Ngā Tauira Pātai e Matapaki ana i te Tūponotanga o ngā Takahanga Whakapūhui
He Kupu Whakataki ki te Tūponotanga o ngā Takahanga Whakapūhui
Ko te tūponotanga he peka o te pāngarau e ako ana i te tūponotanga o te puta o tētahi kaupapa. Ko te tūponotanga o tētahi kaupapa pūhui ko te tūponotanga o tētahi kaupapa e uru ana ki te neke atu i te kotahi te kaupapa. Hei tauira, ko te tūponotanga o te huri i tētahi tau taurite ki runga i tētahi mataono me tētahi āke mai i tētahi kete kāri tākaro he tauira o ngā kaupapa pūhui. Ka matapakihia e tēnei tuhinga ētahi tauira rapanga me te matapaki i te tūponotanga o ngā kaupapa pūhui.
Te Ariā Taketake o te Tūponotanga o ngā Takahanga Whakapūhui
E rua ngā momo huihuinga honohono:
1. Ngā Takahanga Motuhake: E rua ngā takahanga kāore e taea te puta i te wā kotahi. Hei tauira, i te wā e whiua ana he mataono, ko ngā takahanga o te whiunga o te 2 me te 5 he takahanga motuhake nā te mea kāore e taea te whiu i ngā tau e rua i te wā kotahi.
2. Ngā Takahanga Kāore e Tūhonohono ana: E rua ngā takahanga ka puta i te wā kotahi. Hei tauira, i roto i te tānga kāri tākaro, ko ngā takahanga o te whiwhinga kāri ngākau (♥) me te kāri me te tau 10 he takahanga kāore e tūhonohono ana nā te mea he kāri ngākau me te tau 10.
Anei ētahi tātai taketake e whakamahia ana hei tatau i te tūponotanga o ngā takahanga pūhui:
– P(A, B rānei) (mō ngā takahanga kore-motuhake): \(P(A \cup B) = P(A) + P(B) – P(A \cap B)\)
– P(A, B rānei) (mō ngā takahanga motuhake): \(P(A \cup B) = P(A) + P(B)\)
– P(A me B) (mō ngā takahanga motuhake): \(P(A \cap B) = P(A) \times P(B)\)
Ngā Pātai Tauira me te Kōrero
Tauira Pātai 1: Mataono
Pātai:
He aha te tūponotanga kia puta he tau taurite, he tau nui atu rānei i te 4 i runga i tētahi mataono?
Kōrero:
Tuatahi, me tautuhi tātou i ngā kaupapa:
– Takahanga A: Te whiwhi i tētahi tau taurite (2, 4, 6)
– Takahanga B: Te whiwhi i tētahi tau nui atu i te 4 (5, 6)
Muri iho, ka whakatauhia e mātou te tūponotanga o ia huihuinga:
– \(P(A) = \frac{3}{6} = \frac{1}{2}\)
– \(P(B) = \frac{2}{6} = \frac{1}{3}\)
Nā te mea kei roto i ngā takahanga A me B tētahi tau 6, me tatau tātou \(P(A \cap B)\):
– \(P(A \cap B) = \frac{1}{6}\) (nā te mea kotahi anake te tau, arā, ko te 6, kei roto i te A me te B)
Mā te whakamahi i te tātai mō ngā takahanga kore-ā-ira:
\[P(A \cup B) = P(A) + P(B) – P(A \cap B) = \frac{1}{2} + \frac{1}{3} – \frac{1}{6}\]
Me whakarite kia ōrite ngā tauwehenga o ēnei hautau:
\[P(A \cup B) = \frac{3}{6} + \frac{2}{6} – \frac{1}{6} = \frac{4}{6} = \frac{2}{3}\]
Nō reira, ko te tūponotanga o te whiwhi i tētahi tau taurite, i tētahi tau nui atu rānei i te 4 ko \(\frac{2}{3}\).
Tauira Pātai 2: Kāri Tākaro
Pātai:
He aha te tūponotanga kia puta he Āce, he hāpa rānei mai i tētahi kete kāri tākaro?
Kōrero:
Tuatahi, me tautuhi tātou i ngā kaupapa:
– Kaupapa A: Te whiwhi kāri Ace (4 katoa, kotahi mō ia kākahu)
– Kaupapa B: Te whiwhi kāri hāpara (13 katoa)
Muri iho, ka whakatauhia e mātou te tūponotanga o ia huihuinga:
– \(P(A) = \frac{4}{52} = \frac{1}{13}\)
– \(P(B) = \frac{13}{52} = \frac{1}{4}\)
Nā te mea kei roto te Ace of Spades i ngā takahanga A me B, me tatau tātou \(P(A \cap B)\):
– \(P(A \cap B) = \frac{1}{52}\)
Mā te whakamahi i te tātai mō ngā takahanga kore-ā-ira:
\[P(A \cup B) = P(A) + P(B) – P(A \cap B) = \frac{1}{13} + \frac{1}{4} – \frac{1}{52}\]
Me whakarite kia ōrite ngā tauwehenga o ēnei hautau:
\[
P(A \cup B) = \frac{4}{52} + \frac{13}{52} – \frac{1}{52} = \frac{16}{52} = \frac{4}{13}
\]
Nō reira, ko te tūponotanga o te whiwhi i te Āce, i te hāpara rānei, ko \(\frac{4}{13}\).
Tauira Raru 3: Pōro i roto i te Pouaka
Pātai:
I roto i tētahi pouaka e toru ngā pōro whero, e whā ngā pōro kikorangi, me e rima ngā pōro kākāriki. Mēnā ka tangohia matapōkeretia tētahi pōro, he aha te tūponotanga ka puta he pōro whero, he pōro kākāriki rānei?
Kōrero:
Tuatahi, me tautuhi tātou i ngā kaupapa:
– Kaupapa A: Te whiwhinga pōro whero (nama 3)
– Kaupapa B: Te whiwhi i tētahi pōro matomato (nama 5)
Muri iho, ka whakatauhia e mātou te tūponotanga o ia huihuinga:
– Te tapeke o ngā pōro = 3 + 4 + 5 = 12
– \(P(A) = \frac{3}{12} = \frac{1}{4}\)
– \(P(B) = \frac{5}{12}\)
Nā te mea kāore e taea te whero me te kakariki i te wā kotahi, ka tū motuhake ēnei takahanga:
\[P(A \cup B) = P(A) + P(B) = \frac{1}{4} + \frac{5}{12}\]
Me whakarite kia ōrite ngā tauwehenga o ēnei hautau:
\[
P(A \cup B) = \frac{3}{12} + \frac{5}{12} = \frac{8}{12} = \frac{2}{3}
\]
Nō reira, ko te tūponotanga o te whiwhinga o tētahi pōro whero, pōro kakariki rānei ko \(\frac{2}{3}\).
Tauira Pātai 4: Ngā Moni e Rua
Pātai:
Ki te whiua ngā moni e rua i te wā kotahi, he aha te tūponotanga ka puta ake te upoko kotahi, neke atu rānei?
Kōrero:
Ka tautuhia e mātou te Takahanga A: te wheako i te iti rawa i te kotahi ahua.
E whā ngā putanga pea o te whiunga moni e rua:
1. HH
2. HT
3. TH
4. TT
Ko ngā kaupapa kei roto i te iti rawa kotahi te whakaahua ko:
– HT
– TH
– TT
Me tatau te tūponotanga o ia mea:
– Te maha o ngā huihuinga ka taea (katoa): 4
– Te maha o ngā huihuinga kei roto i te iti rawa kotahi te whakaahua: 3
\[
P(A) = \frac{Te maha o ngā takahanga me te upoko kotahi, neke atu rānei}{Te tapeke o ngā takahanga} = \frac{3}{4}
\]
Nō reira, ko te tūponotanga o te putanga mai o tētahi whakaahua, neke atu rānei, ko \(\frac{3}{4}\).
Whakamutunga
Mā te matapakinga o ngā raruraru i runga ake nei ka whakaatuhia te huarahi e taea ai e tātou te tatau i te tūponotanga o tētahi huihuinga pūhui, ahakoa he mea motuhake tetahi ki tetahi, he mea kore motuhake tetahi ki tetahi. Mā te mārama ki ngā ariā taketake me te whakamahi i ngā tātai tika, ka taea e tātou te whakatau i te tūponotanga o ētahi huinga o ngā huihuinga e puta ana i roto i ngā āhuatanga o ia rā. Me haere tonu te whakaharatau i ō pūkenga ki ngā raruraru rerekē kia matatau ake ai tātou ki te whakatau i te tūponotanga o ngā huihuinga pūhui.
