Indlela elula yokuxazulula izinkinga zamathuba
I-Probability yisihloko sezibalo esivame ukuhlangatshezwana naso ezikoleni ngenxa yokuthi siseduze nokuphila kwansuku zonke: kusukela emathubeni okubonakala kwenombolo edayisini, kuya ekudwebeni ibhola elinombala ebhokisini, kuya emathubeni okwenzeka kwesenzakalo ngokusekelwe kudatha. Ngeshwa, abafundi abaningi bathola ukuthi amathuba "anzima" ngoba izinkinga zibonakala zilula, kodwa izimpendulo zingaba yiphutha uma zingacatshangelwa ngokucophelela. Kodwa-ke, kunendlela elula nehlelekile yokuxazulula izinkinga zamathuba: qonda imiqondo eyisisekelo, thola uhlobo lwenkinga, bese ukhetha isu elifanele.
Lesi sihloko sizoxoxa ngezinyathelo ezisebenzayo zokukusiza ukuthi usheshe futhi unembile lapho usebenza ngezinkinga zamathuba.
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1. Qonda umqondo oyisisekelo wamathuba
Ngokuvamile, amathuba omcimbi \(A\) alotshiwe:
\[
P(A) = \frac{\text{inani lemiphumela esekela }A}{\text{inani layo yonke imiphumela engenzeka}}
\]
Isihluthulelo sisezintweni ezimbili:
1. Isikhala sesampula (S): yonke imiphumela engenzeka engenzeka.
2. Umcimbi (A): ingxenye yesikhala sesampula ehambisana nombuzo obuziwe.
Uma yonke imiphumela inamathuba alinganayo (isb., idayisi elifanele, izinhlamvu zemali ezifanelekile), khona-ke ifomula engenhla ingasetshenziswa ngqo. Uma kungenzeka ukuthi kungenzeka alinganayo, amathuba avame ukubalwa kusukela kwamanye amamodeli (isb., idatha yemvamisa noma amathuba anemibandela).
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2. Izinyathelo ezisheshayo zokuxazulula izinkinga zamathuba
Ukuze ugweme ukulahleka, sebenzisa "ifomula yokusebenza" elandelayo:
1. Bhala phansi lokho okubuzwayo (yisiphi isenzakalo esicelwayo?)
2. Nquma isikhala sesampula (mingaki imiphumela engaba khona?)
3. Bala imiphumela esekela umcimbi oceliwe.
4. Ukwabelana ngenzuzo kusekela imiphumela iyonke
5. Yenza kube lula ingxenye bese uqinisekisa ukuthi inengqondo (amathuba kumele abe phakathi kuka-0 no-1)
Amaphutha amaningi ayenzeka ngoba izinyathelo 2 no-3 azicaciswanga.
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3. Sebenzisa amathebula, imidwebo yezihlahla, noma uhlu uma kudingeka.
Ezinkingeni ezihilela izivivinyo ezimbili noma ngaphezulu (isib., phendula uhlamvu lwemali kabili, dweba ikhadi kabili), izindlela ezibonakalayo ziwusizo kakhulu.
a) Ithebula
Ifaneleka ezinguqulweni ezimbili ezilula, isibonelo izinhlamvu zemali kanye nedayisi.
b) Umdwebo wesihlahla
Kufanelekile ekuhlolweni kwesinyathelo ngesinyathelo, ikakhulukazi uma kukhona isimo "sokungabuyi" (inani lezinto liyashintsha).
c) Uhlu lwamalungu esikhala esampula
Ifanele izikhala ezincane zesampula.
Ngale ndlela, akudingeki uqagele.
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4. Imibuzo yamathuba ajwayelekile kanye nezixazululo ezisheshayo
A. Amathuba okushaya idayisi
Amadayisi anezinhlangothi ezi-6: \(\{1,2,3,4,5,6\}\)
Isibonelo: amathuba okuthola inombolo elinganayo.
– Isikhala sesampula \(S = 6\)
– Umcimbi \(A = \{2,4,6\}\) ungo-3
– \(P(A) = 3/6 = 1/2\)
Icebo elisheshayo: njalo bala ukuthi “bangaki abagcwalisayo” esikhundleni sokuthi “inani eliphelele elingu-6”.
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B. Ithuba kuhlamvu lwemali
Uhlamvu lwemali lunezinhlangothi ezimbili: Amakhanda (A) kanye namakhanda (G).
Isibonelo: amathuba okuthola isithombe lapho uhlamvu lwemali luphonswa kanye.
– Isiyonke 2
– Usekelo 1
– Ithuba \(=1/2\)
Uma iphonswe kabili:
– Isikhala sesampula: \(\{AA, AG, GA, GG\}\) isamba esingu-4
Icebiso elisheshayo: ekuhlolweni okuzimele, amathuba aphelele ngokuvamile awukuthi \(2^n\) (ngezinhlamvu zemali) kanye \(6^n\) (ngedayisi).
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C. Amathuba okuthatha ibhola ebhokisini (elinokufakwa esikhundleni futhi ngaphandle kokufakwa esikhundleni)
Isibonelo, ibhokisi liqukethe amabhola abomvu amathathu kanye namabhola aluhlaza okwesibhakabhaka amabili (angu-5 esewonke).
1) Ngokubuyisela
Thatha izikhathi ezimbili, isikhathi ngasinye ibhola libuyiselwa. Amathuba okuthatha obomvu bese kuba luhlaza okwesibhakabhaka:
– Amathuba okubomvu: \(3/5\)
– Bese kuba luhlaza okwesibhakabhaka (ngoba ibuyisiwe, ihlala): \(2/5\)
– \(P = (3/5)(2/5)=6/25\)
2) Akukho ukubuyiselwa kwemali
Thatha izikhathi ezimbili ngaphandle kokubuya. Amathuba okubomvu bese kuba luhlaza okwesibhakabhaka:
– Amathuba okuqala abomvu: \(3/5\)
– Amabhola ama-4 asele, okusele okuluhlaza okwesibhakabhaka okusele okungu-2 → amathuba okwesibili aluhlaza okwesibhakabhaka: \(2/4\)
– \(P = (3/5)(2/4)=6/20=3/10\)
Umehluko oyinhloko: ngaphandle kokufaka esikhundleni, i-denominator iyashintsha ngoba inani eliphelele lamabhola liyancipha.
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5. Sebenzisa imithetho yokwengeza nokugcwalisa
A. Umthetho wokwengeza (OR)
Uma ubuzwa amathuba okuthi “A noma B”, sebenzisa:
\[
P(A \inkomishi B) = P(A) + P(B) – P(A \inkomishi B)
\]
– \(A \inkomishi B\): A noma B (noma kokubili)
– \(A \cap B\): A kanye no-B ngesikhathi esifanayo
Isibonelo esilula: idayisi, amathuba okuthola inombolo elinganayo noma i-multiple ka-3.
– Ilingana: {2,4,6} → 3
– Iziphindaphindo ze-3: {3,6} → 2
– Izingcezu (ezilinganayo neziphindaphindayo zika-3): {6} → 1
– Peluang: \((3/6)+(2/6)-(1/6)=4/6=2/3\)
B. Umphelelisi (ngaphambana)
Ngokuvamile kulula ukubala ukuthi “akwenzekanga” bese uyisusa ku-1.
\[
P(A) = 1 – P(A^c)
\]
Isibonelo: amathuba okuthi “amakhanda” avele okungenani kanye lapho uhlamvu lwemali luphonswa izikhathi ezintathu.
Esikhundleni sokubala inombolo izikhathi ezingu-1, 2, 3, bala i-complement yayo:
– Ukwengeza: azikho izinombolo nhlobo = zonke izibalo (GGG)
– \(P(\text{GGG}) = (1/2)^3 = 1/8\)
– Ngakho-ke \(P(\text{minimum 1 number}) = 1 – 1/8 = 7/8\)
Leli qhinga lithandwa kakhulu emibuzweni ethi “okungenani eyodwa”, “okungenani eyodwa”, “okungenani”.
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6. Amathuba anemibandela: qaphela ulwazi olwengeziwe
Amathuba anemibandela asho amathuba okuthi u-A avele uma nje u-B esevele evele:
\[
P(A|B)=\frac{P(A \cap B)}{P(B)}
\]
Lolu hlobo lombuzo luvame ukuvela uma kukhona ulwazi olwengeziwe, isibonelo "kuyaziwa ukuthi ibhola elibomvu lidonswe" noma "kuyaziwa ukuthi umfundi okhethiwe wayeyintombazane".
Indlela elula:
1. Gxila esimweni B (cabanga ukuthi isikhala sesampula sincipha sibe yi-B kuphela)
2. Kulabo abahlangana no-B, bala labo abahlangana no-A.
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7. Amaphutha avame kakhulu
1. Ukhohlwe ukushintsha isikhala sesampula uma ungasishintshi.
2. Ukuchaza igama elithi “noma” ngendlela engafanele (ngezinye izikhathi lisho kokubili).
3. Ukungacacisi ngokuthi “okungenani” / “okuningi”; kungcono ukusebenzisa izinto eziphelelisayo.
4. Ukucabanga ukuthi imicimbi izimele kanti ayizimele (ngaphandle kokubuya ayizimele).
5. Ungenzi lula futhi ungahloli i-logic (umphumela wamathuba akumelwe ube >1).
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8. Amathiphu okuzijwayeza ukuze ube nekhono ngokushesha
– Jwayela ukubhala: \(S\), \(A\), amalungu amaningi \(n(S)\), \(n(A)\).
- Prakthiza izinkinga "zezigaba ezimbili" ngemidwebo yesihlahla.
– Prakthiza imibuzo “okungenani eyodwa” enezimpendulo eziphelele.
– Uma usunayo impendulo, hlola: ingabe kunengqondo? Uma amathuba esigameko “esingenzeka,” inani laso livame ukuba seduze no-1, hhayi eduze no-0.
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I-Penutup
Ukuxazulula izinkinga zamathuba kulula uma usebenzisa izinyathelo ezihlelekile njalo: chaza isikhala sesampula, unqume izenzakalo, bese ubala isilinganiso. Ngezinkinga eziyinkimbinkimbi kakhulu, sebenzisa izinsiza ezibonakalayo ezifana namathebula noma imidwebo yezihlahla, bese usebenzisa imithetho yokuhlanganisa, ukuphindaphinda, kanye nokugcwalisa. Ngokuzijwayeza okwanele nokufunda ngokucophelela izinkinga, amathuba ngeke esazwakala "eyinkohliso" kodwa esikhundleni salokho abe yingxenye yezibalo ezijabulisa ngempela ngoba ziseduze nemicimbi yansuku zonke.
Uma uthanda, ngingenza nezibonelo eziyi-10 zemibuzo engenzeka kanye nezincazelo zayo (kusukela kokulula kuya kokunzima) ukuze ukwazi ukuzijwayeza ngokushesha.
