Fa'ata'ita'iga o Fesili ma Talanoaga o le Tu'ufa'atasiga Binomial
O le tufatufaina binomial o se tasi o tufatufaga fa'apitoa e masani ona fa'aaogaina. E aoga mo le fa'ata'ita'iina o le aofa'i o manuia i le tele o tofotofoga tutusa ma tuto'atasi, o ia tofotofoga ta'itasi e maua ai se manuia pe toilalo. I totonu o lenei tusiga, o le a matou su'esu'eina atili ai le tufatufaina binomial e ala i le tu'uina atu o ni fa'ata'ita'iga ma se talanoaga auiliili.
Fa'atomuaga i le Tufatufaga Binomial
O uiga autū o le tufatufaina atu o binomial:
1. n : Numera o tofotofoga po'o toe fai.
2. p : Avanoa e manuia ai tofotofoga taʻitasi.
3. q = 1-p : Avanoa e toilalo ai i tofotofoga taitasi.
O le galuega faatino o le tele o le avanoa o le tufatufaina atu o le binomial e faapea:
\[ P(X = k) = {n \choose k} p^k (1-p)^{nk} \]
di mana:
– \( {n \choose k} = \frac{n!}{k!(nk)!} \)
– \( X \): O se fesuia'iga fa'afuase'i e fai ma sui o le aofa'i o manuia.
– \( k \): O le aofaʻi o manuia sa sailia.
Fesili Fa'ata'ita'i ma Talanoaga
Se'i o tatou amata i ni fa'ata'ita'iga o fa'afitauli ina ia malamalama atili ai i le manatu o le binomial distribution.
Faataitaiga 1: Filifiliga mai se Vaega o Tamaiti Aʻoga
Mo se faʻataʻitaʻiga, faapea o loʻo ia i tatou se vaega e toʻa 10 tamaiti aʻoga, ma o le avanoa e filifilia ai tamaiti aʻoga taʻitasi e auai i se tauvaga e 0,3. Tatou te fia iloa le avanoa e filifilia ai tonu tamaiti aʻoga e toʻa 4.
Laasaga 1: Fa'ailoa mai tapula'a o le tufatufaina atu o le binomial.
– \( n = 10 \)
– \( p = 0.3 \)
Laasaga 2: Faaaoga le binomial distribution e fuafua ai le avanoa o le \( X = 4 \).
\[ P(X = 4) = {10 \filifilia 4} (0.3)^4 (0.7)^6 \]
Fuafuaina \( {10 \choose 4} \):
\[ {10 \filifiliga 4} = \frac{10!}{4!(10-4)!} = \frac{10!}{4!6!} = 210 \]
Ia fuafua nei \( (0.3)^4 \) ma le \( (0.7)^6 \):
\[ (0.3)^4 = 0.0081 \]
\[ (0.7)^6 = 0.117649 \]
O lea,
\[ P(X = 4) = 210 \cdot 0.0081 \cdot 0.117649 \approx 0.20012 \]
O lea la, o le avanoa e filifilia ai tamaiti aʻoga e toʻa 4 e tusa ma le 0.20012 poʻo le 20.012%.
Faʻataʻitaʻiga 2: Avanoa e Laʻititi ifo nai lo pe Tutusa ma le 2
O lea la, mo se faʻataʻitaʻiga, ua fesiligia i tatou e uiga i le avanoa e itiiti ifo pe tutusa ma le 2 tamaiti aʻoga o le a filifilia.
Laasaga 1: E tatau ona tatou fuafuaina \( P(X = 0) \), \( P(X = 1) \), ma le \( P(X = 2) \).
– Mo \( P(X = 0) \):
\[ P(X = 0) = {10 \choose 0} (0.3)^0 (0.7)^{10} \]
\[ {10 \choose 0} = 1 \]
\[ (0.7)^{10} = 0.0282475 \]
\[ P(X = 0) = 1 \cdot 1 \cdot 0.0282475 = 0.0282475 \]
– Mo \( P(X = 1) \):
\[ P(X = 1) = {10 \filifilia 1} (0.3)^1 (0.7)^9 \]
\[ {10 \choose 1} = 10 \]
\[ (0.3) \cdot (0.7)^9 = 0.1210608 \]
\[ P(X = 1) = 10 \cdot 0.3 \cdot 0.1210608 = 0.3631824 \]
– Mo \( P(X = 2) \):
\[ P(X = 2) = {10 \filifilia 2} (0.3)^2 (0.7)^8 \]
\[ {10 \choose 2} = 45 \]
\[ (0.3)^2 \cdot (0.7)^8 = 0.2334744 \]
\[ P(X = 2) = 45 \cdot 0.09 \cdot 0.2334744 = 0.2334744 \]
Laasaga 2: Fa'aopoopo fa'atasi avanoa.
\[ P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) \]
\[ P(X \leq 2) = 0.0282475 + 0.3631824 + 0.3826372 = 0.7740671 \]
O lea la, o le avanoa e itiiti ifo pe tutusa ma le 2 tamaiti aʻoga e filifilia e tusa ma le 0.7740671 poʻo le 77.41%.
Faʻataʻitaʻiga 3: Avanoa e le itiiti ifo i le 8
Afai e faia se fa'ata'ita'iga i le 12 taimi, ma o le avanoa e manuia ai i tofotofoga ta'itasi e 0.5, o le a le avanoa e tupu ai le itiiti ifo i le 8 manuia?
Laasaga 1: Seti ia binomial parameters: \( n = 12, p = 0.5 \).
Laasaga 2: Saili le avanoa mo \( X \geq 8 \).
O lenei mea e manaʻomia ai le fuafuaina o ni nai avanoa taʻitasi ma faʻaopoopoina:
\[ P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) \]
Faitau ta'ito'atasi:
– Mo \( P(X = 8) \):
\[ P(X = 8) = {12 \filifilia 8} (0.5)^8 (0.5)^4 \]
\[ {12 \choose 8} = 495 \]
\[ (0.5)^{12} = 0.0002441406 \]
\[ P(X = 8) = 495 \cdot 0.0002441406 = 0.1208496 \]
– Mo \( P(X = 9) \):
\[ P(X = 9) = {12 \filifilia 9} (0.5)^9 (0.5)^3 \]
\[ {12 \choose 9} = 220 \]
\[ P(X = 9) = 220 \cdot 0.0002441406 = 0.05371094 \]
– Mo \( P(X = 10) \):
\[ P(X = 10) = {12 \filifilia 10} (0.5)^{10} (0.5)^2 \]
\[ {12 \choose 10} = 66 \]
\[ P(X = 10) = 66 \cdot 0.0002441406 = 0.01611328 \]
– Mo \( P(X = 11) \):
\[ P(X = 11) = {12 \filifilia 11} (0.5)^{11} (0.5)^1 \]
\[ {12 \choose 11} = 12 \]
\[ P(X = 11) = 12 \cdot 0.0002441406 = 0.002929688 \]
– Mo \( P(X = 12) \):
\[ P(X = 12) = {12 \filifilia 12} (0.5)^{12} \]
\[ {12 \choose 12} = 1 \]
\[ P(X = 12) = 1 \cdot 0.0002441406 = 0.0002441406 \]
Laasaga 3: Fa'aopoopo avanoa uma.
\[ P(X \geq 8) = 0.1208496 + 0.05371094 + 0.01611328 + 0.002929688 + 0.0002441406 \approx 0.1938477 \]
O lea la, o le avanoa e tupu ai le itiiti ifo ma le 8 manuia i tofotofoga e 12 e tusa ma le 0.1938477 poʻo le 19.38%.
I'uga
O le tufatufaina atu o le binomial o se manatu faavae i fuainumera e taua tele i le tele o faʻaoga faatino. I le malamalama i le auala e fuafua ai avanoa mo tulaga eseese o le tufatufaina atu o le binomial, e pei ona faʻaalia i faʻataʻitaʻiga o loʻo i luga, e mafai ona tatou faʻaaogaina lenei manatu i tulaga moni o le lalolagi. O lenei faʻatinoga e faʻamalosia ai foʻi lo tatou malamalama i le auala e galue ai fausaga o avanoa i se tulaga manino ma faʻatulagaina.
