He huarahi ngāwari ki te whakaoti rapanga tūponotanga
He kaupapa pāngarau e kitea pinepinetia ana te tūponotanga i roto i ngā kura nā te mea e tata ana ki te oranga o ia rā: mai i te tūponotanga o te puta mai o tētahi tau i runga i tētahi mataono, ki te tango i tētahi pōro tae mai i tētahi pouaka, ki te tūponotanga o tētahi huihuinga e puta mai ana i runga i ngā raraunga. Engari, he maha ngā ākonga e kite ana he "uaua" te tūponotanga nā te mea he māmā noa iho ngā rapanga, engari ka hē pea ngā whakautu mēnā kāore i te āta whakaarohia. Heoi anō, he huarahi tino ngāwari, he pūnaha hoki hei whakaoti rapanga tūponotanga: kia mārama ki ngā ariā taketake, kia tautuhia te momo rapanga, kātahi ka whiriwhiri i te rautaki tika.
Ka matapakihia e tēnei tuhinga ngā mahi whai hua hei āwhina i a koe kia tere ake, kia tika ake hoki i a koe e mahi ana i ngā rapanga tūponotanga.
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1. Kia mārama ki te ariā taketake o te whai wāhitanga
I te nuinga o te wā, ka tuhia te tūponotanga o tētahi huihuinga \(A\):
\[
P(A) = \frac{\text{te maha o ngā putanga e tautoko ana i te }A}{\text{te maha o ngā putanga katoa}}
\]
Kei roto i ngā mea e rua te kī:
1. Wāhi tauira (S): ngā putanga katoa ka puta pea.
2. Takahanga (A): wāhanga o te wāhi tauira e rite ana ki te pātai i pātaihia.
Mena he ōrite te tūponotanga o ngā putanga katoa (hei tauira, ngā mataono tika, ngā moni tika), ka taea te whakamahi tika i te tātai i runga ake nei. Mena kāore i te ōrite te tūponotanga, ka tatauhia ngā tūponotanga mai i ētahi atu tauira (hei tauira, ngā raraunga auau, ngā tūponotanga ā-tikanga rānei).
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2. Ngā mahi tere hei whakaoti rapanga whai wāhitanga
Hei karo i te ngaro, whakamahia te "tātai mahi" e whai ake nei:
1. Tuhia te mea e tonoa ana (he aha te kaupapa e tonoa ana?)
2. Whakatauhia te wāhi tauira (e hia ngā putanga ka taea?)
3. Tātaihia ngā hua e tautoko ana i te kaupapa i tonoa.
4. Ka tautokohia e te tiritiri hua ngā hua katoa
5. Whakangāwaritia te hautau, ā, kia tika te arorau (me noho te tūponotanga i waenga i te 0 me te 1)
Ko te nuinga o ngā hapa ka puta mai nā te mea kāore i te mārama ngā taahiraa 2 me te 3.
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3. Whakamahia ngā ripanga, ngā hoahoa rākau, ngā rārangi rānei mēnā e tika ana.
Mō ngā rapanga e rua, neke atu rānei ngā whakamātautau (hei tauira, te whiu i te moni kia rua ngā wā, te tango kāri kia rua ngā wā), he tino āwhina ngā tikanga tirohanga.
a) Ripanga
He pai mō ngā taurangi māmā e rua, hei tauira, ngā moni me ngā mataono.
b) Te hoahoa rākau
He pai mō ngā whakamātautau taahiraa-i-te-taahiraa, inā koa mēnā he āhuatanga "kāore e hoki mai" (ka rerekē te maha o ngā mea).
c) Rārangi o ngā mema o te wāhi tauira
He pai mō ngā wāhi tauira iti.
Pēnei, kāore koe e hiahia ki te whakapae.
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4. Ngā pātai tūponotanga matarohia me ngā otinga tere
A. Ngā tūponotanga ki ngā mataono
E ono ngā taha o ngā mataono: \(\{1,2,3,4,5,6\}\)
Tauira: te tūponotanga o te whiwhi i tētahi tau taurite.
– Wāhi tauira \(S = 6\)
– Ko te takahanga \(A = \{2,4,6\}\) he 3
– \(P(A) = 3/6 = 1/2\)
He tinihanga tere: tatauhia "e hia ngā mea e whakakī ana" kaua ko te tatau i te "6 katoa".
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B. Tūponotanga ki tētahi moni
E rua ngā taha o te moni: Ngā Upoko (A) me Ngā Upoko (G).
Tauira: te tūponotanga o te puta o tētahi pikitia ina whiua kotahi te moni.
– Tapeke 2
– Tautoko 1
– Tūponotanga \(=1/2\)
Ki te whiua kia rua ngā wā:
– Wāhi tauira: \(\{AA, AG, GA, GG\}\) katoa 4
Tohutohu tere: mō ngā whakamātautau motuhake, ko te tūponotanga katoa ko \(2^n\) (mō ngā moni) me \(6^n\) (mō ngā mataono).
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C. Te tūponotanga o te tango i tētahi pōro mai i te pouaka (me te whakakapinga me te kore whakakapinga)
Hei tauira, kei roto i tētahi pouaka e 3 ngā pōro whero me ngā pōro kikorangi e 2 (e 5 katoa).
1) Me te hokinga mai
Kia rua ngā wā e whakahokia mai ai te pōro. Ko te tūponotanga ka riro mai i te whero, kātahi ka riro mai i te kikorangi:
– Te tūponotanga o te whero: \(3/5\)
– Kātahi ka kikorangi (nā te mea i whakahokia mai, ka noho tonu): \(2/5\)
– \(P = (3/5)(2/5)=6/25\)
2) Kāore he whakahokinga mai
Tangohia kia rua ngā wā, kaua e hoki mai. Ko te tūponotanga ka whero, kātahi ka kikorangi:
– Tūponotanga whero tuatahi: \(3/5\)
– 4 ngā pōro e toe ana, e 2 ngā pōro kikorangi e toe ana → te tūponotanga o te tuarua o ngā pōro kikorangi: \(2/4\)
– \(P = (3/5)(2/4)=6/20=3/10\)
Ko te rerekētanga matua: ki te kore he whakakapinga, ka huri te taupū nā te mea ka heke te maha katoa o ngā pōro.
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5. Whakamahia ngā ture tāpiri me ngā ture whakakī
A. Ture tāpiri (RĀNEI)
Mena ka pātaihia mō ngā tūponotanga "A, B rānei", whakamahia:
\[
P(A \cup B) = P(A) + P(B) – P(A \cap B)
\]
– \(A \cup B\): A, B rānei (ngā mea e rua rānei)
– \(A \cap B\): A me B i te wā kotahi
Tauira māmā: te mataono, te tūponotanga o te whiwhi i tētahi tau taurite, i tētahi tau maha rānei o te 3.
– Taurite: {2,4,6} → 3
– Ngā tau maha o te 3: {3,6} → 2
– Ngā poro (taurite me ngā tau maha o te 3): {6} → 1
– Peluang: \((3/6)+(2/6)-(1/6)=4/6=2/3\)
B. Tāpiri (kei te ritenga kē)
He māmā ake te tatau i te "kāore i tupu" kātahi ka tangohia mai i te 1.
\[
P(A) = 1 – P(A^c)
\]
Tauira: te tūponotanga o te puta mai o ngā "upoko" i te iti rawa kotahi ina whiua he moni kia toru ngā wā.
Engari i te tatau i te 1, 2, 3 whakarea o tētahi tau, tatauhia tōna tāpiritanga:
– Tāpiri: kāore he tau = ngā tau katoa (GGG)
– \(P(\text{GGG}) = (1/2)^3 = 1/8\)
– Nō reira \(P(\text{tau iti rawa 1}) = 1 – 1/8 = 7/8\)
He tino rongonui tēnei mahi tinihanga mō ngā pātai "kia kotahi te iti rawa", "kia kotahi te iti rawa", me te "iti rawa".
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6. Ngā tūponotanga ā-tikanga: kia aro ki ngā kōrero tāpiri
Ko te tūponotanga ā-tikanga ko te tūponotanga o te puta o A mēnā kua puta kē a B:
\[
P(A|B)=\frac{P(A \cap B)}{P(B)}
\]
Ka puta ake tēnei momo pātai ina he kōrero tāpiri, hei tauira "e mōhiotia ana i tangohia te pōro whero" e mōhiotia ana rānei "e mōhiotia ana he kōtiro te tauira i whiriwhiria".
Ko te ara māmā:
1. Arotahi ki te āhuatanga B (me kī ka whaiti te wāhi tauira ki te B anake)
2. Mai i te hunga e tūtaki ana ki a B, tatauhia te hunga e tūtaki ana hoki ki a A.
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7. Ngā hapa tino noa
1. I wareware ki te whakarerekē i te wāhi tauira i te mea kāore he whakakapinga.
2. Te whakamāori hē i te kupu “or” (i ētahi wā ko te tikanga o ngā mea e rua).
3. Kāore i te tino tika te whakamahi i te "iti rawa" / "tino"; he pai ake te whakamahi i ngā kupu whakakī.
4. Ki te whakaaro he tū motuhake ngā kaupapa, engari kāore (ki te kore he hokinga mai, kāore e tū motuhake).
5. Kaua e whakangawari, kaua hoki e tirotiro i te arorau (kaua te hua tūponotanga e neke ake i te 1).
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8. Whakaharatauhia ngā tohutohu hei tere ake te matatau
– Whakamātauria te tuhituhi: \(S\), \(A\), ngā mema maha \(n(S)\), \(n(A)\).
– Whakaharatauhia ngā rapanga "wāhanga-rua" me ngā hoahoa rākau.
– Whakaharatauhia ngā pātai "kia kotahi te iti rawa" me ngā kupu whakamāmā.
– Kia whiwhi koe i te whakautu, tirohia: he tika? Mena he "tūponotanga pea ka puta" te tūponotanga o tētahi huihuinga, he tata tonu tōna uara ki te 1, ehara i te mea tata ki te 0.
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Te Katinga
He māmā noa iho te whakaoti rapanga tūponotanga mēnā ka whakamahia tonutia ngā mahi pūnaha: te tautuhi i te wāhi tauira, te whakatau i ngā takahanga, kātahi ka tatau i te ōwehenga. Mō ngā rapanga uaua ake, whakamahia ngā āwhina tirohanga pērā i ngā ripanga, ngā hoahoa rākau rānei, me te whakamahi i ngā ture tāpiri, whakarea, me te whakakīkī. Mēnā ka nui te mahi me te pānui tupato i ngā rapanga, kāore te tūponotanga e ahua "tinihanga" engari ka noho hei wāhanga o te pāngarau e tino pārekareka ana nā te mea e tata ana ki ngā takahanga o ia rā.
Ki te hiahia koe, ka taea e au te hanga i ngā tauira pātai tūponotanga tekau me ō rātou whakamārama (mai i te ngāwari ki te uaua) kia taea ai e koe te whakaharatau tonu.
