Mehlala ea Lipotso tse Buisanang ka Mosebetsi oa Kabo ea Binomial
Kabo ea binomial ke kabo e arohaneng ea menyetla e hlalosang palo ea katleho tekong e nang le liteko tse 'maloa tse ikemetseng tse nang le liphetho tse peli tse ka khonehang: katleho le ho hloleha. Teko ka 'ngoe e bitsoa teko, 'me kabo ea binomial hangata e sebelisoa maemong ao palo ea katleho ho tsoa litekong tse 'maloa tse ikemetseng e leng ea bohlokoa. Sehloohong sena, re tla buisana ka likhopolo tsa motheo tsa kabo ea binomial le ho fana ka mehlala le litharollo.
Mehopolo ea Motheo ea Mosebetsi oa Kabo ea Binomial
Pele re kena lipotsong le puisanong ea mohlala, ha re buisaneng ka likhopolo tse ling tsa motheo tse amanang le kabo ea binomial.
1. Tlhaloso: Kabo ea binomial e hlalosoa e le kakaretso ea katleho litekong tse ikemetseng tsa 'n', moo teko ka 'ngoe e nang le liphetho tse peli tse ka khonehang: katleho (ka monyetla oa p) kapa ho hloleha (ka monyetla oa q = 1 - p).
2. Mosebetsi wa Kgonahalo: Mosebetsi wa kgonego wa kabo ya binomial ke:
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{nk}
\]
Moo:
– \( P(X = k) \) ke monyetla wa ho ba le katleho ya k ditekong tsa n.
– \( \binom{n}{k} \) ke motswako wa n take k, e hlaloswang e le \( \frac{n!}{k!(nk)!} \).
– \( p \) ke monyetla oa katleho tekong e 'ngoe le e 'ngoe.
– \( (1-p) \) ke monyetla oa ho hloleha tekong ka 'ngoe.
3. Boleng bo Lebeletsoeng le Phapang:
– Boleng bo lebelletsweng (karolelano) ba kabo ya binomial ke \( \mu = np \).
– Phapang ya kabo ya binomial ke \( \sigma^2 = np(1-p) \).
Jwale, ha re sebediseng mehopolo ena bothateng ba mohlala ho fana ka kutlwisiso e tebileng.
Mohlala Potso ea 1: Lipalo tsa Motheo tsa Kabo ea Binomial
Potso:
Khamphani e hlahisa likarolo tsa elektroniki tse nang le monyetla oa 0.95 oa hore karolo ka 'ngoe e fete tekong ea boleng. Haeba ho hlahisoa likarolo tse 10, bala monyetla oa hore likarolo tse 8 hantle li fete tekong ea boleng.
Puisano:
Re ka sebelisa foromo ea kabo ea binomial ho rarolla bothata bona. Taba ea pele, re khetholla liparamente tse latelang:
– \( n \) (palo yohle ya diteko) = 10
– \( k \) (palo ea katleho) = 8
– \( p \) (monyetla oa katleho) = 0.95
– \( q \) (monyetla oa ho hloleha) = 1 – 0.95 = 0.05
Ebe u nkela boleng bona sebaka ka foromo ea kabo ea binomial:
\[
P(X = 8) = \binom{10}{8} (0.95)^8 (0.05)^2
\]
Taba ea pele, bala motsoako \( \binom{10}{8} \):
\[
\binom{10}{8} = \frac{10!}{8!(10-8)!} = \frac{10!}{8!2!} = \frac{10 \makgetlo a 9 \makgetlo a 8!}{8! \makgetlo a 2!} = \frac{10 \makgetlo a 9}{2 \makgetlo a 1} = 45
\]
Ebe, bala menyetla \( (0.95)^8 \) le \( (0.05)^2 \):
\[
(0.95)^8 \hoo e ka bang 0.6634
\]
\[
(0.05)^2 = 0.0025
\]
Qetellong, atisa boleng boo kaofela ho fumana:
\[
P(X = 8) = 45 \makgetlo 0.6634 \makgetlo 0.0025 \hoo e ka bang 0.0744
\]
Kahoo, monyetla oa hore likarolo tse 8 ho tse 10 li fete teko ea boleng ke hoo e ka bang 0.0744 kapa 7.44%.
Mohlala oa Potso ea 2: Monyetla o Akaretsang
Potso:
Leha ho le jwalo, le ntse le sebetsa le khamphani e tshwanang, bala monyetla wa hore bonyane dikarolo tse 9 ho tse 10 di fete teko ya boleng.
Puisano:
Ho rarolla bothata bona, re hloka ho bala kgonahalo e kopaneng. Kgonahalo ya hore bonyane dikarolo tse 9 ho tse 10 di fete tekong e bolela hore re bala \( P(X \geq 9) \), e ka ngolwang ka tsela ena:
\[
P(X \geq 9) = P(X = 9) + P(X = 10)
\]
Ho sebelisa foromo ea kabo ea binomial:
\[
P(X = 9) = \binom{10}{9} (0.95)^9 (0.05)^1
\]
\[
P(X = 10) = \binom{10}{10} (0.95)^{10} (0.05)^0
\]
Taba ea pele, bala motsoako bakeng sa nyeoe ka 'ngoe:
\[
\binom{10}{9} = \frac{10!}{9!(10-9)!} = 10
\]
\[
\binom{10}{10} = 1
\]
Ebe, bala menyetla ea \( P(X = 9) \) le \( P(X = 10) \):
\[
P(X = 9) = 10 \makgetlo (0.95)^9 \makgetlo 0.05
\]
\[
(0.95)^9 \hoo e ka bang 0.6302
\]
\[
P(X = 9) = 10 \makgetlo 0.6302 \makgetlo 0.05 \hoo e ka bang 0.3151
\]
\[
P(X = 10) = 1 \makgetlo (0.95)^{10} \makgetlo a 1
\]
\[
(0.95)^{10} \hoo e ka bang 0.5987
\]
\[
P(X = 10) = 0.5987
\]
Kakaretso ea monyetla oa \( P(X \geq 9) \):
\[
P(X \geq 9) = 0.3151 + 0.5987 \hoo e ka bang 0.9138
\]
Kahoo, monyetla oa hore bonyane likarolo tse 9 ho tse 10 li fete teko ea boleng ke hoo e ka bang 0.9138 kapa 91.38%.
Mohlala oa Potso ea 3: Boleng bo Lebeletsoeng le Phapang
Potso:
Bala boleng bo lebeletsoeng le phapang ea palo ea likarolo tse fetang teko ea boleng ho tsoa likarolong tse 10 tse hlahisitsoeng, ka monyetla oa ho feta oa 0.95.
Puisano:
Sebelisa foromo e latelang:
– Boleng bo lebelletsweng (bohareng) \( \mu = np \)
– Phapang \( \sigma^2 = np(1-p) \)
Ka \( n = 10 \) le \( p = 0.95 \):
\[
\mu = 10 \makhetlo 0.95 = 9.5
\]
\[
\sigma^2 = 10 \makgetlo a 0.95 \makgetlo a 0.05 = 0.475
\]
Kahoo, boleng bo lebeletsoeng ba palo ea likarolo tse fetang teko ea boleng ke 9.5, 'me phapang ke 0.475.
Qetello
Ka mathata a mehlala e meraro a kaholimo, re buisane ka mokhoa oa ho bala monyetla o sebelisang kabo ea binomial bakeng sa maemo a fapaneng: ho bala monyetla o nepahetseng, monyetla o kopaneng, le ho bala boleng le phapang e lebelletsoeng. Tsebo ea kabo ea binomial e thusa masimong a fapaneng, joalo ka tlhahiso, lipatlisiso tsa bongaka le lipalo-palo tsa sechaba, moo liphetho tsa liteko tse phetoang khafetsa tse nang le liphetho tse peli tse ka khonehang li ka hlahlojoang ho thusa ho etsa liqeto. Re tšepa hore mathata a mohlala le lipuisano tse fanoeng li tla thusa ho ntlafatsa kutloisiso ea hau ea kabo ea binomial.
