B - kula biała
C - kula czarna
Drzewko w załączniku. Losowanie 3 kul na czarno, a na zielono losowanie z trzech wylosowanych wcześniej kul.
[tex]P(A_1)=\dfrac{5}{12}\cdot\dfrac{4}{11}\cdot\dfrac{7}{10}\cdot\dfrac{1}{3}=\dfrac{1}{3}\cdot\dfrac{1}{11}\cdot\dfrac{7}{2}\cdot\dfrac{1}{3}=\dfrac{7}{198}[/tex]
[tex]P(A_2)=\dfrac{5}{12}\cdot\dfrac{7}{11}\cdot\dfrac{2}{5}\cdot\dfrac{1}{3}=\dfrac{1}{6}\cdot\dfrac{7}{11}\cdot\dfrac{1}{1}\cdot\dfrac{1}{3}=\dfrac{7}{198}[/tex]
[tex]P(A_3)=\dfrac{5}{12}\cdot\dfrac{7}{11}\cdot\dfrac{3}{5}\cdot\dfrac{2}{3}=\dfrac{1}{2}\cdot\dfrac{7}{11}\cdot\dfrac{1}{1}\cdot\dfrac{1}{3}=\dfrac{7}{66}[/tex]
[tex]P(A_4)=\dfrac{7}{12}\cdot\dfrac{5}{11}\cdot\dfrac{2}{5}\cdot\dfrac{1}{3}=\dfrac{7}{6}\cdot\dfrac{1}{11}\cdot\dfrac{1}{1}\cdot\dfrac{1}{3}=\dfrac{7}{198}[/tex]
[tex]P(A_5)=\dfrac{7}{12}\cdot\dfrac{5}{11}\cdot\dfrac{3}{5}\cdot\dfrac{2}{3}=\dfrac{7}{2}\cdot\dfrac{1}{11}\cdot\dfrac{1}{1}\cdot\dfrac{1}{3}=\dfrac{7}{66}[/tex]
[tex]P(A_6)=\dfrac{7}{12}\cdot\dfrac{6}{11}\cdot\dfrac{1}{2}\cdot\dfrac{2}{3}=\dfrac{7}{2}\cdot\dfrac{1}{11}\cdot\dfrac{1}{1}\cdot\dfrac{1}{3}=\dfrac{7}{66}[/tex]
[tex]P(A_7)=\dfrac{7}{12}\cdot\dfrac{6}{11}\cdot\dfrac{1}{2}\cdot1=\dfrac{7}{2}\cdot\dfrac{1}{11}\cdot\dfrac{1}{2}=\dfrac{7}{44}[/tex]
[tex]P(A)=P(A_1)+P(A_2)+P(A_3)+P(A_4)+P(A_5)+P(A_6)+P(A_7)[/tex]
[tex]P(A)=3\cdot\dfrac{7}{198}+3\cdot\dfrac{7}{66}+\dfrac{7}{44}=\dfrac{7}{66}+\dfrac{7}{22}+\dfrac{7}{44}=\dfrac{14}{132}+\dfrac{42}{132}+\dfrac{21}{132}=\\=\dfrac{77}{132}=\dfrac{7}{12}[/tex]
