In einer Urne gibt es 5 schwarze und 4 rote Kugeln. Wir ziehen drei mal zufällig
A) alle 3 Kugel rot sind?
B) die ersten zwei schwarz und die dritte rot sind?
C) wir zwei schwarze und eine rote Kugel ziehen?
D) wir zwei rote und eine schwarz Kugel ziehen?
E) das Letztere passiert, wenn wir doch zurücklegen?
A)
4
9
⋅
3
8
⋅
2
7
=
1
21
{\displaystyle {\frac {4}{9}}\cdot {\frac {3}{8}}\cdot {\frac {2}{7}}={\frac {1}{21}}}
B)
5
9
⋅
1
2
⋅
4
7
=
10
63
{\displaystyle {\frac {5}{9}}\cdot {\frac {1}{2}}\cdot {\frac {4}{7}}={\frac {10}{63}}}
C)
5
9
⋅
1
2
⋅
4
7
+
5
9
⋅
1
2
⋅
4
7
+
4
9
⋅
5
8
⋅
4
7
=
10
63
+
10
63
+
10
63
=
10
21
{\displaystyle {\frac {5}{9}}\cdot {\frac {1}{2}}\cdot {\frac {4}{7}}+{\frac {5}{9}}\cdot {\frac {1}{2}}\cdot {\frac {4}{7}}+{\frac {4}{9}}\cdot {\frac {5}{8}}\cdot {\frac {4}{7}}={\frac {10}{63}}+{\frac {10}{63}}+{\frac {10}{63}}={\frac {10}{21}}}
D)
4
9
⋅
3
8
⋅
5
7
+
4
9
⋅
5
8
⋅
3
7
+
5
9
⋅
1
2
⋅
3
7
=
5
42
+
5
42
+
5
42
=
5
21
{\displaystyle {\frac {4}{9}}\cdot {\frac {3}{8}}\cdot {\frac {5}{7}}+{\frac {4}{9}}\cdot {\frac {5}{8}}\cdot {\frac {3}{7}}+{\frac {5}{9}}\cdot {\frac {1}{2}}\cdot {\frac {3}{7}}={\frac {5}{42}}+{\frac {5}{42}}+{\frac {5}{42}}={\frac {5}{21}}}
E)
4
9
⋅
4
9
⋅
5
9
+
4
9
⋅
5
9
⋅
4
9
+
5
9
⋅
4
9
⋅
4
9
=
3
⋅
80
729
=
240
729
{\displaystyle {\frac {4}{9}}\cdot {\frac {4}{9}}\cdot {\frac {5}{9}}+{\frac {4}{9}}\cdot {\frac {5}{9}}\cdot {\frac {4}{9}}+{\frac {5}{9}}\cdot {\frac {4}{9}}\cdot {\frac {4}{9}}=3\cdot {\frac {80}{729}}={\frac {240}{729}}}
