Nā nīnau hoʻohālike e kūkākūkā ana i ke kānāwai no ka hoʻohui ʻana i ʻelua mau hanana kū hoʻokahi A a me B
I ke kumumanaʻo probabilidad, ʻo ke kānāwai huina o nā hanana ʻelua kekahi o nā loina kumu i hoʻohana ʻia e helu i ka probabilidad o nā hanana he nui. Hoʻopili pinepine ʻia kēia manaʻo i nā kūlana like ʻole e hoʻomaopopo i nā hopena hiki o kekahi mau hanana. Ma kēia ʻatikala, e kūkākūkā mākou i ke kānāwai huina o nā hanana ʻelua e kūʻokoʻa ana a hāʻawi i nā laʻana e wehewehe i kēia manaʻo.
ʻO ke Kānāwai o ka Hoʻohui ʻana i ʻElua mau Hanana Kūʻokoʻa
ʻO ka mea mua, he mea nui e hoʻomaopopo i ke ʻano o nā hanana kūʻokoʻa like. Ua ʻōlelo ʻia ʻelua mau hanana he ʻokoʻa a kūʻokoʻa paha kekahi inā ʻaʻole hiki iā lākou ke hana i ka manawa like. I nā huaʻōlelo ʻē aʻe, ʻaʻohe mea i loko o ka hoʻonohonoho o hoʻokahi hanana he mea hoʻi i loko o ka hoʻonohonoho o kekahi hanana ʻē aʻe.
ʻŌlelo ke kānāwai hoʻohui ma ka probability inā he ʻelua mau hanana \(A\) a me \(B\) he kūʻokoʻa kekahi i kekahi, a laila ʻo ka probability o kekahi hanana \(A\) a i ʻole \(B\) ka huina o nā probabilities o nā hanana ʻelua. Ma ke ʻano makemakika, hiki ke hōʻike ʻia kēia lula penei:
P(A B) = P(A) + P(B)
kahi ʻo \(P(A \cup B)\) ka hiki ke hana ʻia o \(A\) a i ʻole \(B\), ʻo \(P(A)\) ka hiki ke hana ʻia o \(A\), a ʻo \(P(B)\) ka hiki ke hana ʻia o \(B\).
Nā nīnau kūkākūkā hoʻohālike
E kūkākūkā kākou i kekahi mau laʻana e wehewehe i ka hoʻopili ʻia ʻana o ke kānāwai no ka hoʻohui ʻana i ʻelua mau hanana kūʻokoʻa kekahi i kekahi.
Laʻana Nīnau 1
Nīnau:
Hoʻolei ʻia hoʻokahi manawa kahi dai ʻeono ʻaoʻao. E huli i ka hiki ke loaʻa ka helu hopena he 2 a i ʻole 4.
Kūkākūkā:
Hiki iā mākou ke wehewehe i ka hanana \(A\) ma ke ʻano he hanana o ka waiwai 2, a me ka hanana \(B\) ma ke ʻano he hanana o ka waiwai 4. Penei:
– ʻO \(P(A)\) ka hiki ke ʻike ʻia ka waiwai 2.
– ʻO \(P(B)\) ka hiki ke ʻike ʻia ka waiwai 4.
ʻOiai he ʻeono ʻaoʻao like ka like o ka make, ʻo ka likelika o ka ʻōwili ʻia ʻana o kahi waiwai kikoʻī he \( \frac{1}{6} \). No laila:
\[ P(A) = \frac{1}{6} \]
\[ P(B) = \frac{1}{6} \]
ʻAʻole pili kekahi i kekahi nā hanana \(A\) a me \(B\) no ka mea ʻaʻole hiki ke ʻike ʻia nā waiwai 2 a me 4 i ka manawa like ma kahi ʻōwili hoʻokahi o ka make. No laila, me ka hoʻohana ʻana i ke kānāwai hoʻohui no ʻelua mau hanana pili kekahi i kekahi:
\[ P(A \cup B) = P(A) + P(B) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \]
No laila, ʻo ka likelika e ʻike ʻia ai ka waiwai he 2 a i ʻole 4 he \( \frac{1}{3} \) a i ʻole ma kahi o 33.33%.
Laʻana Nīnau 2
Nīnau:
I loko o kahi ʻeke he 10 mau pōpō i haku ʻia me 4 mau pōpō ʻulaʻula a me 6 mau pōpō polū. Inā koho mākou i hoʻokahi pōpō ma ke ʻano maʻamau, he aha ka likelika e ʻulaʻula a polū paha ka pōpō i huki ʻia?
Kūkākūkā:
Hiki iā mākou ke wehewehe i ka hanana \(A\) e like me ka lawe ʻana i ka pōpō ʻulaʻula, a ʻo ka hanana \(B\) e like me ka lawe ʻana i ka pōpō polū. Penei:
– ʻO \(P(A)\) ka hiki ke koho i kahi pōpō ʻulaʻula.
– ʻO \(P(B)\) ka hiki ke koho i kahi pōpō polū.
Hiki ke helu ʻia ka hiki ke loaʻa o kēlā me kēia hanana penei:
\[ P(A) = \frac{\text{Helu o nā pōpō ʻulaʻula}}{\text{Helu holoʻokoʻa o nā pōpō}} = \frac{4}{10} = \frac{2}{5} \]
\[ P(B) = \frac{\text{Helu o nā pōpō polū}}{\text{Helu holoʻokoʻa o nā pōpō}} = \frac{6}{10} = \frac{3}{5} \]
ʻAʻole pili kekahi i kekahi nā hanana \(A\) a me \(B\) no ka mea ʻaʻole hiki i ka pōpō ke ʻulaʻula a me ka uliuli. No laila, me ka hoʻohana ʻana i ke kānāwai hoʻohui no ʻelua mau hanana pili kekahi i kekahi:
\[ P(A \cup B) = P(A) + P(B) = \frac{2}{5} + \frac{3}{5} = 1 \]
No laila, ʻo ka likelika e ʻulaʻula a polū paha ka pōpō i huki ʻia he 1, a i ʻole 100%. He mea kūpono kēia no ka mea he ʻulaʻula a polū paha nā pōpō a pau i loko o ka ʻeke.
Laʻana Nīnau 3
Nīnau:
I loko o kahi papa o 20 mau haumāna, 7 o lākou makemake i ka makemakika, 5 o lākou makemake i ka ʻepekema, a ʻaʻohe mea makemake i nā mea ʻelua. Inā koho ʻia kekahi haumāna ma ke ʻano maʻamau, e ʻimi i ka hiki ke makemake ka haumāna i ka makemakika a i ʻole ka ʻepekema.
Kūkākūkā:
Hiki iā mākou ke wehewehe i ka hanana \(A\) e like me ka makemake i ka makemakika, a me ka hanana \(B\) e like me ka makemake i ka ʻepekema. Penei:
– ʻO \(P(A)\) ka likelika e makemake ai kekahi haumāna i ka makemakika.
– ʻO \(P(B)\) ka likelika e makemake ai kekahi haumāna i ka ʻepekema.
Hiki ke helu ʻia ka hiki ke loaʻa o kēlā me kēia hanana penei:
\[ P(A) = \frac{\text{Helu o nā haumāna makemake i ka makemakika}}{\text{Helu o nā haumāna}} = \frac{7}{20} \]
\[ P(B) = \frac{\text{Helu o nā haumāna makemake i ka ʻepekema}}{\text{Helu holoʻokoʻa o nā haumāna}} = \frac{5}{20} = \frac{1}{4} \]
Ua kūʻokoʻa kekahi i kekahi nā hanana \(A\) a me \(B\) no ka mea ʻaʻohe haumāna makemake iā lāua ʻelua. No laila, me ka hoʻohana ʻana i ke kānāwai hoʻohui no ʻelua mau hanana kūʻokoʻa:
\[ P(A \cup B) = P(A) + P(B) = \frac{7}{20} + \frac{5}{20} = \frac{12}{20} = \frac{3}{5} \]
No laila, ʻo ka likelika e makemake ai kekahi haumāna i koho ʻia ma ke ʻano hewa i ka makemakika a i ʻole ka ʻepekema he \( \frac{3}{5} \) a i ʻole 60%.
Ka hopena
ʻO ke kānāwai hoʻohui o ʻelua mau hanana kūʻokoʻa kekahi i kekahi he manaʻo nui ia i ke kumumanaʻo probabili e hoʻomaʻalahi i ka helu ʻana o ka probabili o kahi hanana hui. Ma nā laʻana ma luna, ua ʻike mākou e hiki ke hoʻopili ʻia kēia kumumanaʻo i nā kūlana maoli e like me ka ʻōwili ʻana i ka dice, ka huki ʻana i nā pōpō mai kahi ʻeke, a i ʻole ke koho ʻana i nā haumāna mai kahi papa. Ma ka hoʻomaopopo ʻana a me ka hoʻomaopopo ʻana i kēia manaʻo, hiki iā mākou ke helu pono i nā probabili o nā hanana kūʻokoʻa like ʻole i ke ola o kēlā me kēia lā.
