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8. Stochastik
8.1 Ereignisse
Ereignisraum A ∅P (S) Potenzmenge
|A1⋅ 2 n| 1
⋅n!
2| ∅
1|A ⋅A1 2⋅⋅…⋅ A2⋅…⋅ A
n| A =
n| = |A|
1|A| ⋅1|A n! ⋅2n!
|A | ⋅…⋅
2| n! ⋅…⋅ |A|
nn|A|
n
AA=
= A = A =
sicheres Ereignis,
∅ S, Grundmenge, (n (n−
(n −
k)! − k)! (n−
k)!
unmögliches Ereignis
A∩ AB∪
= B φ
leere n! Menge
A∪
B
n!
A
= A =
A oder B
A∪
B
(n
Vereinigung
−(n k)! − k)! nn
n n
n! n! n! n!
AA= = A = = A
= = =
A∩
B
kk
kk! k! ⋅k(n k! ⋅(n−⋅(n −
k! −⋅k)!
(n−k)!
A∩
B
A und B
A∩
B
Durchschnitt
n
n
n! n!
A
= A = = =
A
k k
k! ⋅
⋅k! (n ⋅−(n −k)!
n− 1+
Gegenereignis
Komplement
n− n1
− + k1
kn+
− k1
+ k
A
AA=
= A = A =
A
nn−
−1
n1
−1n−1
A und B sind unvereinbar
A∩ B=
φ
A n
− und n1
− +
1B k+
sind k elementefremd
A∩ B=
φA
= A =
A∩ B=
φ
−
φ
n−n1
−1
n n−n
−
n −n
n! n! ≈n! ≈nn≈⋅e n! ⋅e⋅≈⋅
e⋅n2
2
⋅ πen
π2
π⋅
n2π
n
Die Anzahl der Elemente der endlichen Mengen
A,A,
11 A,A,
122
…
2
… An
n
Asei
n
|A |A
1|A |,
1|, |A
1 |, |A
2|A |,
21|, |,
2…
|, |A …
n2|A |
n|, | …
n
| |A
n
|
|A1∪ A|
2
= |A
1| + |A|
2
−|A1∩
A|
2
|A1∪ A|
2
= |A
1| + |A| 2
−|A1∩
A|
1∪A2∪ A 2
3| = |A|
1
+ |A
2| + |A|
3
Summenregel
|A
1|A 1|, |, 1|A |,
22|A |, |, 2…
|, |A …
nn|A | |
n
|
|A1∪A 1 |A ∪ A|
2∪ A 2
=
1 A| |A
3| = |A| 1| 2
= |A + |A|
1
+ |A 2
− + |A| |A
2| + |A| 1∩
2
−|A A|
3
−|A1∩A| 2
−|A2∩A| 3
−|A3∩ A|
1
+ |A1∩A2∩A 3
|
2
|A A|
1∪A2∪ A
3| = |A|
1|A+ |A
2| + |A|
3
−|A1∩A| 2
−|A2∩A
1∪A2∪ ∪ A
n| = |A|
1
+
2|
2
|A1∪A2∪ ∪ A
n| = |A|
1
+ |A
2| + …+ |A|,
n
|A 1 |A ∪ A2 1 ∪ A 3| 2 A = |A| 1
+ = |A| |A 2| 1
+ |A + |A| 3
− |A| |A 1∩ 3
−|A A| 2
−
1∩A| |A 2∩ 2
−|A |AA| 1∪ 3
−
2∩A| A|A 2∪3∩ 3
−|A A| ∪
1
+ A| A |A n| =
1A
1
+ |A |A| A A21∩
1∩A + A |A
3
|
2∩A 2| + …+ |A|,
i∩
k
n
3
|
Ai∩
Ak
|A 1∪ |A A 2∪ 1∪A
∪ A n| = ∪ A |A|
1
+ = |A| |A 2| 1
+ |A + …+
|A|,
A A
n
falls i∩
k= für alle i≠
k
+ …+ |A|,
n
i≠
k
AProduktregel
i∩
A
A
i≠
k
k
|A
∩ A
1⋅A2⋅…⋅ A
n| = |A|
1
⋅ |A
2|
⋅…
i
k
|A1⋅A2⋅…⋅ A
n| = |A|
1
⋅ |A
2| ⋅…⋅ |A|
n
i
≠
k
i≠
k
n!
|A A = ⋅ A ⋅…⋅ A | = |A| ⋅ |A | ⋅…⋅
|A|
n
|A ⋅(n A −⋅…⋅ k)! A | = |A| ⋅ |A | ⋅…⋅ |A|
1 2 n
1 2
1 2 n
1 2
n
|A1⋅A2⋅…⋅ A
n| = |A|
1
⋅ |A n! 2| ⋅…⋅ |A|
n
A =
(n−
k)!
A =
n!
23
|A |A |A ∪ ∪ |A| = = ∪ |A |A = A| |A | + | = + | |A| + |A| −| − + |A| − ∩
−A|
∩|AA
1 1 1 2 2 12 121 1 2 21 2 1 12
1 2 2
|A |A ∪ A| ∪ A| = |A = |A | | + |A| | + |A| ∪ −A A∪ |A −A∪ |A A ∩∪A | A| ∩A | = = ∪ |A| = A |A| | + + = |A| + | + | + | |A|
+
−
11 1 22 2 11 1 1212 122 21 12 3132
23 2 131 1 2 21 2 3 32
|A |A ∪ A∪ ∪ A A ∪ | | A = | |A| = |A| + ∪ |A A+ A∪|A | | A∪+ ∪|A| | + A |A| −A ∪ A |A | −A | = |A ∩= ∪ | = |A| ∩A |A| | −+ = +
−
|A +
|A ∩
+ | A
11 1 22 2 33 3 11 11 1 2 2 12 332 3n
n11n
1 221n1 21
22
212
Wahrscheinlichkeitssprache Symbol Mengensprache
∅
|A11|A ∪1 A∪22 ∪A2 ∪ ∪ A ∪
n
n| Ai| i= n∩A| i|A| = ∩Ak
1k1A|A| ik+ ∩1 |A +
2k2|A | | +
2
| …+ + …+
|A|,
nn|A|,
n
Ergebnisraum S (auch Ω) Grundmenge
A∪
B
Ai
i
∩A
iA∩
kk
Ak
i≠i≠
kik≠
ki≠
k
Ereignis A Teilmenge
Elementarereignis
A∩
B
{a i
} ii
≠
ik≠
einelementige k |A |ATeilmenge
⋅|AA⋅ A ⋅A⋅…⋅ |A⋅…⋅ ⋅ A | ⋅…⋅ A | = = | = A |A| | |A| = ⋅ ⋅ |A| ⋅ | | ⋅…⋅
⋅|
|A ⋅
8.2 Kombinatorik
1 1 212 21 n2n
n n1 1 1 2 21
2
n −n−
nn
−n
n! ≈n! n≈⋅ ⋅ne ⋅⋅e⋅
2⋅
π
2n
π
n A,A,
A,A, … A…
A A…
A
1 1 12
2 21 n n2
n
|A ∪ A| = |A | + |A| −|A ∩ A
1 2 1 2 1 2
n