Nā nīnau hoʻohālike a me ke kūkākūkā ʻana o ka hoʻokaʻawale ʻana o Binomial
ʻO ka hoʻokaʻawale binomial kekahi o nā hoʻokaʻawale probabilidad discrete i hoʻohana pinepine ʻia. He mea pono ia no ke kumu hoʻohālike i ka helu o nā holomua ma waena o nā hoʻokolohua like a kūʻokoʻa, kahi e hana ai kēlā me kēia i kahi holomua a i ʻole ka hāʻule ʻana. Ma kēia ʻatikala, e luʻu hohonu mākou i ka hoʻokaʻawale binomial ma ka hāʻawi ʻana i kekahi mau laʻana a me kahi kūkākūkā kikoʻī.
Hoʻolauna i ka Binomial Distribution
Nā ʻano nui o ka hoʻokaʻawale binomial:
1. n : Helu o nā hoʻāʻo a i ʻole nā hana hou ʻana.
2. p : Ka hiki ke kūleʻa i kēlā me kēia hoʻokolohua.
3. q = 1-p : Ka hiki ke hāʻule i kēlā me kēia hoʻokolohua.
ʻO ka hana nuipa probabilidad o ka binomial distribution penei:
\[ P(X = k) = {n \koho k} p^k (1-p)^{nk} \]
Ma hea:
– \( {n \koho k} = \frac{n!}{k!(nk)!} \)
– \( X \): He loli kaulele e hōʻike ana i ka helu o nā holomua.
– \( k \): Ka helu o nā holomua i ʻimi ʻia.
Nā Nīnau Laʻana a me ke Kūkākūkā
E hoʻomaka kākou me kekahi mau pilikia hoʻohālike e hoʻomaopopo pono ai i ke kumumanaʻo o ka hoʻokaʻawale binomial.
Laʻana 1: Ke koho ʻana mai kahi hui o nā haumāna
Eia kekahi laʻana, ke manaʻo nei he hui ko kākou o 10 mau haumāna, a ʻo ka likelika e koho ʻia ai kēlā me kēia haumāna e komo i kahi hoʻokūkū he 0,3. Makemake mākou e ʻike i ka likelika e koho pololei ʻia ai he 4 mau haumāna.
KaʻAnuʻu Hana 1: E ʻike i nā palena o ka hoʻokaʻawale binomial.
– \( n = 10 \)
– \( p = 0.3 \)
KaʻAnuʻu Hana 2: E hoʻohana i ka hoʻokaʻawale binomial e helu i ka hiki ke loaʻa o \( X = 4 \).
\[ P(X = 4) = {10 \koho 4} (0.3)^4 (0.7)^6 \]
Ke helu nei \( {10 \choose 4} \):
\[ {10 \koho 4} = \frac{10!}{4!(10-4)!} = \frac{10!}{4!6!} = 210 \]
I kēia manawa e helu iā \( (0.3)^4 \) a me \( (0.7)^6 \):
\[ (0.3)^4 = 0.0081 \]
\[ (0.7)^6 = 0.117649 \]
No laila,
P(X = 4) = 210 \cdot 0.0081 \cdot 0.117649 \approx 0.20012 \]
No laila, ʻo ka likelika e koho pololei ʻia ai he 4 mau haumāna ma kahi o 0.20012 a i ʻole 20.012%.
Laʻana 2: Ka Pahiki i emi iho a i ʻole like me 2
I kēia manawa, no ka laʻana, ua nīnau ʻia mākou e pili ana i ka hiki ke koho ʻia he 2 mau haumāna a emi mai paha.
KaʻAnuʻu Hana 1: Pono mākou e helu iā \( P(X = 0) \), \( P(X = 1) \), a me \( P(X = 2) \).
– No \( P(X = 0) \):
\[ P(X = 0) = {10 \koho 0} (0.3)^0 (0.7)^{10} \]
\[ {10 \koho 0} = 1 \]
\[ (0.7)^{10} = 0.0282475 \]
\[ P(X = 0) = 1 \cdot 1 \cdot 0.0282475 = 0.0282475 \]
– No \( P(X = 1) \):
\[ P(X = 1) = {10 \koho 1} (0.3)^1 (0.7)^9 \]
\[ {10 \koho 1} = 10 \]
\[ (0.3) \cdot (0.7)^9 = 0.1210608 \]
\[ P(X = 1) = 10 \cdot 0.3 \cdot 0.1210608 = 0.3631824 \]
– No \( P(X = 2) \):
\[ P(X = 2) = {10 \koho 2} (0.3)^2 (0.7)^8 \]
\[ {10 \koho 2} = 45 \]
\[ (0.3)^2 \cdot (0.7)^8 = 0.2334744 \]
\[ P(X = 2) = 45 \cdot 0.09 \cdot 0.2334744 = 0.2334744 \]
KaʻAnuʻu Hana 2: E hoʻohui i nā mea hiki ke hana.
P(X 2) = P(X = 0) + P(X = 1) + P(X = 2)
P(X \leq 2) = 0.0282475 + 0.3631824 + 0.3826372 = 0.7740671
No laila, ʻo ka likelika e koho ʻia ai he 2 mau haumāna a emi mai paha ma mua o 0.7740671 a i ʻole 77.41%.
Laʻana 3: Ka hiki ke loaʻa ma ka liʻiliʻi he 8
Inā hana ʻia kahi hoʻokolohua he 12 mau manawa, a ʻo ka likelika o ka holomua ma kēlā me kēia hoʻokolohua he 0.5, he aha ka likelika e hiki mai ana ma ka liʻiliʻi he 8 mau holomua?
KaʻAnuʻu Hana 1: Hoʻonohonoho i nā palena binomial: \( n = 12, p = 0.5 \).
KaʻAnuʻu Hana 2: E huli i ka hiki ke loaʻa no \( X \geq 8 \).
Pono kēia i ka helu ʻana i kekahi mau mea hiki ke hoʻohālikelike ʻia a hoʻohui iā lākou:
\[ P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) \]
E helu pākahi:
– No \( P(X = 8) \):
\[ P(X = 8) = {12 \koho 8} (0.5)^8 (0.5)^4 \]
\[ {12 \koho 8} = 495 \]
\[ (0.5)^{12} = 0.0002441406 \]
\[ P(X = 8) = 495 \cdot 0.0002441406 = 0.1208496 \]
– No \( P(X = 9) \):
\[ P(X = 9) = {12 \koho 9} (0.5)^9 (0.5)^3 \]
\[ {12 \koho 9} = 220 \]
\[ P(X = 9) = 220 \cdot 0.0002441406 = 0.05371094 \]
– No \( P(X = 10) \):
\[ P(X = 10) = {12 \koho 10} (0.5)^{10} (0.5)^2 \]
\[ {12 \koho 10} = 66 \]
\[ P(X = 10) = 66 \cdot 0.0002441406 = 0.01611328 \]
– No \( P(X = 11) \):
\[ P(X = 11) = {12 \koho 11} (0.5)^{11} (0.5)^1 \]
\[ {12 \koho 11} = 12 \]
\[ P(X = 11) = 12 \cdot 0.0002441406 = 0.002929688 \]
– No \( P(X = 12) \):
\[ P(X = 12) = {12 \koho 12} (0.5)^{12} \]
\[ {12 \koho 12} = 1 \]
\[ P(X = 12) = 1 \cdot 0.0002441406 = 0.0002441406 \]
KaʻAnuʻu Hana 3: E hoʻohui i nā mea hiki ke hana ʻia a pau.
P(X 8) = 0.1208496 + 0.05371094 + 0.01611328 + 0.002929688 + 0.0002441406 kokoke 0.1938477
No laila, ʻo ka likelika e loaʻa ma ka liʻiliʻi he 8 mau holomua i 12 mau hoʻokolohua ma kahi o 0.1938477 a i ʻole 19.38%.
Ka hopena
ʻO ka hoʻokaʻawale binomial kahi manaʻo nui i nā helu helu he mea nui ia i nā noi hana he nui. Ma ka hoʻomaopopo ʻana pehea e helu ai i nā probabilities no nā hihia like ʻole o ka hoʻokaʻawale binomial, e like me ka mea i hōʻike ʻia ma nā laʻana ma luna, hiki iā mākou ke hoʻopili i kēia manaʻo i nā kūlana honua maoli. Hoʻoikaika pū kēia hana i ko mākou ʻike pehea e hana ai nā ʻano probability i kahi pōʻaiapili maopopo a hoʻonohonoho pono ʻia.
