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N T The T T H Lam/ Ren luyen ndng lUc giai toan cho smh vien ngd,nh suphgrn, toon REN LUYEN NANG LlJC GIAI T O A N CHO SINK VIEN N G A N H STJ P H A M TOAN HOC T R O N G DAY H O C K I E N THLfC XAC SU[.]
N T The T T H Lam/ Ren luyen ndng lUc giai toan cho smh vien ngd,nh suphgrn, too R E N L U Y E N N A N G LlJC G I A I T O A N C H O SINK V I E N N G A N H STJ P H A M T O A N H O C T R O N G DAY H O C K I E N THLfC XAC SUAT Nguyin Thi The, Thai Thi Hdng Lam Khoa Sii pham Toan hoc, Trudng Dai hoc Vinh Ngay nhan bai 28/7/2016, nhan dang 19/11/2016 Tdm tat: Bai bao dg cap den viec ren luygn nang luc giai toan cho sinh vien nganh Su pham Toan hoc day hpc mon Xac suat th6ng qua viec to cbiic, budng dan sinb vien khai thac kien thiic vh xac suat dg giai mot so bai toan tbong Mc( dau Trong Taxa K hpc, nang lUc la dac digm tam If cua ca nhan phd hop vdi nhiing yeu cau cua mOt boat dong nbat dinh, dam bao cbo lioat d6ng co k^t qua [6] Do co the hieu nang luc giai toan (NLGT) la dSc diim tam 11 ca nhan ciia ngUcii dap ilng dupe yeu cau cua boat dgng giai toan va la dieu kien cS,n thilt dg boan t6t boat dgng giai toan Mot ngUcfi duoc coi la c6 nang lUc giai toan ngu ngubi nam viing tri thiic toan hoc, c6 ki nang, ki xao cac boat dgng giai toan Nhu v|Ly, NLGT la mot bS phan quan tao nen nang luc toan hpc cho mOi sinh viSn sU pham va dUOc the hien thong qua be thong cac kl nang boat dgng giai toan Ren luyen NLGT chigm vi tri quan trpng d^y bpc cac hpc phan m6n Toan ca vg phuong dien dac thu va khoi lupng thdi gian Ren luyen NLGT cho sinh vien sU pham Toan se giiip sinh vien nam viing kign thiic va bigt van dimg kien tbiic da hpc vao giai toan, dong thSi gop phan ren luyen nang luc nghe nghigp cho cac em, giup cac em sau trudng day toan t6t hon Tim bilu nhiing kho khan cua giao vign Toan pb6 thong van dung cac kiln thiic xac suat da dupe hpc d trUcfng Dai hoc vao day bpc xac sudt ciing nbu giai cac bai toan co be ngoai khong lien quan d6n xac suat, chung toi nhan thay: kha nhigu giao vign toan chua higu thUc sU cac khai niem cd ban va cac tinh chat cua xac suat dupe hpc d trudng Dai hpc, nhat la ap dung H thuygt dg giai cac bai tap; kbo khan viec van dung cac phudng phap giai cac bai toan xac suat d Dai bpc vao day hgc giai bai tap phan xac suat d THPT; Dac biet, nhilu giao vign ph6 tbong chua biet hoac chUa co y thiic khai thac kien thiic xac suat de giai cac bai toan Dai s6, Hinh hgc, Giai tich TtC do, khong dam bao duoc tinh tich hop npi m6n day bpc mon Toan trucing phd thong, Nhu vay, viec ren luygn NLGT cbo sinh vign Su pham Toan bgc thong qua day hgc xac suat d trUcfng Dai bpc c^n dupe quan tam Bai vilt dg c$.p den viec ren luyen NLGT cho sinh vien Su pbam Toan hoc tap mSn xac suit, cii the la khai tbac m6t so kiln thiic ve xac suat de giai cac bai toan npi bo xac suit ciing nbu nhflng bai toan co ve be ngoai khong lien quan den xac suat Pham vi ren luyen NLGT tbeo ba trpng tam: ki ' Email: nhanhthe@gmail.com (N T The) Tntdng Dai hoc Vmli Tcip chi khoa hoc, Tap 45, Sd 3A (2016), tr 9-2 nang dinb budng tim phUdng phap giai; ki nang thuc bign Idi giai; ki nang khai thac, ling dung kit qua phUdng phap giai D&i vdi mOi kl nSng can ren luyen cho sinh vign nhiing thao tac ki thuat cu the khai thac bai toan Day hoc xac suit cho sinh vien nganh Sit pham Toan hoc theo hudng gan vdi viec nang cao tay nghe day hoc Viec doi mdi phUdng phap bign xac dinh giao vien co vai tro la ngUdi thigt kg, iiy tbac, dilu khign va thi chi hoa qua trinh nhan thiic cua hgc sinh Van de quan trgng la qua trinh dieu khign giao vign phai bilt dinh hudng giup hpc sinb phat hien each thiic giai quylt vin dg, tim toi kiln thiic mdi, ciing nhu danh gia va thg chi boa kien thiic Dl giai quygt duoc van dl do, giao vign phai diing tren tam cao dg nhin nhan vin dg thSng, ngoai viec khai tbac trigt dl tai lieu sach giao khoa phai bilt sii dung cac kiln tbiic toan da dupc bpc d trudng dai hpc vao viec nhm nhan kien tbiic toan lien quan d pbo tbdng mot each toan dien hOn Chinh vi vay cac muc tigu cbinh cua viec day hpc hgc phan xac suat cho sinh vien sU pham nganh Toan d trudng dai bpc la: +) Trang bi cbo sinh vign cac kiln thiic cd scf, nang cao vg xac suat; -I-) Ren luyen cho sinh vien ki nang van dung cac kiln tbiic da hgc vao giai cac bai tap Cling nhu giai quylt cac van dg thiic tiln; -i-) Lam cho sinh vign bilt dua vao cac kiln tbiic xac suat d trudng Dai hgc dl nhin nhan cac kiln thiic xac suit d php thong mot each day du va sau sic bdn; +) Lam cbo sinh vign bilt van dung cac kiln thiic xac suat d trirdng Dai hoc dl dinb hudng tim toi Idi giai cho cac bai toan xac suit d tbong, cung nbU cac bai toan cac phan mon khac nhu dai so, hinh hoc, giai tich; +) Lam cho sinb vien bilt van dung cac kiln thiic xac suit da hoc trudug Dai hpc dl sang tao cac bai toan d thong Ren luyen NLGT cho sinh vien nganh su* pham Toan hoc Ren luyen NLGT la viec giao vign thUc hien chii dong viec luyen tap cac ki nang giai toan cho sinh vign dat din sU thao co tinh sang tao Trong ren luyen NLGT, sinh vign can thuc bign cac giai doan khac nbau gOm: xac dinh hufing giai, tim kiem Idi giai, thuc hien giai va ngbien ciiu, khai tbac ling dung ket qak giai Nhan thiic ve xac dmh bitdng giai phai gdn v6i viec nam vflng cac dang, cac loai bai toan, pham vi kiln thiic trUc tilp va gian tilp co lign quan Tim Idi giai la xac dinh cac pbUdng phap giai khac theo cac dinh hudng giai Viec tim dudc phUdng hudng giai din viec giai hoan cbinh bai toan la ca mot qua trinh, tit sii dung chinh xac cac kiln thiic cd ban vl npi dung ly thuylt din viec ap dung, chuyln dOi tbanh thao cac quy trinh va cac thao tac co tinh ki tbuat Nhung dinh hudng tim Idi giai co tinh chat quyet dinh toan b5 cdng viec giai toan, dinh hUdng chinh xac va hieu qua thg hien kba nang nam bat kiln thiic va kha nang tu Nghien cdu, khai tbac kit qua giai se hinh cbo sinb vien y thiic nhin nhan lai qua trinh giai, danh gia phudng phap giai va phat hien kit qua mdi Ren luyen NLGT la ren luygn kha 33 A^ T The, T T H Lam/ Ren luyen nang lUc giai toan cho sinh vien nganh sit pham toan , nang sang tao tim Idi giai va ren luygn cac kl nang, thao tac, ki thuat giai Vigc ren luyen NLGT cho sinh vien ng^nh sU pham Toan khOng chi la kit qua giai bai toan ma doi boi tiiib sang tao viec tao cac bai toan mdi tii bai toSn ban dau, ciing nhu the hien d vigc tim each giai doc dao cho bai toan Ren luyen NLGT cho sinh vien nganh sU pham Toan hoc day hoc kiln thiic xac suat 4.1 Nhiera vu cua giao vien Cung mot noi dung day bpc, cac giao vien c6 thi thilt kl cac tinh huSng day hpc khac Tuy nhien, theo chiing toi dl ren luygn NLGT cho sinh vign sU pbam Toan thdng qua day hpc mot ngi dung cu thg vl xac suat, giao vien cin phai: - Xac dinh ro kiln thiic, Id nang, thai va tu sinh vign cin dat qua trinh day hpc npi dung do; - Can cii vao ndi dung day hpc, xay dung cac bai toan ndi bp mdn Xac suat va cac bai toan thudc cac phan mdn khac co thi giai dUpc bang viec sii dung cac kiln thiic vl xac suit viia dUdc hpc, giao vign cin quan tam hai dang bai toan: dang bai toan thuan chuong trinh dai hgc va dang bai toan gan giii vdi chUdng trinh xac suit d pbo thdng; - Dieu kbidn boat dong giai toan cua sinh vien; - Lap kl hoach danh gia ket qua giai toan cua sinh vign 4.2 Nhtem vu cua sm/i vien Dg giii dupe bai toan giao vign yeu ciu, qua dat dUdc muc tieu day hgc ma giao vien da dat hpc mdt noi dung nao dd, sinh vien cin thuc liign theo quy trinh sau: Budc 1: Phan tich cac diu hieu cua bai toan cd lign quan din cac ddi tUdng khdng gian xac suit, Budc Xay dung md hinh khong gian xac suit va ddi tUdng tbich bdp cbo bai toan; Budc 3: Giai bai toan ndi bd kiln thiJc xac suit; Budc 4: Ngbign ciiu sau kit qua (tim each giai khac cua bai toan; tim bai toan tong quat, bai toan dac biet, bai toan tUdng tu, ) Luu y ring, nlu bai toan la bai toan thuan xac suit thi sinh vien kbdng cin thdc hien Budc va Budc 4-3 Mot sd vi du mm/t hoa cho viec ren luyen NLGT cho sinh vien nganh Sit pham Toan HQC day hoc hin thiic xac suit Muc nham neu mdt so bai toan minh hoa cbo quy trinh da neu trgn Chang ban, sau day cbo sinh vign cac kiln thiic cO ban vl biln c6; bg diy d i cac biln co; , giao vien CO the yen cau sinb vign giai bai toan sau Chiing minh ring vdi cac sd tu nhien m,A;,ni, ,nm,{m,}z > 1), ta cd 2^ ^ni^^.^ (.Tt.ri, .,rm]eN"',Ti+r2+-+r„=k C^2 = ^ni+n2+-+nm- Bai toan nlu khai triln true tilp di giai thi se kha khd khan Tii dd budc sinh vien Trudng Dai hoc Vmh Tap chi khoa hoc, Tap 45, Sd SA (2016), tr 9-i phai suy nghi din vigc biln ddi bai tofin bai toan tUdng duong sau ^ ^ ^ ^ ^ = E (1) {rur2, ,rm)eAf"',J-i+T-2+-+7-m=fe ^"1+^24- •+1m Giao vign cd thg td chlic hudng din sinh vign giai bai toan nhU sau: Budc 1: Phan tich cac diu hieu cua bai toan co lien quan din cac doi tUdng khong gian xac suat Ta thiy vl trai ciia (1) la tdng cua cac sd hang khdng am va cd cic ciii sd vet hit cac klia nang cd thd Tdng cac s6 lai bang Tii dd gpi cbo sinb vien lien tUdng din xac suit cua nhdm day dii cac bien co Van de dat la xay dung md hinh xac suit nhu thi nao cho phii hop? Budc 2: Xay duug md hinh khdng gian xac suit vh, ddi tUdng thich hop cho bai toan Dua vao sU xuat hien ciia cac sd hang cd dang — | —, cbiing ta xem man sd chinh la sd each cbgn k phan tii tap hop co ni -I- n2 H h Jim phan tii, xem tii so la s5 each chpn lien tilp ri phan tii tif tap rzi phin tii, , chon Vk phan tii tii tap rifc phin tii Do ta cd thg xay duog mo hinh phep thii nhu sau: Xet mdt bdp chiia ni qua ciu loai 1, 7^2 qua ciu loai 2, , Ji„, qua ciu loai m Tii h6p lay k qua ciu Budc 3: Giai bai toan npi bd kiln thiic xac suit: Ung vdi mdi bp k so nguyen khdng am (ri,r2, ,rniy,ri+r2-\ \-rm = k, ta cd mdt biln co vdi xac suit la ^ni+n2+- +nm Cac bien cd tren lap ho diy dii cac biln cd, dd Z^ {ri,r2, ,T)N"';ri+T2+- +T^=k nk ^ã "l+ô2+ +nm Do dd ta CO dieu pbai chiing minh Nhu vay, dl giai bai toan sinh vign pbai bilt chon phfip thii va kbai tbac tinh chat ciia nhdm diy dh cac bien cd Budc '• Ngbien cilu sau kit qua Bang each xet trudng hpp dac biet ciia bai toan tren chi cd bai chi so n = jzi, m :^ 712, ta cd bai toan sau: Bai toan 1.1: Chiing minh rang vdi < m < A; < TI thi clfil + clfil-^ + • • • + czc^"^ = cl^^ Nlu tCr Bai toan 1.1, xet cbo trUdng hdp m ^ n ^ fc, ta cd bai toan sau: N T Thi, T T H Lam/ Ren luyen nang lUc gidi toan cho smh vi^n nganh siipham toan Bai toan 1.2: Chiing minh rang vdi mgi sd tu nhien n thi Trf Bai toan 1.2, nlu cho n bing mdt gii tri ci;i thi nao dd, ta dUdc cac bai toan dac biet cua Bai toan 1.2 Cung tii Bai toan 1.1, ting vdi m = 1, ta cd bai toan quen thudc sau day Bai toan 1.3: Chiing minb rang vdi cac sd tu nhien < fc < n, ta co Day chinh la dang thiic Pascal quen thuoc d6i vdi hpc sinh sach giao khoa Dai s6 va Giai tich 11 Tuy nhien d day viec chiing minb dUdc xay dung theo each khac Dilu it nhilu tao nen si;r hiing tbu cbo sinh vign qua trinh bpc tap mdn xac suit d trudng dai bpc Nhu vay, tif mdt bai toan tdng quat dUdc giai bang kiln thulc xac suit dupe hpc d triidng dai hoc, sinh vien da dl xuit dUdc mot sd bai toan cbo hoc sinh d trudng phd thdng Qua dd, sinh vign se phan nao bilt each tao bai toan mdi vdi nguon goc xuit phat cua no, tvf dinh hudng dupe each giai cho hpc sinb Vdi bai toan sau day, sinh vign cin bilt xay dUng md binh xac suit phii hdp va bilt van dung tinh chit cua cac biln co dpc lap, biln cd ddi cua mot bien cd Cho x,y la cac sd thuc khong am co tdng bing va m,Ti la cac so tu nhien Chiing minh rang (i-i"r+(i-!/'"r>i (2) Cd thg md ta qua trinh suy nghi va each giai bai toan nhU sau: - Dua vao sir xuit bign ciia hai sd x, y khdng am vk cd tdng bang gdi cbo sinh vien lign tudng din x va y la xac suit ciia hai bien cd ddi - Mat khac, cac so hang d vl trai cua bieu thiic (2) xuat hien tich cua cac so kbdng Sm va khong Idn hdn gpi cho sinh vien nghi din xac suit cua tich cac biln cd doc lap - Hon niia, cin chiing minb t6ng ciia hai so hang ve trai ciia biiu thiic (2) Idn hdn hoac bang Do do, sinh vien se xem mdt so hang la xac xuit cua biln co A, va sd hang lai la xac suit cua biln cd D nao cho D D A Van de dat la cd xay dung dupc mo binh phep thii phii hdp hay khdng Vdi each phan tich bai toan nhU tren, sinh vign cd the xay dUng mo hinh xac suit tUOng thich VA giai bai toan nhu sau: Giasiici,C2, ,Cn la cac ddng xusao cbo vdi moi i = l, ,n, xac suit dl cac ddngxu c, mat sip la x Tung cac ddng xu mot each ddc lap m lin • Goi A la biln co "mCi mot m lin tung, co it nbit mdt ddng xu mat ngiia" Khidd P(A) = ( l - r ' ' ) ' " That vay, goi Aj la bien cd lin tung thii j cd it nhat mdt ddng xu m5t ngiia (j == 1,2, , m) Kbi cac A_, la dpc lap va A = A1A2 .A^ Tmdng Dat hoc Vinh Tcip chi khoa hoc Tap 45, S6 3A (2016), tr 9-20 Suy ra, ^{A) = V{Ai)^{A2) f{A^) Rd rang Aj la biln co Iln tung thii j ca n ddng xu diu sip Do dd IP(ij) = r" Nhu vay, P(A) {\-^Ai)) {\-f{A,n)) • Gpi B la biln co mdi mot dong xu xuit bien it nbit mdt lin sip m lin tung Gpi Di la biln cd ddng xu thii i xuit bign it nhat mot lan sap m lan tung (t = 1,2, ,n) Khi dd cac biln co D^, dgc lap, va D= DlD2 Dn Ma^t khac, ta cd D^ la biln cd ddng xu thii i xuit hign mat ngiia ca m lan tung Do dd, P(A) = ( l - x ) ( l - i ) ( l - x ) - ( l - ; ) " ^ = y'^ Suy ra, n c ) = (1 - j / ^ j a - D ••• (1-»"•) = (1 - »")"• Xet biln c6 doi cua biln cd D: dd la biln cd tdn tai it nbit mdt dong xu xuit bign mat ngiia ca TTI lin tung Nhu vay nlu D xay thi A xay ra, tiic la DC A Theo tinh chit ciia xac suit ta cd 1P(£J) > P(A) Tii dd ta cd (l-3:X + ( l - r r ¥{A)+F{D) > ¥{A) + F{A) Diu bang xay va cbi kbi ¥{D) - W(A) Khi dd n = Bing vide nhin lai each giai bai toan, ta thiy cdc ddng xu dupc ebon cd xac suit xuit hiSn mat sip la nhu (bang x) Tit dd, giao vign cd the dat van dl cbo sinh vign xem xet: nlu xac suit cua cac ddng xu xuit hign mat sip la khac thi sao? Co the xay dung dUdc bai toan t6ng quat nao hay khdng? Bai toan sau day cbinb la mot bai toan tong quat cua bai toan trgn 37 N T Thi, T T H Lam/ R&n luyen n&ng lUc gidi todn cho smh viSn nganh sUphctm toa (IMOBungari, 19S4) Chox^,y^,{^ = 1,2, ,n) Ia2ns6 thuckhdngamsaocboa;,+yi = Chiing minh rang vdi mgi so ti; nhign m, ta deu co (1 - iiX2-x„r+(1 - !/r)(i - »2"')-(i - y") > 1- Viec giai bai toan thuan xac suit ciing chinh la da thuc hien theo qui trinh trgn nhung da bd qua Budc va Budc (va cd tbi ca bUdc 4) Cu thi ta xet bai toan sau Cho A la tap cua tap ^ {1,2, , ri} Hay tinh xkc suit cua biln co E ma tap A xuit hign la doan dau mdt boan vi cua Giai O day phep tbii la viec chon mdt hoan vi cua S Do so kha nang cd thg la TI! Bay gid ta tim so trudng bpp thuan Idi cho biln cd E Gia sii (? = {xi, X2, , Xn) la mdt hoan vi ciia tap {1,2, , n} Ky higu a la s6 phan tii cua A Kbi dd E la biln c6 ma A xuit hien nbu doan diu ciia a, nen {3;i,a;2, • ,Xa} = A Vl vay, so cac trudng hdp thuan loi cbo bien co E la ai{n — a)\ Do dd xac suit cua E la ' ( - ) - ^ ^ = ^Cuoi ciing ta md rong bai toan tren dl chiing minh bit ding thiic Lubell-YamamotoMeshalkin bing phudng phap xac suit Cho 41,^2, , As la cac tap cua {1,2, , n}, va gia sii ring kbdng cd tap Aj nao la tap cua cac tap lai Ky higu a, la sd phin tii cQa tap A, Cbiing minh ring U