Espaces probabilisés 1 Définition 2 Propriétés élémentaires des

Ω
Ω = {0,1}0 1
Ω =
Ω = R+
Ω = {1,2,3,4,5,6}
Ωσ
Ω∈
(An)nSS
n
An∈S
A S A ∈S
SΩP(Ω) S
ΩS
∅ ∈ S
(An)nT
n
An∈S
ΩP(Ω) Ω
ΩR R B(R)
ΩSPS
[0,1]
P(Ω) = 1
(An)nS Ai∩Aj=∅
i6=j
P([
n
An) = X
n
P(An).
(Ω, S, P)
A B A, B ∈S
P(A)=1−P(A)
P(∅)=0
P(B) = P(A∩B) + P(A∩B)
P(A∪B) = P(A) + P(B)−P(A∩B)

A A
P(A) + P(A) = P(A∪A) = P(Ω) = 1.
A= Ω
P(Ω) + P(Ω) = 1 = P(∅)+1.
B∩A B ∩A B =B∩Ω = B∩(A∪A)=(B∩A)∪(B∩A)
P(B) = P((B∩A)∪(B∩A)) = P(B∩A) + P(B∩A).
A B ∩A
P(A∪B) = P(A∪(B∩A)) = P(A) + P(B∩A).
P(B∩A) = P(B)−P(A∩B)
P(A∪B) = P(A) + P(B)−P(A∩B).
(Ai)1≤i≤nnΩ
P n
[
i=1
Ai!=
n
X
i=1
P(Ai)−X
1≤i<j≤n
P(Ai∩Aj) + X
1≤i<j<k≤n
P(Ai∩Aj∩Ak) + ··· + (−1)nP n
\
i=1
Ai!
PΩ
P(Ω) Ω
P({ω})ω∈ΩA⊂Ω
P(A) = X
ω∈A
P(ω).
1 = P(Ω) = Pω∈ΩP(ω)
Ω1={pile, f ace}Ω1
P P(pile) = P(f ace)=1/2
k k ≥2 Ωk= Ωk
1k
(Ωk)=2kkP(ω)=2−kω∈Ωk
Ω
P(ω) = p ω ∈Ωp= 1/(Ω)
P(A) = X
ω∈A
P(ω) = X
ω∈A
1
(Ω) =(A) (Ω) ,
P(A)

N
N∗
P(1) = 1/6
P(2) = P( ) = 5
36
k≥2
P(k) = P(k−1 ) = 5
6k−11 6.
N∗Pk≥1P(k) = 1
f:R→R{x, f(x)≤
y}∈B(R)y∈R R
A⊂R
P(A) = ZA f(x)dx.
[a, b]a<b
P(A) = ZA∩[a,b]
1
b−adx.
R
f(x) = 1
√2πσ exp (x−m)2
2σ2
m∈Rσ∈R+∗
A B
A∩B A P(A∩B)=1/6
P(B) = 5/6
B
P(A∩B)/P(B)
A B P(B)6= 0 A
B B P(A/B)
P(A/B) = P(A∩B)
P(B).
BP(B)Q(A) = P(A/B)
S

Q
A∈S
0≤Q(A) = P(A/B)≤1
P(B) = P(A∩B) + P(A∩B)≥P(A∩B)
Q(Ω) = P(Ω/B) = P(Ω ∩B)
P(B)=P(B) P(B)= 1.
(An)n
Q(S
n
An) = P((S
n
An)/B) =
P((S
n
An)∩B)
P(B)=
P(S
n
(An∩B))
P(B)=X
n
P(An∩B)
P(B)=X
n
P(An/B)
QS
(Ai)i∈IP(Ai)>0
i
[
i∈I
Ai= Ω et Ai∩Aj=∅,∀i6=j.
(Ai)i∈IB
P(B) = X
i∈I
P(B∩Ai).
(Ai)i∈IS
i∈I
Ai= Ω
P(B) = P(B∩Ω) = P(B∩(S
i∈I
Ai)) = P(S
i∈I
(B∩Ai))
AiB∩Ai
P(B) = X
i∈I
P(B∩Ai).
(Ai)i∈IBP(B)6= 0
P(Ai)6= 0 ∀i∈I
P(B) = X
i∈I
P(B/Ai)P(Ai).
P(Ak/B) = P(Ak)P(B/Ak)
P
i∈I
P(Ai)P(B/Ai)∀k∈I.
A B P(B)6= 0 0 <P(A)<1
P(A/B) = P(B/A)P(A)
P(A)P(B/A) + P(A)P(B/A).

P(B) = X
i∈I
P(B∩Ai).
P(B/Ai) = P(B∩Ai)
P(Ai)
P(B) = X
i∈I
P(B/Ai)P(Ai).
P(Ak/B) = P(Ak∩B)
P(B)
P(Ak/B) = P(B/Ak)P(Ak)
P(B).
P(B)
P(Ak/B) = P(Ak)P(B/Ak)
P
i∈I
P(Ai)P(B/Ai).
1N A
B
A B
A B
A B
P(A/B) = P(A)
P(B/A) = P(B)
P(A∩B) = P(A)P(B).
P(A/B) = P(A)⇐⇒ P(A∩B)
P(B)=P(A)
⇐⇒ P(A∩B) = P(A)P(B)
⇐⇒ P(B∩A)
P(A)=P(B)
⇐⇒ P(B/A) = P(B)
A B A `B
P(A∩B) = P(A)P(B).
A B
A`A⇐⇒ P(A) = 0 ou P(A) = 1
A`B⇐⇒ A`B⇐⇒ A`B⇐⇒ A`B
A`A⇐⇒ P(A∩A) = P2(A)⇐⇒ P(A)(1 −P(A)) = 0 ⇐⇒ P(A) = 0 P(A)=1