TRUONG DAI Hcpc D6NG THAP Tap chf Khoa hoc sd 16 (11 -2015)
\J M O T sfi Ll/U Y VE XAC SUAT c6 DIEU KIEN
• ThS. Nguyen Dinh Inh'*' Tdm xii
Xdc sudt cd diiu kien Id mgt trong nhitng ngi dung ca bdn cda Ly thuyet xdc sudt. Bdi viit ndy phdn tick nhiing hgn ehi trong mgt si cdch tii'p cgn vi xdc sudt cd diiu kien thucmg ggp. Dgc biet, bdi viit dua ra mdt edeh md phdng cho cdng thiie xdc sudt cd dieu kiin trong trudng hgp tong qudt (cdc biin cosa dp khdng nhdt thiit ddng khd ndng ciing nhu sdbiin edsa cdp khdng nhdt thiit hOu hgn).
TUkhda: cdng thiic xdc sudt ed dieu ki$n, mo hinh eo dien, ddng khd ndng.
vdng la 0,6. Tinh xae sud^t ngffdi nay ddu vong 2 bie't anh ta da dau vdng 1.
Bdi todn ndy khdng thudc md hinh cd didn. Thtfc chit cde xae suat ngtfdi dtf thi ddu vdng 1 cung nhtf dau cd hai vdng chi ed dtfdc bdng cdch thdng ke ma khong phai do tinh toan tiT dinh nghia xdc suaft cd dien, d day ta khong bidt sd trtfdng hdp ddng kha nSng co the xay ra hay sd" trtfdng hdp thudn ldi nen khong tbd tfnh trtfc tidp dtfdc xdc sud^t ngtfdi dtf thi dau vdng 2 khi da ddu vdng 1 ndu chi dffng djnb nghia
"b^ng ldi" thuin tffy. Dd "giai quyet", ede tde gia theo trtfdng phdi dinh nghta "b^ng ldi" ([1]) thtfdng phdt bidu thdm cdng thffc (1), dtfdi dang mpt dinh 1;^ vd "chffng minh" nhtf sau:
Xet trudng hgp khdng gian mdu gdm hOu hgn biin c6 sa cdp ddng khd ndng (md hinh ed diin), ggi sd trudng hop ed thi xdy ra, so trudng hgp thudn lgi cho biin co A, sd trUdng hgp thugn lgi eho biin cd' B, sd trudng hgp thudn lgi cho biin cd AB trong phip thit ldn lugt Id n,n^,ng,n^. Do dd, trong tinh hud'ng bii'n cd A da xdy ra. sd trudng hop cd the xdy ra bi thu hep thdnh n^ cdn sd trudng hgp thugn lgi cho B chinh Id n^ nin xdc sudt cda biin cd B tinh trong diiu ki$n bii'n cd A da xdy ra se
P(B\A)= (1)
1. Hai each t r i n h bay vd xac suS't cd didu kidn thtfdng gdp
C^c tdi Udu vd xdc sud^t thtfdng dinh nghia xde su^t ed didu kidn bang mdt trong bai cdch sau ddy:
Cach thi? nh^t: Xdc sudt cua bii'n cd B tinh trong diiu ki$n bii'n ed' A da xdy ra dugc ggi Id xdc sudt cua biin cd B vdi diiu kien bii'n ed'A vd ky hi$u la P(B\A).
Cach thrf hai: Cho A Id biin eo ed xdc sudt ldn han 0. Xdc sudt cda bii'n cd B vdi diiu ki$n biin cd A, k^ hi$u ldP(B\A), Id so duge xdc dinh bdi edng thdc:
P(AB) P(.A)
Cdng thUc (1) ggi la cdng thUe xdc sudt cd diiu ki$n.
Ta tam gpi dinh nghia theo edeh thff nhat Id dinh nghia bang •$ nghia, hay dinh nghia
"bSng ldi", dinh nghia theo cdch thff hai Id dinh nghia bdng edng thffc.
Dinh nghia "bdng ldi" dam bdo tinh trtfc quan, the hidn dffng ban chS't effa xdc suat cd didu kidn nhutig ehi dp dung dtfde eho mdt sd bdi todn ddn gian vdi mo hinh xac sud^t cd didn.
Trong nhidu trtfdng hdp, dffng dinh nghia thufin tdy bdng ldi khdng gidi quydt dtfdc, chang han dang bdi sau ddy:
Vi dy 1. Mpt ngtfdi tham dtf 2 vong tbi.
Khd ndng ddu vdng 1 la 0,8; kha ndng dau cd 2
bdng ^ . Mdtkhdccung cd PJ^^) ^ n ^t^As
"A PiA) n^ n^
nin xdc sudt cua biin cd B tinh trong diiu kien '''TrueiDg Dai hoc Cong nghiSp Thtfc ph^m Thanh phS"
Ho Chf Minh.
biin cd A dd xdy ra chinh Id P(AB) P(A) '
TRUONG DAI HOC £X3NG THAP Tap chi Khoa hpc so 16 (11-2015) phii cua (1). Quay ve Vt du 1. ggi AvdBldn
Um Id cdc biin cd ngodi du tM ddu vang 1 vd 2, dp dung cdng thdc (1) cd xdc sudt cdn tinh Id
^ ' ' P{A) 0,8
Ta thay stf bd't hdp 1^ trong edeh giai quydt trdn: chi gidi thieh dtfde cdng thffc trong md hinh cd didn nhffhg lai dp dung cdng thffc dd gidi bdi todn khdng thupc md hinh ed dien.
Ddi vdi cdch tiep can thff hai (dinh nghia xdc sud't cd didu kidn bdng edng thffc), b^t hdp li ditu tidn Id nd khdng giai thich dtfdc ban eh^t effa xdc su^t cd didu kien. Tai sao xac sud't xdc dinh bdi cdng thtfc (1) lai mang ^ nghia la xdc su^t effa bidn ed B vdi didu kidn bidn cd -4 da xdy ra? Cdc tde gid trinh bdy xac suat cd didu kidn theo edeh ndy thffdng dSn d^t ra edng thffc (1) bdng mpt vi du md d i u thudc md hinh cd didn rdi dp d^t xdc suat cd didu kidn trong mpi md hinh ddu dinh nghia bdng edng thffc (1) CI3], [4], [6], [7]). Cfing ed tdc gia ([5]) khdng dilng vi du md d i u md dtfa ra ngay dinh nghia theo cdng thffc (1), sau dd cd g^ng li giai J nghia eiia c6ng thffc (1) nhtf sau: biin cdA cdng vdi biin cdB chinh Id biin cdAB tdc Id "cd hai ciing xdy ra'. Ta cd thi coi A vd B Id hai tap con cda mdt khdng gian xdc sudt H ban ddu.
Cdc tdp con (do dugc theo dg do xdc sudt P) cua A chinh Id cdc biin cdvdi dieu kiin A duge thda man. Khi dgt diiu ki$n A tUe Id dd hgn che khdng gian xdc sudt tU CI xudng cdn A vd hgn chi biin cdB xudng cdn AB. Xdc sudt cda B vdi diiu Hfn A chinh Id xdc sudt cua AB trong khdng gian xdc sudt mdi vdi mgt do do xdc sudt P,:P{B\A) = P^(AB). Trong dd, do do xdc sudt / | dugc sinh ra bdi dg do xdc sudt P ban ddu theo nguyin tdc 'binh qudn': niu C vd D Id hai t4p con cua A (tUc hai biin cd thoa mdn diiu ki^n A) vdi cdng xdc xudt P{C) = P{D), thi cOng phdi coi rdng chiing cd ciing xdc sudt cd
dieu kien P^(C) = P^{D). Mdt cdch tong qudt hem, niu CvdDld hai tgp con cua A, ta cd cdng thiic ty 1$ thudn
PiC)^P,(C) P{D) P,{D) TUdd suy ra
PiAB).m^ = M ^ = P^(AB).PiB\A) P(A) PM) 1 A ^ V ! I
Cdch li giai ndy dffng tdi If thuydt dd do vd dinh nghia xdc sud't tdng quat theo hd ti6n dd. Tuy nhien nd khdng thuyd't phuc vi nguygn tde "binh qudn" xdc sud^t hodn todn do tdc gii tff ddt ra. Ndi chung, khdng the dffng dioli nghia xdc sulft theo tien de giai thich ^ ngbia cua cdng thtfc (1), bdi dinh nghia xde sud^t theo tidn dd khdng he ndu cdng thffc tinh cu thd cbo xdc suat md chi phdt bieu xac suat Id b^t k^
hdm tap ndo thda man cdc dieu kidn effa he tidndd.
Mdt khdc, ngay ed khi chffng minh (trong trffdng hdp long qudt) dffdc xde su^t effa bidn cd B xdt trong didu kien bid^n cd A da xay ra bdng vd phai cua (1) thi cung khdng thd suy ludn ngffcfc chidu rdng xde sud^t xde dinh bdi vd phdi ciia (1) Id xdc suat eua B vdi di^n kien A. Didu ndy cung gidng nhtf didn tich hlnh thang cong Id mdt tich phdn xdc dinh nhffhg khdng the ndi moi tich phdn xde dinh, b^t kd am hay dffdng, deu la didn tich cua mot hinh thang cong.
Nhff vdy, cdng thffc (1) khdng bao h&m dffdc bdn chat cua xdc suat cd didu kien, dilng cdng thffc (1) dd dinh ngMa cho xdc sud't cd didu kidn chi ldm eho nd cd ve "todn hoc"
nhitag lai khdng cd tinh trffc quan. Vide xSy dtfng cd sd todn chdt che cho li thuydt xdc su^l Id c i n thid't, tuy nhidn khong ndn "todn hoc hda" qud mffc ldm m^t di linh trtfc quan cia van dd.
Ngodi ra dinh ngMa theo edng thffc cdn ban chd Id khdng xdt dffdc trtfdng hdp P{A) = 0: cdng thffc (1) chi cd nghia trong
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TRtrdNG DAI HQC D 6 N G T H A P Tap chi Khoa hpc s6 16 (11-2015)
P(B\A) =
trtfdng hdp P(A) > 0 trong khi xdc sud't cd dieu kidn v i n tdn tai trong trffdng hdp didu k i d n ^ cd xdc suit bdng 0. Chffng ta xdt vi du sau day:
Vf di^ 2. Chpn n g l u nhidn mpt didm trong midn phlng gidi ban bdi hinb vudng MNPQ.
Thih xdc su^t didm ndy ndm trdn canh MN ndu bidt nd ndm trdn mot trong bdn canh cua hinh vudng?
Giai
Goi A Id bidn eo chpn dffde didm tren canh hinh vudng, B Id bidn ed ehpn dtfdc didm trdn canh MN. Ta e i n tMi P(B\A).TAC6
/ \ _ "didn tich" cde canh
"didn tich" hinh vudng (Ltfu ^ rdng bie'n c6 A d ddy cd xdc suat bdng 0 nhuiig v i n cd thd xay ra, ta gpi bidn cd dang ndy Id bidn cd^ hdu khdng). Tiep theo ta t i n h P ( 5 | ^ ) :
dp ddi canh MN _ 1 tdng dd ddi 4 canh hinh vudng 4 Vide khdng xdt dtfcfc xdc su^t effa B vdi didu kidn A khi P(A) = 0 khdng chi ldm "mit trtfdng hdp" md cdn gdy ra nhffng ke hd trong cdc edng thffc lidn quan, chIng ban cdng thffc nhan: P(^AB) = P(A).P(B \ A). Ro rdng, cdng thffc nhan v i n dffng trong trffdng hdp P(A) = 0. Vi vdy, ndu khdng xdt dffdc P{B\A) trong trtfdng hdp P(A)=0 md v i n phdt bidu edng thffc nhan cho hai bidn cd A,B bat k j Id khdng phff hdp.
2. Mdt each md phong cdng thtfc xac su^t cd didu kidn
Nhtf da phdn tich, edeh trinh bdy thff nhat vd xde sud^t ed didu kidn chi dp dung dtfdc vdi md hinh ed didn cdn cdch thff hai thi khong lot td dtfdc bdn chSit effa xdc sud't cd didu kidn. Van dd se dffde gidi quydt bdng cdch md phdng cdng thffc (1) trong tntifng hdp tdng qudt.
Trtfdc bet ta nhdc lai mot dinh nghia xdc sud't theo t i n sud't (dinh nghia th^ng kd): xac
sud't cua bien ed A chinh la gidi han cua day t i n su^l xuat hidn bidn co^ A khi s6' i l n thff d i n ra vd ban (ludt sd ldn Bernoulli da cho th^y gidi ban ndy tdn tai); tffe Id
P ( ^ ) = l i m / ( ^ ) = l i m ^
"-*^ „^„ „
trong dd n^ Id sd lan xuaft hidn bidn ed' A trong
« lan thff.
Ve 1^ thuydt, xac suat effa mpi bidn ed deu cd thd xae dinh bdng dinh nghia ndy (xac sud't ngffdi dtf Ihi ddu vdng 1 vd dau cd 2 vdng trong Vl du 1 cung chi dtfdc tim bang thdng ke). Ta se dffng dinh nghia xdc sud't theo t i n sud't dd md phdng cong thffc (1). Thdy vdy:
Thtfe hien phdp thff n iln, gpi n^,nB>"AB ttfdng ffng Id sd i l n edc bien cd A, B, AB xay ra. Ta ed:
P{A) = lim ^ ; P(B) = lim ^ ; P{AB) = hm ^ Neu bien cd A da xay ra, tffe chi xdt trong n^ i l n ed A xuat hidn. Khi dd sd iln B xnit hidn chinh Id sd i l n cd ^ vd 5 xu^t hidn, dd chinh Id rt^. Vdy xdc sud't effa bidn cd B vdi didu kidn A Id:
P(B\A)= l i m ^ ^ .
Tff hd thffc
cho « -> 00, (dd J rang day w^ khdng gidm vd P(A) = l i m ^ ^ > 0 ndn w^ ^ x ) ta dffdc:
lim
l i m ^ ^ '^AB _ "-'"' n
hay
P(B\A). P(AB) P(A)-
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TROONG DAI HOC £ ) 6 N G T H A P Tap chi Khoa hoc s o 16 (11-2015]
3. K^t luSn n^y se dam bao tinh tnrc quan v^ logic, Tif n h ^ g phSn tich tren, chung ta co the Tren day la mpt v4i di^m can liIu ^ ve xic trinh b&y v l xac suat c6 dieu kien nhU sau: su^t c6 d i l u kiln. Hy vpng b&i viet n^y phin phat bieu dinh nghia "bang ISi" sau do phat nao giup cho ban dpc quan t i m tdi L^ thuydt bieu cong thlJc (i) va mo phong bang dinh xac su^t CO each tiep cSn chat che, logic htfti nghia xic su£t theo tin sua^t. Cach trinh bay ve x i c su§ft co dieu kien,
Tai lilu tham khao
[ 11. La SI Bong (2010), Xdc sudt Thdng ke vd ling dung, NXB G i i o Due.
[2]. W. Feller (1971), An introduction to probability theory and its applications, volume I third edition, John Wiley and Sons, New York.
[3]. M. Lolve (1977), Probability Theory 1,4" edidon. Springer - Veriag.
[4]. Sheldon M. Ross (2004), Introduction to probability and statistics for engineers and scientists, third edition, Elsevier Academic press.
[5]. Do DiJc Thii, Nguyen Tien Dung (2010), Nhdp mon hien dgi xdc sudt vd Ihdng ke, NXB Dai hoc SU pham Ha Npi.
[6]. DSng Hung Thang (2009), Md ddu vi li thuyet xdc sudi, NXB Gido Due.
[7], Nguyen B4c VSn (1997), Md ddu Ihdng ki xdc sudt, NXB Gido Due.
SOME NOTES ON C O N D m O N A L PROBABILITY Summary
Conditional probabiUty is one of the basic contents of the Probabihty Theory. This paper analyzes limitations in some approaches to the common conditional probabiUty. Especially, it offers a simulation for the condiUonal probabihty formula in general (the primary events are iiol necessarily of the same ability and the number of primary events is not necessarily finite).
Keywords: conditional probability formula, classical model, the same ability.
Ngdy nhdn bdi: 03/9/2015; Ngdy nhdn lgi: 12/10/2015; Ngdy duyet ddng: 23/10/2015.
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