Izehlakalo ezizimele ziyizehlakalo ezingenamthelela komunye nomunye, okusho ukuthi ukwenzeka kwesinye isigameko akuthinti ithuba lesinye. Kulo mongo, imiboniso, izibonelo, nokuzivocavoca kungamathuluzi abalulekile okuqonda nokusebenzisa kahle imiqondo ehlobene nemicimbi ezimele. Kulesi sihloko, sizohlola ukuthi singazihlonza kanjani izehlakalo ezizimele, sethule izibonelo ezisebenzayo ukukhombisa ukusetshenziswa kwazo, futhi siphakamise izivivinyo zokuhlola nokwenza ngcono ukuqonda isihloko. Sizojulisa ukuqonda kwethu izenzakalo ezizimele kanye nendlela ezingahlaziywa futhi zisetshenziswe ngayo ezimweni ezahlukene.
Izibonelo zezenzakalo ezizimele: qonda ukuthi zisebenza kanjani futhi ubone izimo ezingokoqobo.
Izehlakalo ezizimele ziyizehlakalo ezingenamthelela komunye, okusho ukuthi ukwenzeka kwesinye isigameko akuthinti amathuba okuthi esinye senzeke. Ukuze uqonde kangcono ukuthi imicimbi ezimele isebenza kanjani, ake sibheke izibonelo ezithile ezisebenzayo.
Isibonelo esilula sezehlakalo ezizimele umqulu wefa fair. Uma sigingqa idayi bese sithola u-4, amathuba okuthola inombolo elinganayo kumqulu olandelayo ahlala engu-1/2, njengoba izehlakalo zizimele zodwa.
Esinye isibonelo esivamile ukupheqa uhlamvu lwemali. Uma siphenyisisa uhlamvu lwemali bese luqhamuka phezulu, amathuba okuthi luqhamuke umsila ku-flip elandelayo ahlala engu-1/2, njengoba izehlakalo zizimele.
Isibonelo esisebenzayo semicimbi ezimele singatholakala kugeyimu yekhadi. Uma sidweba ikhadi emphemeni futhi kuwujeke, amathuba okudweba inkosi kumdwebo olandelayo ahlala engu-1/13, njengoba imicimbi izimele.
Kubalulekile ukwazi ukuthi ungazihlonza kanjani lezi zehlakalo ukuze wenze izibalo zamathuba ngendlela efanele.
Ukuhlonza ubudlelwano phakathi kwezehlakalo: ukuncika noma ukuzimela ezimeni ezingenzeka.
Lapho ubhekana nezimo ezingenzeka, ukuhlonza ubudlelwano phakathi kwezehlakalo kubalulekile ukuze kuhlaziywe kahle. Kunezinhlobo ezimbili eziyinhloko zobudlelwane phakathi kwezenzakalo: ukuthembela nokuzimela.
Izehlakalo ezizimele yilezo lapho ukwenzeka kwesinye isigameko kungabi namthelela ukwenzeka kwesinye. Ngamanye amazwi, amathuba okuthi esinye isigameko senzeke awathintwa ukwenzeka noma cha kwesinye isigameko. Isibonelo, lapho ugingqa idayi bese upheqa uhlamvu lwemali, imiphumela izimele komunye nomunye.
Ukuze sibonise ukuzimela phakathi kwemicimbi, singasebenzisa ifomula: P(A no-B) = P(A) * P(B), lapho u-P emele amathuba okuba kwenzeke. Ngamanye amazwi, amathuba azo zombili izehlakalo ezenzekayo alingana nomkhiqizo wamathuba omcimbi ngamunye ngamunye.
Isibonelo esilula sezehlakalo ezizimele umqulu wamadayisi amabili. Amathuba okugoqa u-4 kufayizi yokuqala angu-1/6, futhi amathuba okugoqa u-3 kufayizi yesibili nawo angu-1/6. Ukuphindaphinda lawa mathuba, sithola u-1/36, okungamathuba okugoqa u-4 kufayizi yokuqala kanye no-3 kufa kwesibili.
Ukuzijwayeza ukuhlonza nokubala imicimbi ezimele, kubalulekile ukuxazulula izivivinyo ezithile. Isibonelo, ukubala amathuba okudweba amakhadi amabili edekhini futhi womabili abe izinhliziyo, noma ithuba lokukhetha ngokungahleliwe amabhola amabili ku-urn futhi womabili abebomvu.
Izehlakalo ezizimele yilezo lapho ukwenzeka kwesinye isigameko kungaphazamisi ukwenzeka kwesinye, futhi amathuba okuba zombili zenzeke awumkhiqizo walokho okungenzeka ngakunye.
Zitholele ukuthi unganquma kanjani amathuba emicimbi emibili ezimele enye kwenye.
Ukuze unqume ukuthi kungenzeka yini izenzakalo ezimbili ezizimele komunye nomunye, kubalulekile ukuqonda umqondo wezenzakalo ezizimele. Izehlakalo ezimbili zithathwa njengezizimele lapho ukwenzeka kwesinye kungaphazamisi ukwenzeka kwesinye.
Ukuze ubale amathuba emicimbi emibili ezimele, vele uphindaphinde amathuba omcimbi ngamunye. Okusho ukuthi, uma u-A no-B kuyizehlakalo ezimbili ezizimele, amathuba akho kokubili okwenzeka ngesikhathi esisodwa anikezwa ngu-P(A no-B) = P(A) * P(B).
Isibonelo, uma amathuba okuthi line ngosuku olunikeziwe angu-30% (P(A) = 0.3) futhi amathuba okuthi othile osebenzisa isambulela ngalolo suku angama-40% (P(B) = 0.4), amathuba okuthi line futhi othile osebenzisa isambulela ngesikhathi esifanayo angu-30% * 40% = 12%.
Ukuzilolonga, masixazulule umsebenzi. Uma amathuba okuthi iqembu lebhola likanobhutshuzwayo liwine umdlalo angu-60% futhi amathuba okuba imvula ibe ngama-20% ngosuku lomdlalo, mangakanani amathuba okuthi iqembu liwunqobe umdlalo futhi line ngosuku lomdlalo? Ngokusebenzisa ifomula ethi P(A no B) = P(A) * P(B), sithola ukuthi impendulo ithi 60% * 20% = 12%.
Lena indlela elula nesebenzayo yokubala amathuba emicimbi ezimele.
Ukuhlaziywa kokuzimela kwemicimbi ngamabhangqa athile.
Ukuhlaziya ukuzimela kwezehlakalo ngamabhangqa athile kuyingxenye ebalulekile yethiyori yamathuba. Izehlakalo ezimbili zithathwa njengezizimele lapho ukwenzeka kwesinye kungaphazamisi amathuba okuthi esinye senzeke. Ukukhombisa ukuzimela kwemicimbi ngamabhangqa athile, singasebenzisa incazelo yamathuba anemibandela.
Uma izehlakalo ezimbili u-A no-B zizimele, khona-ke amathuba azo zombili izenzakalo ezenzeka ngesikhathi esisodwa alingana nomkhiqizo wamathuba angawodwana omcimbi ngamunye. Okungukuthi, P(A no B) = P(A) * P(B).
Isibonelo sakudala sezehlakalo ezizimele ukuphenywa kohlamvu lwemali kanye ne-roll of a die. Amathuba okuthola amakhanda ohlamvini lwemali awanawo umthelela emathubeni okuthola inombolo ethile ku-die.
Ukuzijwayeza ukuhlaziya ukuzimela kwemicimbi ngamabhangqa athile, singakwazi ukuxazulula ezinye izivivinyo. Isibonelo, bala amathuba okudweba i-ace kusuka kudeki yamakhadi ezehlakalweni ezimbili ezilandelanayo ngaphandle kokushintshwa. Izehlakalo zizimele ngoba amathuba okudweba i-ace esehlakalweni sesibili awathintwa ngokuthi udwebe u-ace esehlakalweni sokuqala.
Ukwazi ukuhlonza izehlakalo ezizimele kubalulekile ekwenzeni izibalo ezinembile nokwenza izinqumo ezinolwazi ezimweni ezingaqinisekile.
Imicimbi Ezimele: Imiboniso, Izibonelo, Ukuzivocavoca
Kumele imicimbi izimele , lapho amathuba okuthi enye yazo yenzeke ingathonywa ukuthi enye iyenzeka noma cha, kucatshangelwa ukuthi lezi zenzakalo zenzeka ngokungahleliwe.
Lesi simo senzeka noma nini lapho inqubo ekhiqiza umphumela womcimbi 1 ingaguquli nganoma iyiphi indlela amathuba emiphumela engaba khona yesehlakalo sesi-2. Kodwa uma lokhu kungenzeki, izenzakalo kuthiwa zincike.
Isimo semicimbi ezimele simi kanje: ake sithi amadayisi amabili anezinhlangothi eziyisithupha, elilodwa eliluhlaza okwesibhakabhaka nelilodwa eliphinki. Amathuba okuthi 1 kudayi eluhlaza okwesibhakabhaka ancike emathubeni okuthi 1 - noma cha - kuday epinki.
Esinye isibonelo sezehlakalo ezimbili ezizimele ukupheqa uhlamvu lwemali kabili ngokulandelana. Umphumela we-flip yokuqala ngeke uncike kumphumela wesibili, futhi ngokuphambene nalokho.
Ake sibheke isibonelo sezenzakalo ezilandelayo ezimele : isikhwama esinamabhola amabili amhlophe namabhola amabili amnyama. Amathuba okudweba ibhola elimhlophe noma elimnyama ayafana ekuzameni kokuqala.
Ake sithi umphumela kube yibhola elimhlophe. Uma ibhola elikhishiwe libuyiselwa esikhwameni, isimo sokuqala siyaphindwa: amabhola amabili amhlophe namabhola amabili amnyama.
Ngakho, emcimbini wesibili noma ukudweba, amathuba okudweba ibhola elimhlophe noma elimnyama afana nalawo okuqala. Ngakho-ke, ziyimicimbi ezimele.
Kodwa uma ibhola elimhlophe elikhishiwe emdlalweni wokuqala lingashintshwa, ukutonyulwa kwesibili kunethuba eliphakeme lokudweba ibhola elimnyama. Amathuba okuthi umdwebo wesibili uzobuyela kumhlophe ahlukile kulelo lomcimbi wokuqala futhi ahambisana nomphumela wangaphambilini.
Ukuboniswa kwemicimbi emibili ezimele
Ukuze siqinisekise ukuthi izehlakalo ezimbili zizimele yini, sizochaza umqondo wamathuba anemibandela womcimbi othile ohlobene nomunye. Ukuze senze lokhu, sidinga ukuhlukanisa phakathi kwemicimbi ekhethekile nebandakanyayo:
Imicimbi emibili ikhethekile uma amanani angenzeka noma izici zomcimbi A azifani ngalutho namanani noma izici zomcimbi B.
Ngakho-ke, ezenzakalweni ezimbili ezikhethekile, isethi yokuhlangana engu-A eno-B ayinalutho:
Imicimbi engabaliwe: A∩B = Ø
Ngokuphambene, uma izehlakalo zihlanganisiwe, kungase kwenzeke ukuthi umphumela womcimbi A uphinde uqondane nalowo wesinye isenzakalo B, u-A no-B kube izehlakalo ezihlukene. Esimweni esinjalo:
Imicimbi ehlanganisayo: A∩B ≠ Ø
Lokhu kusiholela ekuchazeni amathuba anemibandela ezenzakalo ezimbili ezibandakanya bonke abantu, okungukuthi, amathuba okuba umcimbi A wenzeke noma nini uma kwenzeka umcimbi B:
P (A¦B) = P (A∩B) / P (B)
Ngakho-ke, amathuba anemibandela yithuba lokuthi kokubili u-A no-B kwenzeke kuhlukaniswe ithuba lokuthi u-B enzeka. Ungaphinda uchaze amathuba okuthi u-B enzeka enemibandela kokuthi A:
P (B¦A) = P (A∩B) / P (A)
Imibandela yokwazi ukuthi izehlakalo ezimbili zizimele yini
Ngezansi, sizohlinzeka ngemibandela emithathu yokunquma ukuthi imicimbi emibili izimele yini. Kwanele ukuthi eyodwa kwezintathu ihlangatshezwe ukuze kuboniswe ukuzimela kwemicimbi.
1.- Uma amathuba okuthi u-A enzeka noma kunini uma u-B evela elingana namathuba ka-A, kusho ukuthi izehlakalo ezizimele:
P(A¦B) = P(A) => A uzimele ku-B
2.- Uma amathuba okuba kwenzeke okuthi B anikezwe u-A alingana namathuba okuthi B, khona-ke kukhona izehlakalo ezizimele:
P(B¦A) = P(B) => B izimele ku-A
3.- Uma amathuba okuthi u-A no-B enzeke elingana nomkhiqizo wamathuba okuthi u-A enzeka ngamathuba okuthi u-B enzeka, khona-ke lezi izehlakalo ezizimele. Ingxoxo nayo iyiqiniso.
P (A∩B) = P (A) P (B) <=> U-A no-B yimicimbi ezimele.
Izibonelo zezehlakalo ezizimele
Amasoli enjoloba akhiqizwa abahlinzeki ababili abahlukene ayaqhathaniswa. Amasampula avela kumkhiqizi ngamunye angaphansi kokuhlolwa okuningana ukuze anqume ukuthi ayahlangabezana yini nezicaciso noma cha.
Isifinyezo esiwumphumela samasampula angama-252 simi kanje:
Umkhiqizi 1; 160 ukuhlangabezana nokucaciswa; 8 ayihlangabezani nemibandela.
Umkhiqizi 2; 80 ukuhlangabezana nokucaciswa; 4 ayihlangabezani nemibandela.
Umcimbi A: “Ukuthi isampula ngeyomkhiqizi 1”.
Umcimbi B: "Ukuthi isampula ihlangabezana nokucaciswa."
Sifuna ukwazi ukuthi le micimbi engu-A no-B izimele noma cha, lapho sisebenzisa enye yemibandela emithathu eshiwo esigabeni esandulele.
Imibandela: P(B¦A) = P(B) => B izimele ku-A
P(B) = 240/252 = 0,9523
P(B¦A) = P(A ⋂ B) / P(A) = (160/252) / (168/252) = 0,9523
Isiphetho: Imicimbi A no-B izimele.
Ake sithi umcimbi C: "ukuthi isampula livela kumkhiqizi 2"
Ingabe umcimbi B uzimele kumcimbi C?
Sisebenzisa eyodwa yemibandela.
Imibandela: P(B¦C) = P(B) => B izimele ku-C
P(B¦C) = (80/252) / (84/252) = 0,9523 = P(B)
Ngakho-ke, ngokuya ngedatha etholakalayo, amathuba okuthi i-rubber ekhethwe ngokungahleliwe izohlangabezana nemininingwane azimele kumkhiqizi.
Ukuzivocavoca umzimba
– Isivivinyo 1
Ebhokisini, sibeka amamabula ayi-10 emfanekisweni woku-1, oku-2 kuwo aluhlaza, 4 aluhlaza okwesibhakabhaka, futhi 4 amhlophe. Kuzokhethwa amamabula amabili angahleliwe, eyodwa kuqala neyodwa kamuva. Ucelwa ukuthi uthole i-
Amathuba okuthi awekho kuwo aluhlaza okwesibhakabhaka, ngaphansi kwezimo ezilandelayo:
a) Ngokufaka esikhundleni, okungukuthi, ukubuyisela imabula yokuqala ebhokisini ngaphambi kokukhethwa kwesibili. Khomba ukuthi lezi izehlakalo ezizimele noma zincike.
b) Ngaphandle kokushintshwa, ukuze imabula yokuqala ekhishiwe ibe ngaphandle kwebhokisi lapho kukhethwa okwesibili. Ngokufanayo, bonisa ukuthi lezi izehlakalo ezincikile noma ezizimele.
Isixazululo se
Sibala amathuba okuthi imabula yokuqala ekhishiwe ayiluhlaza okwesibhakabhaka, okungu-1 susa amathuba okuthi ibe blue P(A), noma ngokuqondile ukuthi ayiluhlaza okwesibhakabhaka, ngoba ibiluhlaza noma imhlophe:
P(A) = 4/10 = 2/5
P (hhayi eluhlaza okwesibhakabhaka) = 1 – (2/5) = 3/5
Okuhle:
P (okuluhlaza noma okumhlophe) = 6/10 = 3/5.
Uma imabula ekhishiwe ibuyiswa, konke kuzoba njengakuqala. Kulesi sizinda sesibili, kunethuba elingu-3/5 lokuthi imabula ekhishiwe ngeke ibe luhlaza okwesibhakabhaka.
P(hhayi eluhlaza okwesibhakabhaka, hhayi eluhlaza okwesibhakabhaka) = (3/5).(3/5) = 25/9.
Imicimbi izimele, njengoba imabula ekhishiwe ibuyiselwe ebhokisini futhi isehlakalo sokuqala asinawo umthelela ekutheni kwenzeke isehlakalo sesibili.
Isixazululo b
Ngomdwebo wokuqala, qhubeka njengasesigabeni esidlule. Amathuba okuthi ingabi luhlaza okwesibhakabhaka ngu-3/5.
Ngokukhishwa kwesibili, sinamabula angu-9 esikhwameni, njengoba eyokuqala ayizange ibuye, kodwa yayingeyona eluhlaza okwesibhakabhaka; ngakho-ke, esikhwameni kukhona amamabula angu-9 kanye nama-5 angewona aluhlaza okwesibhakabhaka:
P (okuluhlaza noma okumhlophe) = 5/9.
P(none are blue) = P(first not blue). P(owesibili ongewona oluhlaza okwesibhakabhaka / owokuqala wawungewona oluhlaza okwesibhakabhaka) = (3/5). (5/9) = 1/3
Kulokhu, akuzona izehlakalo ezizimele, ngoba umcimbi wokuqala ubeka owesibili.
– Isivivinyo 2
Isitolo sinamahembe angu-15 ngosayizi abathathu: amancane ama-3, angu-6 amaphakathi, angu-6 amakhulu. Amahembe ama-2 akhethwa ngokungahleliwe.
a) Angakanani amathuba okuthi womabili amahembe akhethiwe mancane uma elilodwa likhishwa kuqala futhi ngaphandle kokushintsha elinye eqeqebeni?
b) Angakanani amathuba okuthi womabili amahembe akhethiwe mancane uma elilodwa likhishwa kuqala, lishintshwa eqeqebeni, futhi elesibili likhishwe?
Isixazululo se
Nansi imicimbi emibili:
Umcimbi A: Ihembe lokuqala elikhethiwe lincane
Umcimbi B: Ihembe lesibili elikhethiwe lincane
Amathuba omcimbi A yilawa: P(A) = 3/15
Amathuba omcimbi B yilokhu: P(B) = 2/14, ngoba ihembe elilodwa selisusiwe (14 kusale), kodwa futhi umcimbi kufanele ufezeke. Ihembe lokuqala elikhishiwe kufanele libe lincane futhi kusele ama-2 amancane.
Ngamanye amazwi, amathuba okuthi u-A no-B bawumkhiqizo wamathuba yilawa:
P(A kanye no-B) = P(B¦A) P(A) = (2/14) (3/15) = 0,029
Ngakho-ke, ithuba lokwenzeka kwesehlakalo A no-B lilingana nomkhiqizo wesenzeko A, ngamathuba omcimbi B ukwenzeka uma umcimbi A.
Qaphela ukuthi:
P (B¦A) = 2/14
Amathuba omcimbi B, kungakhathaliseki ukuthi umcimbi A wenzeka noma cha, kuzoba:
P(B) = (2/14) uma eyokuqala incane, noma P(B) = 3/14 uma eyokuqala ingencane.
Ngokuvamile, okulandelayo kungaphethwa:
U-P(B¦A) akalingani no-P(B) => B akazimele ku-A
Isixazululo b
Futhi, kunemicimbi emibili:
Umcimbi A: Ihembe lokuqala elikhethiwe lincane
Umcimbi B: Ihembe lesibili elikhethiwe lincane
P (A) = 3/15
Khumbula ukuthi noma ngabe yini umphumela, ihembe elikhishwe ku-batch lishintshwa, futhi futhi ihembe likhishwa ngokungahleliwe. Amathuba omcimbi B, uma kwenzeka isigameko A, yilawa:
P (B¦A) = 3/15
Amathuba emicimbi A no-B azoba:
P(A kanye no-B) = P(B¦A) P(A) = (3/15) (3/15) = 0,04
Qaphela ukuthi:
I-P(B¦A) ilingana no-P(B) => B izimele ku-A.
– Isivivinyo 3
Cabangela izehlakalo ezimbili ezizimele u-A no-B. Kuyaziwa ukuthi amathuba okuba kwenzeke isehlakalo A ngu-0,2 futhi amathuba okuba kwenzeke isehlakalo B ngu-0,3. Ayoba yini amathuba azo zombili izenzakalo?
Isixazululo 2
Ukwazi ukuthi izehlakalo zizimele, kuyaziwa ukuthi amathuba azo zombili izehlakalo enzeka awumphumela wamathuba angawodwana. Lokho kusho ukuthi,
P(A∩B) = P(A) P(B) = 0,2 * 0,3 = 0,06
Qaphela ukuthi lokhu amathuba aphansi kakhulu kunethuba lokuthi umcimbi ngamunye uzokwenzeka ngaphandle komphumela womunye. Noma, ngamanye amazwi, aphansi kakhulu kunamathuba omuntu ngamunye.
Izinkomba
- Berenson, M. 1985. Izibalo zokuphatha nezomnotho. I-Interamerican SA 126-127.
- Monterrey Institute. Amathuba Wezehlakalo Ezizimele. Ithathwe ku: monterreyinstitute.org
- Izehlakalo Ezizimele Zothisha Wezibalo Zibuyiswe: youtube.com
- Izinhlobo Zomcimbi We-Superprof, Imicimbi Encike. Kubuyiswe ku-: superprof.es
- Umfundisi obonakalayo Amathuba Atholakale ku: vitutor.net
- I-Wikipedia Independence (amathuba). Kutholwe ku: wikipedia.com