOBTBMTBZ HFMNFPMBTMôLBÀUS &OB[CJSLF[HFMNFTJPMBTMôLBÀUS \" 1 # 1 $ 1 % 1 & 3 12 3 6 4 8 \" 5 # 11 $ 1 % 2 & 35 18 36 3 3 36 \"WF#CBóNT[JLJPMBZPMNBLÐ[FSF #JSUPSCBEBCFZB[ NBWJ LSN[CJMZFWBSES P^ A h = 1 WFP^ A , B h = 2 33 5PSCBEBOZFSJOFLPONBLT[OBSUBSEBCJMZF¿FLJ- MJZPS PMEVôVOBHÌSF 1 # LBÀUS ·ÀÑOÑOEFBZOSFOLUFPMNBPMBTMôLBÀUS \" 1 # 1 $ 1 % 2 & 5 6 4 23 6 \" 1 # 1 $ 1 % 2 & 3 15 5 35 5 {0, 1, 2 ,3} #JSUPSCBEBTBSWFNBWJUPQWBSES#VUPSCBEBO LÐNFTJOJO FMFNBOMBS LVMMBOMBSBL ZB[MBCJMFO SB- LBNMBSGBSLMUÐNTBZMBSCJSFSLBSUBZB[MQCJSUPS- BSUBSEBUPQ¿FLJMJZPS CBZBLPOVMVZPS5PSCBEBOBSUBSEBJLJLBSU¿FLJMJZPS ¦FLJMFO UPQMBS SFOHJOF CBLMQ UFLSBS UPSCBZB ÷LJ LBSUUB EB ÑÀ CBTBNBLM CJS TBZOO ÀFLJMNJö LPOVMEVôVOB HÌSF UPQMBSEBO JML JLJTJOJO NBWJ PMNBPMBTMôLBÀUS ÑÀÑODÑOÑOTBSPMNBPMBTMôLBÀUS \" 1 # 1 $ 4 % 2 & 2 \" 3 # 51 $ 29 % 1 & 3 3 49 392 98 3 4 27 9 27 9 E B A C 24 D E C B
www.aydinyayinlari.com.tr 0-\"4*-*, .0%·- 11. SINIF %&/&:4&-7&5&03÷,0-\"4*-*, ÷MJöLJMJ,B[BONMBS 11.7.2.1 : %FOFZTFMPMBTMLJMFUFPSJLPMBTMóJMJõLJMFOEJSJS TANIM %m/*m #JSPMBTMLEFOFZJTPOVDVOEBFMEFFEJMFOWFSJ- #JSËSOFLV[BZEBEFOFZTFMPMBTMLEFóFSJ EF- MFSF HËSF IFTBQMBOBO PMBTMóB EFOFZTFM PMB- OFNF TBZT BSUUL¿B UFPSJL PMBTML EFóFSJOF TMLEFOJS#JSPMBZOEFOFZTFMPMBTMó PMBZO ZBLMBõS HFS¿FLMFõNF TBZTOO EFOFZ TBZTOB PSBO- ES ÖRNEK 3 #JS EFOFZEF PSUBZB ¿LBCJMFDFL UÐN TPOV¿MBS HË[ ËOÐOEF CVMVOEVSVMBSBL ZBQMBO NBUFNB- UJLTFMIFTBQMBNBZBUFPSJLPMBTMLEFOJS #JSBMõWFSJõNFSLF[JIFEJZFPMBSBLPUPNPCJMWFSJMFDFLCJS ¿FLJMJõEÐ[FOMJZPS ÖRNEK 1 #JMFUTBZT Burak \"MJ Ceren Arda #JS [BSO LF[ BSU BSEB BUMNBT EFOFZJOEF ÐTU ZÐ[F IBOHJTBZOOLB¿LF[HFMEJóJBõBóEBLJUBCMPEBWFSJMNJõ- 5BCMPEB ¿FLJMJõF LBUMBO EËSU LJõJOJO BMEó CJMFU TBZMB- UJS SWFSJMNJõUJS ¶TUZÐ[FHFMFOTBZ 1 2 3 4 5 #VOBHÌSF CVEÌSULJöJEFOIBOHJTJOJOPUPNPCJMLB- [BONBPMBTMôUFPSJLPMBTMôBEBIBZBLOES ,B¿LF[HFMEJóJ 22314 #VTPOVÀMBSBHÌSF $FSFOhJO EFOFZ TBZT EBIB ÀPL PMEVôVOEBO UFPSJL PMBTM- ôBEBIBZBLOES a) ¶TUZÐ[FHFMNFPMBZOOEFOFZTFMPMBTMóLB¿- US C ¶TUZÐ[FHFMNFPMBZOOUFPSJLPMBTMóLB¿US Olay›n gerçekleflme say›s› a) Deney say›s› 21 = = PMVS 18 9 ‹stenen durum 1 C = PMVS Tüm durumlar 6 ÖRNEK 2 ÖRNEK 4 5PMHBCJS[BSLF[BUBSBLFMEFFEJMFOTPOV¿MBSBHËSF #JS PL¿V IFEFG UBIUBTOB LF[ BUõ ZBQZPS WF JOEF [BSOHFMNFPMBTMóOIFTBQMZPS IFEFGJWVSVZPS #VOBHÌSF CVEVSVNIBOHJÀFöJUPMBTMôBÌSOFLUJS #VOBHÌSF CVPLÀVOVOIFEFGJWVSNBTOOEFOFZTFM %FOFZTFMPMBTML WFUFPSJLPMBTMLMBSUPQMBNLBÀUS 81 DFOFZTFMPMBTML= = 16 2 1 5FPSJLPMBTML= 2 11 5PQMBN= + = 1PMVS 22 11 %FOFZTFMPMBTML 25 $FSFO 1 a) C 96
TEST - 13 %FOFZTFMWF5FPSJL0MBTML #JSUPSCBEBTBS NBWJUPQWBSES #JSNBEFOJQBSBBSUBSEBLF[BUMEóOEBLF[ \"SUBSEBJLJUPQÀFLJMEJôJOEFUPQMBSOGBSLMSFOL- ZB[ LF[UVSBHFMJZPS UFPMNBTOOUFPSJLPMBTMôLBÀUS %FOFZTFM PMBTMôB HÌSF ZB[ HFMNF PMBTMô ZÑ[EFLBÀUS \" 1 # 8 $ 5 % 2 & 37 \" # $ % & 2 15 9 5 45 4FEBUCJS[BSLF[BUBSBLÐTUZÐ[FHFMFOTBZMBS #BõBSCJS[BSLF[BUQÐTUZÐ[FHFMFOTBZMBSCJS UBCMPZBZB[NõUS UBCMPZBZB[ZPS5BCMPZBHËSFÐTUZÐ[FHFMFOTBZ- OOUFOCÐZÐLPMNBPMBZOOEFOFZTFMPMBTMóUF- [BS [BS [BS [BS [BS [BS PSJLPMBTMóOEBO 2 GB[MBES 345123 5 #VOBHÌSF #BöBShOUBCMPZBZB[EôTBZMBSEBO LBÀUBOFTJUFOLÑÀÑLUÑS \" # $ % & #VTPOVÀMBSBHÌSFÑTUZÑ[FBTBMTBZHFMNFTJ- OJOEFOFZTFMPMBTMôLBÀUS \" 1 # 1 $ 1 % 2 & 3 6 3 23 4 LJõJMJLCJSËóSFODJHSVCV BZOTBBUUFZBQMBDBLGJ- [JL LJNZBWFCJZPMPKJEFSTMFSJOEFOCJSJOFHJSFDFLMFS- EJS LJõJ GJ[JL EFSTJOJ LJõJ LJNZB EFSTJOJ LJõJ JTFCJZPMPKJEFSTJOJUFSDJIFUNJõUJS #VOB HÌSF CJS ÌôSFODJOJO LJNZB EFSTJOJ UFS- DJI FUNJö PMNBTOO EFOFZTFM WF UFPSJL PMBTM- ôBöBôEBLJMFSEFOIBOHJTJEJS * %FOFZTFMPMBTMLEFóFSJEFOCÐZÐLPMBCJMJS %FOFZTFM 5FPSJL 1 ** %FOFZTFM PMBTML EFóFSJ EFOFZ TBZT BSUUL¿B 1 2 UFPSJLPMBTMLEFóFSJOFZBLMBõS \" 3 2 *** %FOFZTFMPMBTMLEFóFSJUFPSJLPMBTMLEFóFSJO- 1 3 EFOCÐZÐLUÐS # 2 1 JGBEFMFSJOEFOIBOHJMFSJEBJNBEPôSVEVS 1 3 $ 3 \" :BMO[* # :BMO[** $ :BMO[*** 1 5 3 % *WF** & **WF*** % 6 1 2 3 & 3 B D B 26 D A C
0MBTML KARMA TEST - 1 ·ÀNBEFOJQBSBCJSMJLUFBUMEôOEBJLJTJOJOUVSB #JS TPSVZV &LSFNJO ¿Ë[NF PMBTMó 4 3[BOO CJSJOJOZB[HFMNFPMBTMôLBÀUS ¿Ë[NePMBTMó 2 UÐS 5 \" 1 # 1 $ 3 % 5 & 3 3 2 4 88 4 BuOBHÌSF &LSFNWFZB3[BOOTPSVZVÀÌ[NF PMBsMôLBÀUS \" 14 # 13 $ 4 % 3 & 2 15 15 5 5 5 [BSBZOBOEBBUMZPS #JSLVUVEBCFZB[ TJZBICJMZFWBSES\"SLBBSLB- ·TUÑZÑ[FHFMFOTBZMBSOUPQMBNOOEBOCÑ- ZBWF¿FLJMFOCJMZFZFSJOFLPONBLT[OCFZB[CJM- ZFCVMBOBLBEBSLVUVEBO¿FLJNZBQMZPS ZÑLPMNBPMBTMôLBÀUS #JMZFÀFLJNJOJOEFOFNFEFCJUNJöPMNBPMBT- \" 1 # 1 $ 1 % 5 & 1 MôLBÀUS 18 12 9 36 6 \" 1 # 1 $ 3 % 2 & 1 10 5 10 5 2 QP[JUJGWFOFHBUJGTBZBSBTOEBOSBTUHFMFTF- #JSLVUVEBCFZB[ TJZBICJMZFWBSESöLJLJõJBSU ÀJMFOTBZOOÀBSQNOOQP[JUJGPMNBPMBTMô BSEBCJSFSCJMZF¿FLNFLUFEJS4JZBICJMZFZJJML¿FLFO LB[BOBDBLUS LBÀUS ¦FLJMFO CJMZF HFSJ LPONBNBL Ñ[FSF PZVOV CJ- \" 10 # 3 $ 8 % 1 & 2 SJODJPZVODVOVOLB[BONBPMBTMôLBÀUS 21 7 21 3 7 \" 4 # 41 $ 3 % 9 & 23 7 70 5 14 35 #JSTOGUBFSLFL L[ËóSFODJWBSES&SLFLMFSJO öLJ[BSBUMZPS J L[MBSOTJNBWJHË[MÐEÐS ;BSMBSO ÑTU ZÑ[MFSJOF HFMFO TBZMBSO UPQMBN- 4OGUBOTFÀJMFOCJSÌôSFODJOJONBWJHÌ[MÑWFZB OOEFOLÑÀÑLPMEVôVCJMJOEJôJOFHÌSFÑTUZÑ- FSLFLPMNBPMBTMôLBÀUS \" 5 # 8 $ 17 % 9 & 19 [FHFMFOTBZMBSOIFSJLJTJOJOEFBTBMTBZPMNB 11 11 22 11 22 PMBTMôLBÀUS \" 1 # 2 $ 1 % 4 & 1 15 15 5 15 3 C E A E 27 A C E D
KARMA TEST - 2 0MBTML #JS[BSBSLBBSLBZBLF[BUMZPS –3 –2 –1 0 1 2 3 #VÑÀBUöUBOJLJTJOJO CJSJOJOHFMNFPMBTMô LBÀUS :VLBSEBLJTBZEPôSVTVOEBOTFÀJMFOUBNTB- \" 1 # 1 $ 1 % 1 & 1 ZOONVUMBLEFôFSDFGBSLOOUFOCÑZÑLPMNB 6 36 72 128 216 PMBTMôLBÀUS \" 1 # 1 $ 2 % 3 & 2 3 2 37 7 *UPSCBEBTBS MBDJWFSU **UPSCBEBTBS MBDJ- WFSUCJMZFWBSES4F¿JMFOCJSUPSCBEBOCJSCJMZF¿FLJ- MJZPS ¦FLJMFOCJMZFOJOTBSPMNBPMBTMôLBÀUS AB \" 1 # 1 $ 2 % 3 & 4 \"UPSCBTOEBTJZBIWFCFZB[ #UPSCBTOEBTJ- 2 3 34 5 ZBIWFCFZB[UPQWBSES5PSCBMBSOIFSCJSJOEFO õFSUPQ¿FLJMJQEJóFSUPSCBZBBUMZPS )FSCJSUPSCBEBOÀFLJMFOUPQVOBZOSFOLUFPM- EVôV CJMJOEJôJOF HÌSF ZFS EFôJöUJSNFEFO TPO- SB\"WF#UPSCBMBSOEBLJUPQMBSOSFOLMFSJOFHÌ- SFEBôMNOEBEFôJöJNPMNBNBPMBTMôLBÀUS L[WFFSLFLZBOZBOBTSBMBOEôOEBIFSIBO- \" 2 # 1 $ 1 % 1 & 3 3 24 HJJLJFSLFôJOZBOZBOBPMNBNBPMBTMôLBÀUS \" 1 # 1 $ 1 % 1 & 1 42 35 28 24 15 D6 C \"#$%ZBNVL ôFLJMEFLJBóB¿EJZBHSBNOEBCJSQBSBOOBSUBSEB 12 A [AB] // [CD] LF[ BUMNBT TPOVDV ZB[ WF UVSB HFMNF EVSVNMBS WFSJMNJõUJS | |AB =DN | |B DC =DN : : T : : T T \"#$% ZBNVôVOVO JÀ CÌMHFTJOEF TFÀJMFO CJS : : OPLUBOO [AB] LFOBSOB PMBO V[BLMôOO [CD] T LFOBSOB PMBO V[BLMôOEBO EBIB LÑÀÑL PMNB T : T T PMBTMôLBÀUS \" 1 # 5 $ 1 % 7 & 2 #VOBHÌSF ZB[UVSBHFMNFPMBTMôLBÀUS 9 36 6 12 9 \" 1 # 1 $ 3 % 1 & 5 8 4 82 8 C A B D 28 E D C
0MBTML KARMA TEST - 3 #JS¿JGU[BSBUMEóOEBÐTUZÐ[MFSFHFMFOTBZMBSUPQ- d1 MBNOOEBOLпÐLPMEVóVCJMJOJZPS A B #VOBHÌSF ÑTUZÑ[FHFMFOTBZMBSOÀBSQNOO FGD C HI J d2 UFLPMNBPMBTMôLBÀUS E \" 4 # 1 $ 2 % 1 & 2 15 5 5 3 15 õFLJMEFLJ OPLUBEBO ÑOÑ LÌöF LBCVM FEFO EÌSUHFOMFS BSBTOEBO TFÀJMFO CJS EÌSUHFOJO CJS LÌöFTJOJO\"PMNBPMBTMôLBÀUS )BWB (ÐOFõMJ :BóNVSMV 3Ð[HºSM ,BSM \" 2 # 1 $ 1 % 1 & 1 3 2 34 5 %VSVNV .D]D\\DSPD2ODV×O×NODU× * ** *** :VLBSEBLJ UBCMPEB ZPMB ¿LBDBL * ** WF *** OPMV BSB¿MBSOZPMEBCVMVOBDBóHÐOPMBCJMFDFLIBWBEV- SVNVOVO PMBTMLMBS WF CV PMBTMLMBS EBIJMJOEF LB- [BZBQNBPMBTMLMBSWFSJMNJõUJS #VOBHÌSF CVBSBÀMBSOLB[BZBQNBPMBTMLMB- ôFLJMEF WFSJMFO IFEFG UBIUBTOB BUõ ZBQBO CJSJOJO SOB HÌSF LÑÀÑLUFO CÑZÑôF EPôSV TSBMBOöMBS FOEõCËMHFEFOJ¿FSJZFEPóSVCËMHFMFSJWVSNBPMB- BöBôEBLJMFSEFOIBOHJTJEJS TMLMBSTSBTZMB WFEVS \" *<**<*** # **<*<*** $ **<***<* % ***<*<** & ***<**<* \"UöZBQBOLJöJOJOLPZVCÌMHFMFSJWVSBNBZBDB- ôCJMJOEJôJOFHÌSF FOJÀCÌMHFZJWVSNBPMBTMô LBÀUS \" 1 # 2 $ 7 % 2 & 7 10 15 25 7 20 #JSJNLBSFMFSJOLËõFMF- SJOEFWFSJMFOOPLUB- EBOUBOFTJTF¿JMJZPS #JSUPSCBEBTBS MBDJWFSUWFCFZB[CJMZFWBSES 5PSCBEBOZFSJOFLPONBLT[OBSUBSEBCJMZF¿FLJ- MJZPS ¦FLJMFO CJMZFMFSJO GBSLM SFOLUF PMNB PMBTMô #V OPLUBMBS BSBT V[BLMôO CS PMNB PMBTMô LBÀUS LBÀUS \" 1 # 2 $ 3 % 4 & 2 \" 3 # 9 $ 9 % 1 & 15 25 25 50 75 75 54 56 28 2 28 A C E 29 B D C
YAZILI SORULARI 0MBTML #JSUPSCBEBCFZB[ NBWJCJMZFWBSES#VUPSCB- #JSNBEFOJQBSBOOLF[IBWBZBBUMNBTEFOF- EBO¿FLJMFOCJMZFHFSJLPONBNBLÐ[FSFBSUBSEBJLJ ZJOEFÑTUZÑ[FFOB[LF[ZB[HFMEJôJCJMJOEJôJ- CJMZF¿FLJMJZPS OFHÌSF LF[ZB[HFMNFPMBTMôLBÀUS ¦FLJMFO CJSJODJ CJMZFOJO CFZB[ JLJODJ CJMZFOJO YYYTTT + YYYYTT + YYYYYT + YYYYYY NBWJPMNBPMBTMôLBÀUS 6! 6! 6! #JSJODJCFZB[ JLJODJNBWJPMEVôVOEBOTSBTCFMMJEJS#V- + + +1 43 2 3! .3! 4! .2! 5! OBHÌSF · = PMVS 76 7 + 15 + 6 + 1=PMVS A = {0, 1, 2, 3, 4}LÐNFTJOJOFMFNBOMBSLVMMBOMB- ÷TUFOFOPMBTML= YYYYYT 6 1 = = PMVS SBLFMEFFEJMFCJMFDFLUÐNпCBTBNBLMTBZMBSCJ- 42 42 7 SFSLBSUBZB[MQCJSUPSCBZBLPOVZPS &ËSOFLV[BZOEB\"WF#PMBZMBSWFSJMJZPS 5PSCBEBO SBTUHFMF TFÀJMFO CJS TBZOO ÀJGU PM- EVôVCJMJOEJôJOFHÌSF SBLBNMBSOOGBSLMPMNB 1 \"h = 1 , P^ B h = 1 WF1 \"b# = 3 PMBTMôLBÀUS 23 4 &WSFOTFMLÑNFÀJGUÑÀCBTBNBLMTBZMBSES |PMEVôVOBHÌSF 1 \" # EFôFSJLBÀUS 4 5 3 = 60 0 11 ÷TUFOFOSBLBNMBSGBSLMPMBOES P(A') = ise P(A) = 2 4 22 4 3 1 + 3 3 2 = 30 02 P(A b B) = P(A) + P(B) - P(A a B) 4 30 1 53 1 P(A a B) = - = PMVS JTUFOFOPMBTML= = PMVS 60 2 6 4 12 öLJUPSCBEBË[EFõCJMZFMFSWBSES#JSJODJUPSCBEB 1 NBWJ CFZB[CJMZF JLJODJUPSCBEBNBWJ CFZB[ 1 \"=# = P^ A + B h 12 1 CJMZFWBSES#JSJODJUPSCBEBOCJSCJMZF¿FLJMJQJLJODJ == UPSCBZBBUMZPS÷LJODJUPSCBEBOBMOBOCJSCJMZF- P^ B h 14 OJONBWJPMNBPMBTMôOBôBÀEJZBHSBNZÌOUF- NJJMFCVMVOV[ 3 4 ôFLJMEFLJ EJLEËSUHFO CJ¿JNJOEFLJ IFEFG UBIUBT Fõ 9B 2B LBSFMFSEFOPMVõNVõUVS 5 5M 9 #VIFEFGUBIUBTOBBSUBSEBBUöZBQBOCJSBU- DOOJLJBUöUBEBIFEFGJWVSEVôVCJMJOEJôJOFHÌ- 3 SF FOB[CJSJOEFUBSBMCÌMHFZJWVSNVöPMNBPMB- 3 9B TMôLBÀUS 5M 1 - BUöUBWVSBNBNBPMBTMô 6M 9 16 16 22 45 1- · =1- · =1- = 2 5 3 6 28 24 24 33 99 ·+·= 5 9 5 9 45 2 1 28 1 1 5 7 2 45 7 4 9
0MBTML <(1m1(6m/6258/$5 1 ôFLJMEFLJ ZFM EFóJS- 10 m B A 2 NFOJOJOLPMMBSGBSL- 12 m 12 m 6 M SFOLMFSEF CPZBO- File Nõ WF LPMMBSB CJSFS :VLBSEB FOJ N WF CPZV N PMBO EJLEËSUHFO OVNBSB WFSJMNJõUJS CJ¿JNJOEFCJSUFOJTLPSUVWFSJMNJõUJS 3Ð[HBS CFMMJ CJS I- [O BMUOB EÐõUÐ- r \" OPLUBTOEBLJ UFOJT¿J LFOEJ BMBOOO BóSML óÐOEFZFMEFóJSNF- NFSLF[JOEF CVMVOBO # OPLUBTOEBLJ UFOJT¿JZF 5 4 3 OJEVSNBLUBES)FS CJSTFSWJTBUõZBQBDBLUS A EVSEVóVOEB LPMMB- SOEBO CJSJ \" OPLUB- r #OPLUBTOEBLJUFOJT¿J LFOEJ¿FWSFTJOFNWF TOOIJ[BTOBHFMNFLUFEJS:FMEFóJSNFOJHÐOJ¿JO- NEFOEBIBB[V[BLMLUBEÐõFOUPQMBSLBSõ- EFEFGBEVSVZPS MBZBCJMNFLUFEJS #VOBHÌSF \"OPLUBTOEBEVSBOLPMMBSOBSEöL OVNBSBMPMNBPMBTMôLBÀUS \" 5 # 1 $ 2 % 1 & 5 36 6 9 4 18 #VOB HÌSF TFSWJT BUö ZBQBO \" OPLUBTOEBLJ UFOJTÀJOJOUPQVLBSöBMBOBEÑöÑSEÑôÑCJMJOEJôJ- OFHÌSF QVBOBMNBPMBTMôLBÀUS \" 1- π # 1- 2π $ 1 - π 15 15 5 % 1- π & 1- π 3 2 Okul Ev AB .VSBUZPMVOCBõOEBLJFWJOEFO PLZËOÐOEF HF¿UJ- LBUMCJSBQBSUNBOO\"WF#HJSJõLBQMBSWBSES óJZPMEBOUFLSBSHF¿NFNFLÐ[FSFZPMB¿LZPS.V- \"LBQTOEBOHJSEJLUFOTPOSBBTBOTËSUFLOVNBSB- MLBUMBSB #LBQTOEBOHJSEJLUFOTPOSBBTBOTËS¿JGU SBUhO TPMVOB EËONF PMBTMó 1 TBóOB EËONF OVNBSBMLBUMBSBHJUNFLUFEJS 3 r \"WF#LBQMBSOEBHJSJõõJGSFTJUÐS PMBTMó 1 WFEÐ[HJUNFPMBTMóEB 1 ES r )FSJLJLBQEBLJUVõMBSOUVõVCP[VLUVSWF 1 26 3 PMBTMLMBUVõVOVBMHMBZQ 2 PMBTMLMBUVõV 3 BMHMBNBNBLUBES \"ZSDB \" LBQTOEBO HJSJMEJ- óJOEFLJ BTBOTËSÐO EF UVõV BZO LVSBMB HËSF CP[VLUVS r \"LBQTOEBOHJSFO\"INFU#FZ LBUUB #LBQ- TOEBOHJSFO0SIBO#FZEFLBUUBPUVSVZPSMBS #VOB HÌSF .VSBUhO PLVMB HJUNF PMBTMô LBÀ- ÷LJTJOJOEFJMLEFOFNFEFEBJSFMFSJOFVMBöNBPMB- TMôLBÀUS US \" 1 # 1 $ 1 % 1 & 1 \" 1 # 1 $ 1 % 1 & 1 36 18 12 6 3 3 9 27 81 243 B B 31 E C
<(1m1(6m/6258/$5 0MBTML y A B CDE A(–2, 2) D F G x H O B C(14, –6) A(–2, 2) y x ,-MN B C' ôFLJMEFLJPUPQBSLUBIFSQBSLZFSJCJSIBSGJMFHËTUF- D SJMNJõUJS 0UPQBSLB HJSFO п BSB¿ CV QBSL ZFSMFSJOF O SBTUHFMFQBSLFEFDFLMFSEJS %JL LPPSEJOBU TJTUFNJOEF HËTUFSJMFO LFOBSMBS FL- #VOB HÌSF QBSL FUUJLMFSJ IBSGMFSJ CJSFS EPôSV TFOMFSF QBSBMFM \"#$% EJLEËSUHFOJ õFLMJOEFLJ CJS JMF CJSMFöUJSEJôJNJ[EF ÑÀHFO PMVöUVSNB PMBTMô LºóU QBS¿BT #% EPóSVTV CPZVODB $ OPLUBT $h LBÀUS OPLUBTÐ[FSJOFHFMFDFLõFLJMEFLBUMBOQHFSJB¿M- \" 27 # 31 $ 9 % 10 & 41 ZPS 44 44 11 11 44 #VOB HÌSF EJLEÌSUHFO JÀJOEFO TFÀJMFDFL CJS OPLUBOO ÑTU ÑTUF HFMFO CÌMHFEF CJS OPLUB PM- NBPMBTMôLBÀUS \" 5 # 5 $ 1 % 7 & 1 18 8 3 18 2 4D C AK B ôFLJMEF ¿FNCFS CJ¿JNJOEFLJ CJS BSTBOO FUSBGOB 2 2 –2 L ZBQMNõFWHËTUFSJMNJõUJS A ,BSFõFLMJOEFLJCJSNBTBZBEBJSFõFLMJOEFCJSNBTB FB ËSUÐTÐ TFSJMJZPS .BTB ËSUÐTÐ NBTBOO EËSU LFOB- E C | |SOEBOEB KL = 2 2 - 2 CSPMBDBLõFLJMEF NB- D TBOO LËõFMFSJ BSBTOEB FõJU BMBOM EBJSF LFTNFMFSJ PMVõUVSBSBLTBSLZPS | |AD =CSPMEVôVOBHÌSF ÌSUÑÑ[FSJOEFTFÀJ- MFOCJSOPLUBOOTBSLBOQBSÀBZBBJUCJSOPLUBPM- #VFWMFSBSBTOEBIFSIBOHJUBOFTJOJCJSCJSJOFCBó- NBPMBTMôLBÀUS MBZBOEËSUHFOõFLMJOEFZPMZBQMBDBLUS \" 1 - 4 # 1 - 3 $ 1 - 2 #VZPMVOCJSLÌöFTJOJO&OPLUBTOEBLJFWPMNB π π π PMBTMôLBÀUS % 1 - 1 & 1 - 2 π 3π \" 1 # 2 $ 3 % 4 & 5 3 3 45 6 1. A 2. AB 3. AB 32 4. A 5.AE 6. AC
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