Su'aalo Tusaale ah oo Ka Hadlaya Shaqada Qaybinta Laba-geesoodka ah
Qaybinta laba-geesoodka ah waa qaybin kala go'an oo qeexaysa tirada guulaha tijaabada ah oo ka kooban tiro tijaabo oo madax-bannaan oo leh laba natiijo oo suurtagal ah: guul iyo guuldarro. Tijaabo kasta waxaa loo yaqaan tijaabo, qaybinta laba-geesoodka ahna waxaa badanaa loo isticmaalaa xaaladaha ay xiisaha leeyihiin tirada guulaha ee tiro tijaabo oo madax-bannaan. Maqaalkan, waxaan ka hadli doonnaa fikradaha aasaasiga ah ee qaybinta laba-geesoodka ah waxaanan bixin doonnaa tusaalooyin iyo xalal.
Fikradaha Aasaasiga ah ee Shaqada Qaybinta Laba-geesoodka ah
Kahor inta aynaan guda gelin su'aalaha iyo doodda tusaalaha ah, aan ka wada hadalno qaar ka mid ah fikradaha aasaasiga ah ee la xiriira qaybinta laba-geesoodka ah.
1. Qeexitaan: Qaybinta laba-geesoodka ah waxaa lagu qeexaa wadarta guulaha tijaabooyinka 'n' ee madaxa-bannaan, halkaas oo tijaabo kasta ay leedahay laba natiijo oo suurtagal ah: guul (oo leh suurtogalnimada p) ama guuldarro (oo leh suurtogalnimada q = 1 - p).
2. Shaqada Suurtagalnimada: Shaqada Suurtagalnimada ee qaybinta binomial waa:
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{nk}
\]
Halkee:
– \( P(X = k) \) waa fursadda lagu guuleysan karo k tijaabooyinka n.
– \( \binom{n}{k} \) waa isku-darka n qaadashada k, kaas oo lagu qeexo \( \frac{n!}{k!(nk)!} \).
– \( p \) waa suurtogalnimada guusha tijaabo kasta.
– \( (1-p) \) waa suurtogalnimada guuldarada tijaabo kasta.
3. Qiimaha la filayo iyo Kala duwanaanshaha:
– Qiimaha la filayo (celceliska) ee qaybinta binomial waa \( \mu = np \).
– Kala duwanaanshaha qaybinta binomial waa \( \sigma^2 = np(1-p) \).
Hadda, aan fikradahan ku dabaqno masalo tusaale ah si aan u helno faham qoto dheer.
Su'aal Tusaale 1aad: Xisaabinta Aasaasiga ah ee Qaybinta Laba-geesoodka ah
Su'aal:
Shirkad ayaa soo saarta qaybo elektaroonik ah oo leh fursad 0.95 ah in qayb kastaa ay ka gudubto imtixaan tayo leh. Haddii 10 qaybood la soo saaro, xisaabi suurtagalnimada in 8 qaybood ay ka gudbaan imtixaanka tayada.
Dood:
Waxaan isticmaali karnaa qaacidada qaybinta laba-geesoodka ah si aan u xallino dhibaatadan. Marka hore, waxaan aqoonsaneynaa xuduudaha soo socda:
– \( n \) (wadarta tirada tijaabooyinka) = 10
– \( k \) (tirada guulaha) = 8
– \( p \) (suurtagalnimada guusha) = 0.95
– \( q \) (suurtagalnimada guuldarada) = 1 – 0.95 = 0.05
Kadibna qiimahan ku beddel qaacidada qaybinta binomial-ka:
\[
P(X = 8) = \binom{10}{8} (0.95)^8 (0.05)^2
\]
Marka hore, xisaabi isku-darka \( \binom{10}{8} \):
\[
\binom{10}{8} = \frac{10!}{8!(10-8)!} = \frac{10!}{8!2!} = \frac{10 \times 9 \times 8!}{8! \times 2!} = \frac{10 \times 9}{2 \times 1} = 45
\]
Kadib, xisaabi fursadaha \( (0.95)^8 \) iyo \( (0.05)^2 \):
\[
(0.95)^8 \qiyaastii 0.6634
\]
\[
(0.05)^2 = 0.0025
\]
Ugu dambeyntii, ku dhufo dhammaan qiimayaashaas si aad u hesho:
\[
P(X = 8) = 45 \ jeer 0.6634 \ jeer 0.0025 \ qiyaastii 0.0744
\]
Markaa, suurtogalnimada in 8 ka mid ah 10-ka qaybood ay ka gudbaan imtixaanka tayada waa qiyaastii 0.0744 ama 7.44%.
Su'aal Tusaale 2: Su'aal Isugeyn ah
Su'aal:
Wali shirkad isku mid ah, xisaabi suurtagalnimada in ugu yaraan 9 ka mid ah 10-ka qaybood ay ka gudbaan imtixaanka tayada.
Dood:
Si aan u xallino dhibaatadan, waxaan u baahanahay inaan xisaabino suurtogalnimada isugeynta. Suurtagalnimada in ugu yaraan 9 ka mid ah 10-ka qaybood ay ka gudbaan imtixaanka waxay la macno tahay inaan xisaabinno \( P(X \geq 9) \), kaas oo loo qori karo sidan:
\[
P (X \geq 9) = P (X = 9) + P (X = 10)
\]
Iyadoo la adeegsanayo qaacidada qaybinta laba-geesoodka ah:
\[
P(X = 9) = \binom{10}{9} (0.95)^9 (0.05)^1
\]
\[
P(X = 10) = \binom{10}{10} (0.95)^{10} (0.05)^0
\]
Marka hore, xisaabi isku-darka kiis kasta:
\[
\binom{10}{9} = \frac{10!}{9!(10-9)!} = 10
\]
\[
\binom{10}{10} = 1
\]
Kadib, xisaabi fursadaha \( P(X = 9) \) iyo \( P(X = 10) \):
\[
P(X = 9) = 10 \jeer (0.95)^9 \jeer 0.05
\]
\[
(0.95)^9 \qiyaastii 0.6302
\]
\[
P(X = 9) = 10 \ jeer 0.6302 \ jeer 0.05 \ qiyaastii 0.3151
\]
\[
P(X = 10) = 1 \jeer (0.95)^{10} \jeer 1
\]
\[
(0.95)^{10} \qiyaastii 0.5987
\]
\[
P(X = 10) = 0.5987
\]
Wadarta suurtogalnimada ee \( P(X \geq 9) \):
\[
P(X \geq 9) = 0.3151 + 0.5987 \qiyaastii 0.9138
\]
Markaa, suurtogalnimada in ugu yaraan 9 ka mid ah 10-ka qaybood ay ka gudbaan imtixaanka tayada waa qiyaastii 0.9138 ama 91.38%.
Su'aal Tusaale ah 3: Qiimaha la Filayo iyo Kala Duwanaanshaha
Su'aal:
Xisaabi qiimaha la filayo iyo kala duwanaanshaha tirada qaybaha ka gudbaya tijaabada tayada ee 10 qaybood oo la soo saaray, iyadoo suurtogalnimada ay tahay inay dhaafto 0.95.
Dood:
Isticmaal qaacidada soo socota:
– Qiimaha la filayo (celcelis) \( \mu = np \)
– Kala duwanaansho \( \sigma^2 = np(1-p) \)
Iyadoo \( n = 10 \) iyo \( p = 0.95 \):
\[
\mu = 10 \ jeer 0.95 = 9.5
\]
\[
\sigma^2 = 10 \ jeer 0.95 \ jeer 0.05 = 0.475
\]
Markaa, qiimaha la filayo ee tirada qaybaha ka gudbaya tijaabada tayada waa 9.5, kala duwanaanshahana waa 0.475.
Gabagabo
Saddexda dhibaato ee tusaalaha ah ee kor ku xusan, waxaan ka wada hadalnay sida loo xisaabiyo itimaalka iyadoo la adeegsanayo qaybinta laba-geesoodka ah xaalado kala duwan: xisaabinta itimaalka saxda ah, itimaalka isugeynta, iyo xisaabinta qiimaha la filayo iyo kala duwanaanshaha. Aqoonta qaybinta laba-geesoodka ah waxay faa'iido u leedahay dhinacyo kala duwan, sida wax soo saarka, cilmi-baarista caafimaadka, iyo tirakoobka bulshada, halkaas oo natiijooyinka tijaabooyinka soo noqnoqda ee leh laba natiijo oo suurtagal ah lagu falanqeyn karo si loo caawiyo go'aan qaadashada. Waxaan rajeyneynaa, dhibaatooyinka tusaalaha ah iyo doodaha la bixiyay waxay kaa caawin doonaan inaad sii fahanto qaybinta laba-geesoodka ah.
