Su'aalo Tusaale ah iyo Doodda Qaybinta Laba-geesoodka ah
Qaybinta laba-geesoodka ah waa mid ka mid ah qaybinta fursadaha kala duwan ee ugu badan ee la isticmaalo. Waxay faa'iido u leedahay qaabaynta tirada guulaha ee tiro tijaabo oo isku mid ah oo madax-bannaan, kuwaas oo mid walba uu keeno guul ama guuldarro. Maqaalkan, waxaan si qoto dheer u dhex geli doonnaa qaybinta laba-geesoodka ah annagoo bixinayna tusaalooyin dhowr ah iyo dood faahfaahsan.
Hordhac ku saabsan Qaybinta Laba-geesoodka ah
Astaamaha ugu muhiimsan ee qaybinta laba-geesoodka ah:
1. n: Tirada tijaabooyinka ama ku celcelinta.
2. p: Suurtagalnimada guusha tijaabo kasta.
3. q = 1-p: Suurtagalnimada in lagu guuldareysto tijaabo kasta.
Shaqada tirada suurtogalnimada ee qaybinta laba-geesoodka ah waa:
\[ P(X = k) = {n \dooro k} p^k (1-p)^{nk} \]
Halkee:
– \( {n \dooro k} = \frac{n!}{k!(nk)!} \)
– \( X \): Doorsoome aan kala sooc lahayn oo matalaya tirada guulaha.
– \( k \): Tirada guulaha la raadinayo.
Su'aalo iyo Doodo Tusaale ah
Aan ku bilowno tusaalooyin su'aalo ah si aan si faahfaahsan u fahanno fikradda qaybinta laba-geesoodka ah.
Tusaale 1: Ka xulashada Koox Arday ah
Tusaale ahaan, waxaan haynaa koox ka kooban 10 arday, fursadda arday kasta loo doorto inuu ka qayb galo tartanka waa 0,3. Waxaan rabnaa inaan ogaano suurtagalnimada in 4 arday la doorto.
Tallaabada 1: Aqoonso xuduudaha qaybinta binomial-ka.
– \( n = 10 \)
– \( p = 0.3 \)
Tallaabada 2: Adeegso qaybinta laba-geesoodka ah si aad u xisaabiso suurtogalnimada \( X = 4 \).
\[ P(X = 4) = {10 \ dooro 4} (0.3)^4 (0.7)^6 \]
Xisaabinta \( {10 \ dooro 4} \):
\[ {10 \dooro 4} = \frac{10!}{4!(10-4)!} = \frac{10!}{4!6!} = 210 \]
Hadda xisaabi \( (0.3)^4 \) iyo \( (0.7)^6 \):
\[ (0.3)^4 = 0.0081 \]
\[ (0.7)^6 = 0.117649 \]
Haddaba,
\[ P(X = 4) = 210 \cdot 0.0081 \cdot 0.117649 \ qiyaastii 0.20012 \]
Markaa, suurtogalnimada in 4 arday la doorto waa qiyaastii 0.20012 ama 20.012%.
Tusaale 2: Suurtagalnimada Ka Yar ama La Mid ah 2
Hadda, tusaale ahaan, waxaa naloo weydiinayaa suurtagalnimada in wax ka yar ama la mid ah 2 arday la dooran doono.
Tallaabada 1aad: Waa inaan xisaabinnaa \( P(X = 0) \), \( P(X = 1) \), iyo \( P(X = 2) \).
– Loogu talagalay \( P(X = 0) \):
\[ P(X = 0) = {10 \ dooro 0} (0.3)^0 (0.7)^{10} \]
\[ {10 \ dooro 0} = 1 \]
\[ (0.7)^{10} = 0.0282475 \]
\[ P(X = 0) = 1 \cdot 1 \cdot 0.0282475 = 0.0282475 \]
– Loogu talagalay \( P(X = 1) \):
\[ P(X = 1) = {10 \ dooro 1} (0.3)^1 (0.7)^9 \]
\[ {10 \ dooro 1} = 10 \]
\[ (0.3) \cdot (0.7)^9 = 0.1210608 \]
\[ P(X = 1) = 10 \cdot 0.3 \cdot 0.1210608 = 0.3631824 \]
– Loogu talagalay \( P(X = 2) \):
\[ P(X = 2) = {10 \ dooro 2} (0.3)^2 (0.7)^8 \]
\[ {10 \ dooro 2} = 45 \]
\[ (0.3)^2 \cdot (0.7)^8 = 0.2334744 \]
\[ P(X = 2) = 45 \cdot 0.09 \cdot 0.2334744 = 0.2334744 \]
Tallaabada 2: Isu gee fursadaha.
\[ P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) \]
\[ P(X \leq 2) = 0.0282475 + 0.3631824 + 0.3826372 = 0.7740671 \]
Markaa, fursadda ah in wax ka yar ama la mid ah 2 arday la doorto waa qiyaastii 0.7740671 ama 77.41%.
Tusaale 3: Suurtagalnimada ugu yaraan 8
Haddii tijaabo la sameeyo 12 jeer, oo fursadda guusha ee tijaabo kasta ay tahay 0.5, waa maxay fursadda ugu yaraan 8 guulood ay dhacaan?
Tallaabada 1: Deji xuduudaha laba-geesoodka ah: \( n = 12, p = 0.5 \).
Tallaabada 2aad: Soo hel fursadda \( X \geq 8 \).
Tani waxay u baahan tahay xisaabinta dhowr fursadood oo shaqsiyeed iyo isku darka:
\[ P (X \geq 8) = P (X = 8) + P (X = 9) + P (X = 10) + P (X = 11) + P (X = 12) \]
Mid mid u tiri:
– Loogu talagalay \( P(X = 8) \):
\[ P(X = 8) = {12 \ dooro 8} (0.5)^8 (0.5)^4 \]
\[ {12 \ dooro 8} = 495 \]
\[ (0.5)^{12} = 0.0002441406 \]
\[ P(X = 8) = 495 \cdot 0.0002441406 = 0.1208496 \]
– Loogu talagalay \( P(X = 9) \):
\[ P(X = 9) = {12 \ dooro 9} (0.5)^9 (0.5)^3 \]
\[ {12 \ dooro 9} = 220 \]
\[ P(X = 9) = 220 \cdot 0.0002441406 = 0.05371094 \]
– Loogu talagalay \( P(X = 10) \):
\[ P(X = 10) = {12 \ dooro 10} (0.5)^{10} (0.5)^2 \]
\[ {12 \ dooro 10} = 66 \]
\[ P(X = 10) = 66 \cdot 0.0002441406 = 0.01611328 \]
– Loogu talagalay \( P(X = 11) \):
\[ P(X = 11) = {12 \ dooro 11} (0.5)^{11} (0.5)^1 \]
\[ {12 \ dooro 11} = 12 \]
\[ P(X = 11) = 12 \cdot 0.0002441406 = 0.002929688 \]
– Loogu talagalay \( P(X = 12) \):
\[ P(X = 12) = {12 \ dooro 12} (0.5)^{12} \]
\[ {12 \ dooro 12} = 1 \]
\[ P(X = 12) = 1 \cdot 0.0002441406 = 0.0002441406 \]
Tallaabada 3: Ku dar dhammaan fursadaha.
\[ P(X \geq 8) = 0.1208496 + 0.05371094 + 0.01611328 + 0.002929688 + 0.0002441406 \ qiyaastii 0.1938477 \]
Markaa, suurtogalnimada in ugu yaraan 8 guulood ay ka dhacaan 12 tijaabo waa qiyaastii 0.1938477 ama 19.38%.
Gabagabo
Qaybinta laba-geesoodka ah waa fikrad aasaasi ah oo ku jirta tirakoobka oo muhiim u ah codsiyo badan oo wax ku ool ah. Marka la fahmo sida loo xisaabiyo fursadaha kiisaska kala duwan ee qaybinta laba-geesoodka ah, sida ku cad tusaalooyinka kor ku xusan, waxaan ku dabaqi karnaa fikraddan xaaladaha dhabta ah ee adduunka. Layligani wuxuu sidoo kale xoojinayaa fahamkeenna sida qaab-dhismeedka suurtogalnimada uu ugu shaqeeyo xaalad cad oo habaysan.
