Nā nīnau hoʻohālike e kūkākūkā ana i ke kānāwai o ka hoʻohui ʻana i ʻelua mau hanana A a me B i kū ʻole kekahi i kekahi
Pendahuluan
I ka probability, he mea koʻikoʻi ka hoʻomaopopo ʻana i nā manaʻo kumu a me kā lākou noi. ʻO kahi manaʻo e kū pinepine mai ana, ʻo ia ke kānāwai o ka hoʻohui probability. Ma kēia ʻatikala, e ʻeli hohonu mākou i loko o ke kānāwai o ka hoʻohui no ʻelua mau hanana ʻaʻole exclusive. Hana ʻia nā hanana ʻaʻole exclusive i ka wā e kaʻana like ai nā hanana ʻelua i kahi mea hiki, a i ʻole ma nā huaʻōlelo ʻē aʻe, hiki i nā hanana ʻelua ke hana like. No ka hoʻomaʻamaʻa ʻana i ka hoʻomaopopo ʻana, e nānā kākou i kahi pilikia laʻana a me kāna wehewehe ʻana.
Kumumanaʻo Kumu
Inā loaʻa iā mākou ʻelua mau hanana A a me B ʻaʻole i kū hoʻokahi kekahi i kekahi, a laila hiki ke hōʻike ʻia ke kānāwai hoʻohui no ka helu ʻana i ka hiki ke loaʻa o ka hanana A a i ʻole B penei:
P(A \cup B) = P(A) + P(B) – P(A \cap B) \]
Maanei:
– ʻO \( P(A \cup B) \) ka hiki ke kū mai ʻo A a i ʻole B (a i ʻole nā mea ʻelua).
– ʻO \( P(A) \) ka hiki ke hana ʻia ka hanana A.
– ʻO \( P(B) \) ka hiki ke hana ʻia ka hanana B.
– ʻO \( P(A \cap B) \) ka hiki ke kū mai ʻo A lāua ʻo B i ka manawa like.
Laʻana pilikia
Nīnau 1: Ma ke kula, makemake ka 70% o nā haumāna e pāʻani i ka pôpeku (hanana A), makemake ka 40% o nā haumāna e pāʻani i ka basketball (hanana B), a makemake ka 20% o nā haumāna e pāʻani i nā haʻuki ʻelua. He aha ka likelika e makemake ai kahi haumāna e pāʻani i ka pôpeku a i ʻole ka basketball?
Kūkākūkā:
Ua ʻike ʻia:
– \( P(A) = 0.70 \)
– \( P(B) = 0.40 \)
– \( P(A \cap B) = 0.20 \)
Ke hoʻohana nei i ke kānāwai o ka hoʻohui ʻana i ʻelua mau hanana kūʻokoʻa like ʻole:
P(A \cup B) = P(A) + P(B) – P(A \cap B) \]
\[ P(A \cup B) = 0.70 + 0.40 – 0.20 \]
\[ P(A \cup B) = 0.90 \]
No laila, ʻo ka likelika e makemake ai kekahi haumāna i ka pāʻani pôpeku a i ʻole ka basketball he 0.90 a i ʻole 90%.
Nīnau 2: Ma kahi noiʻi, makemake ka 35% o ka poʻe pane i ke kofe (hanana A), makemake ka 50% o ka poʻe pane i ke kī (hanana B), a makemake ka 10% o ka poʻe pane i nā mea ʻelua. He aha ka likelika e makemake ai ka mea pane i ke kofe a i ʻole ke kī?
Kūkākūkā:
Ua ʻike ʻia:
– \( P(A) = 0.35 \)
– \( P(B) = 0.50 \)
– \( P(A \cap B) = 0.10 \)
Ke hoʻohana nei i ke kānāwai o ka hoʻohui ʻana i ʻelua mau hanana kūʻokoʻa like ʻole:
P(A \cup B) = P(A) + P(B) – P(A \cap B) \]
\[ P(A \cup B) = 0.35 + 0.50 – 0.10 \]
\[ P(A \cup B) = 0.75 \]
No laila, ʻo ka likelika e makemake ai ka mea pane i ke kofe a i ʻole ke kī he 0.75 a i ʻole 75%.
Nīnau 3: Mai ka ʻikepili helu, ua ʻike ʻia he 60% o ka poʻe makemake i ka heluhelu ʻana i nā puke (hanana A), 30% o ka poʻe makemake i ka nānā ʻana i nā kiʻiʻoniʻoni (hanana B), a he 15% o ka poʻe makemake i ka hana ʻana i nā mea ʻelua. He aha ka likelika e makemake ai ke kanaka i ka heluhelu ʻana i nā puke a i ʻole ka nānā ʻana i nā kiʻiʻoniʻoni?
Kūkākūkā:
Ua ʻike ʻia:
– \( P(A) = 0.60 \)
– \( P(B) = 0.30 \)
– \( P(A \cap B) = 0.15 \)
Ke hoʻohana nei i ke kānāwai o ka hoʻohui ʻana i ʻelua mau hanana kūʻokoʻa like ʻole:
P(A \cup B) = P(A) + P(B) – P(A \cap B) \]
\[ P(A \cup B) = 0.60 + 0.30 – 0.15 \]
\[ P(A \cup B) = 0.75 \]
No laila, ʻo ka likelika e makemake ai ke kanaka i ka heluhelu ʻana i nā puke a i ʻole ka nānā ʻana i nā kiʻiʻoniʻoni he 0.75 a i ʻole 75%.
Nīnau 4: Mai kahi noiʻi ma kahi hale ʻaina, ua kauoha ʻo 55% o nā mea kūʻai aku i ka pizza (hanana A), ua kauoha ʻo 35% o nā mea kūʻai aku i ka pasta (hanana B), a ua kauoha ʻo 20% o nā mea kūʻai aku i nā mea ʻelua. He aha ka likelika e kauoha ai kahi mea kūʻai aku i ka pizza a i ʻole ka pasta?
Kūkākūkā:
Ua ʻike ʻia:
– \( P(A) = 0.55 \)
– \( P(B) = 0.35 \)
– \( P(A \cap B) = 0.20 \)
Ke hoʻohana nei i ke kānāwai o ka hoʻohui ʻana i ʻelua mau hanana kūʻokoʻa like ʻole:
P(A \cup B) = P(A) + P(B) – P(A \cap B) \]
\[ P(A \cup B) = 0.55 + 0.35 – 0.20 \]
\[ P(A \cup B) = 0.70 \]
No laila, ʻo ka likelika e kauoha ai ka mea kūʻai aku i ka pizza a i ʻole ka pasta he 0.70 a i ʻole 70%.
Ka hopena
I ke aʻo ʻana i ka hiki ke hoʻohālikelike ʻia, he mea nui loa ke kānāwai o ka hoʻohui ʻana o ʻelua mau hanana ʻaʻole kū hoʻokahi ʻo A a me B no ka hoʻoholo ʻana i ka hiki ke hana ʻia kekahi a i ʻole nā hanana ʻelua i ka manawa like. ʻO ka hoʻohana ʻana i ke ʻano hana \( P(A \cup B) = P(A) + P(B) – P(A \cap B) \) e maʻalahi ai ka helu ʻana i ka hiki ke hui pū ʻia o nā hanana ʻelua i loaʻa kahi intersection a i ʻole kahi mea like.
Ma o nā hiʻohiʻona ma luna, hiki iā kākou ke ʻike i ka hoʻomaopopo ʻana i nā ʻanuʻu kumu o kēia helu ʻana e hiki ai iā kākou ke hoʻopili iā ia ma nā kūlana like ʻole, inā paha ma ka hoʻonaʻauao ʻana, nā anamanaʻo, a i ʻole ke ola o kēlā me kēia lā. ʻO ka maopopo a me ka pololei i ka ʻike ʻana i nā hanana e kaʻana like i nā mea he mea nui ia no ka helu pololei ʻana i ka hiki. No laila, ʻo ka hoʻomaʻamaʻa mau a me ka hoʻomaopopo ʻana i nā manaʻo kumu e kōkua nui i ka hoʻopaʻa ʻana i kēia kumuhana.
