Ngā Tauira Pātai e Matapaki ana i te Mahi Tohatoha Binomial
Ko te tohatoha rua-ira he tohatoha tūponotanga motuhake e whakaahua ana i te maha o ngā angitu i roto i tētahi whakamātautau kei roto ko te maha o ngā whakamātautau motuhake me ngā putanga e rua: te angitu me te kore angitu. Ka kiia ia whakamātautau he whakamātautau, ā, he maha ngā wā ka whakamahia te tohatoha rua-ira i ngā āhuatanga e whai pānga ana ki te maha o ngā angitu puta noa i te maha o ngā whakamātautau motuhake. I roto i tēnei tuhinga, ka matapakihia e mātou ngā ariā taketake o te tohatoha rua-ira, me te whakarato tauira me ngā otinga.
Ngā Ariā Taketake o te Mahi Tohatoha Binomial
I mua i te urunga atu ki ngā tauira pātai me te matapakinga, me matapaki tātou i ētahi ariā matua e pā ana ki te tohatoha rua-ira.
1. Whakamāramatanga: Ko te tohatoha rua-ira te tapeke o ngā angitu i roto i ngā whakamātautau motuhake 'n', e rua ngā putanga pea o ia whakamātautau: angitu (me te tūponotanga p) me te kore rānei (me te tūponotanga q = 1 – p).
2. Mahi Tūponotanga: Ko te mahi tūponotanga o te tohatoha rua ko:
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{nk}
\]
kāore i te mana:
– Ko te P(X = k) te tūponotanga o te angitu o ngā whakamātautau e k.
– Ko \( \binom{n}{k} \) he huinga o n take k, e tautuhia ana ko \( \frac{n!}{k!(nk)!} \).
– Ko te tūponotanga o te angitu i ia whakamātautau ko \( p \).
– Ko te tūponotanga o te korenga i ia whakamātautau ko \( (1-p) \).
3. Uara me te Rerekētanga e Tumanakohia ana:
– Ko te uara e tumanakohia ana (te toharite) o te tohatoha rua ko \( \mu = np \).
– Ko te rerekētanga o te tohatoha rua-ira ko \( \sigma^2 = np(1-p) \).
Nā, me whakamahi ēnei ariā i roto i tētahi tauira rapanga hei whakarato i tētahi māramatanga hōhonu ake.
Tauira Pātai 1: Ngā Tātaitanga Taketake o te Tohatoha Binomial
Pātai:
He kamupene e hanga ana i ngā wāhanga hiko, ā, ko te tūponotanga 0.95 ka paahitia e ia wāhanga te whakamātautau kounga. Mena ka hangaia he 10 ngā wāhanga, tatauhia te tūponotanga ka paahitia e 8 tonu ngā wāhanga te whakamātautau kounga.
Kōrero:
Ka taea e tātou te whakamahi i te tātai tohatoha rua-ira hei whakaoti i tēnei raruraru. Tuatahi, ka tautuhia e tātou ngā tawhā e whai ake nei:
– \( n \) (te tapeke o ngā whakamātautau) = 10
– \( k \) (te maha o ngā angitu) = 8
– \( p \) (tūponotanga o te angitu) = 0.95
– \( q \) (tūponotanga o te rahunga) = 1 – 0.95 = 0.05
Kātahi ka whakakapia ēnei uara ki te tātai tohatoha binomial:
\[
P(X = 8) = \binom{10}{8} (0.95)^8 (0.05)^2
\]
Tuatahi, tatauhia te huinga \( \binom{10}{8} \):
\[
\binom{10}{8} = \frac{10!}{8!(10-8)!} = \frac{10!}{8!2!} = \frac{10 \times 9 \times 8!}{8! \times 2!} = \frac{10 \times 9}{2 \times 1} = 45
\]
Kātahi, tatauhia ngā tūponotanga \( (0.95)^8 \) me \( (0.05)^2 \):
\[
(0.95)^8 \tata ki te 0.6634
\]
\[
(0.05)^2 = 0.0025
\]
Hei whakamutunga, whakareatia aua uara katoa kia whiwhi ai:
\[
P(X = 8) = 45 \times 0.6634 \times 0.0025 \approx 0.0744
\]
Nō reira, ko te tūponotanga e 8 tonu i roto i te 10 ngā wāhanga ka paahitia te whakamātautau kounga he tata ki te 0.0744, arā, 7.44%.
Tauira Pātai 2: Tūponotanga Whakaemi
Pātai:
Me te whakamahi tonu i te kamupene kotahi, tatauhia te tūponotanga e 9 o te 10 ngā wāhanga ka paahi i te whakamātautau kounga.
Kōrero:
Hei whakaoti i tēnei raruraru, me tatau tātou i te tūponotanga tāpiripiri. Ko te tūponotanga kia eke te 9 o te 10 o ngā wāhanga ki te whakamātautau, ko te tikanga ka tatau tātou \( P(X \geq 9) \), ka taea te tuhi penei:
\[
P(X \geq 9) = P(X = 9) + P(X = 10)
\]
Mā te whakamahi i te tātai tohatoha rua:
\[
P(X = 9) = \binom{10}{9} (0.95)^9 (0.05)^1
\]
\[
P(X = 10) = \binom{10}{10} (0.95)^{10} (0.05)^0
\]
Tuatahi, tatauhia te huinga mō ia take:
\[
\binom{10}{9} = \frac{10!}{9!(10-9)!} = 10
\]
\[
\binom{10}{10} = 1
\]
Kātahi, tatauhia ngā tūponotanga mō \( P(X = 9) \) me \( P(X = 10) \):
\[
P(X = 9) = 10 \times (0.95)^9 \times 0.05
\]
\[
(0.95)^9 \tata ki te 0.6302
\]
\[
P(X = 9) = 10 \times 0.6302 \times 0.05 \approx 0.3151
\]
\[
P(X = 10) = 1 \times (0.95)^{10} \times 1
\]
\[
(0.95)^{10} \tata ki te 0.5987
\]
\[
P(X = 10) = 0.5987
\]
Te tūponotanga katoa mō \( P(X \geq 9) \):
\[
P(X \geq 9) = 0.3151 + 0.5987 \approx 0.9138
\]
Nō reira, ko te tūponotanga kia eke te 9 o roto i te 10 o ngā wāhanga ki te whakamātautau kounga he tata ki te 0.9138, arā, 91.38%.
Tauira Pātai 3: Uara me te Rerekētanga e Tumanakohia ana
Pātai:
Tātaihia te uara e tumanakohia ana me te rerekētanga o te maha o ngā wāhanga i puta i te whakamātautau kounga mai i ngā wāhanga 10 i hangaia, me te tūponotanga o te putanga o te putanga he 0.95.
Kōrero:
Whakamahia te tātai e whai ake nei:
– Uara e tumanakohia ana (toharite) \( \mu = np \)
– Rerekētanga \( \sigma^2 = np(1-p) \)
Me \( n = 10 \) me \( p = 0.95 \):
\[
\mu = 10 \whakareatia ki te 0.95 = 9.5
\]
\[
\sigma^2 = 10 \whakareatia ki te 0.95 \whakareatia ki te 0.05 = 0.475
\]
Nō reira, ko te uara e tumanakohia ana mō te maha o ngā wāhanga e paahitia ana te whakamātautau kounga ko 9.5, ā, ko te rerekētanga ko 0.475.
Whakamutunga
Mā roto i ngā tauira raruraru e toru i runga ake nei, kua matapakihia e tātou te huarahi ki te tatau i te tūponotanga mā te whakamahi i te tohatoha rua-ira mō ngā āhuatanga rerekē: te tatau i te tūponotanga tika, te tūponotanga tāpiripiri, me te tatau i te uara me te rerekētanga e tumanakohia ana. He whai hua te mōhiotanga ki te tohatoha rua-ira i roto i ngā mara rerekē, pērā i te hanga, te rangahau hauora, me ngā tatauranga pāpori, ka taea te tātari i ngā hua o ngā whakamātautau maha me ngā putanga pea e rua hei āwhina i te whakatau. Ko te tumanako, mā ngā tauira raruraru me ngā kōrero kua whakaratohia e āwhina i tō māramatanga ki te tohatoha rua-ira.
