Généralités sur les probabilités Fichier

(Ω,F)
PF[0,1]
P(Ω) = 1
(An)n≥1FP(S
n≥1
An) = P
n≥1
P(An)
Fω∈Ω{ω} ∈ F{ω}
P(Ac) = 1 −P(A)P(A) + P(Ac) = P(A∪Ac) = P(Ω) = 1
∀A, B ∈FP(A) + P(B) = P(A∪B) + P(A∩B) 1A+ 1B= 1A∩B+ 1A∪B
(An)n≥1FP(S
n≥1
An)≤P
n≥1
P(An)
(An)n≥1Flim
n→∞P(An) = P(S
n≥1
An)
(An)n≥1Flim
n→∞P(An) = P(T
n≥1
An)

ν(Ω,F)P=ν
ν(Ω)
(An)n≥1Fn≥1P(An) = 1
P(T
n≥1
An) = 1
(Ω,F)aΩ
a δa
∀A∈Fδa(A)=1A(a).
f(Ω,F)RfRfd(δa) = f(a)
P(A∪B∪C)
B(Ω,F)P(B)>0A
A B
P(A|B) = PB(A) = P(A∩B)
P(B).
PBP(B)=1 P(Bc)=0
A∈FPB(A)∈[0,1] P(A∩B)≤P(B)
P(Ω) = P(Ω∩B)
P(B)= 1

(An)n≥1F
(An∩B)n≥1
PB(∪n≥1An) = P(B)−1P[([
n≥1
An)∩B)]
=P(B)−1P[([
n≥1
(An∩B)]
=P(B)−1X
n≥1
P(An∩B)
=X
n≥1
PB(An).
P(B)>0P(A∩B) = P(A|B)×P(B)
P(A∩B)>0
P(A∩B∩C) = P(C|A∩B)×P(A∩B) = P(C|A∩B)×P(A|B)×P(B).
(Ω,F, P )
Ω
(B1, ..., Bn)
∀i∈[1..n]Bi∈F
∀i6=j∈[1..n]Bi∩Bj=∅
∀i∈[1..n]P(Bi)>0
(B1, ..., Bn)A
P(A) =
n
X
i=1
P(A|Bi)×P(Bi).
A=
n
S
i=1
(A∩Bi)
P(A) =
n
X
i=1
P(A∩Bi) =
n
X
i=1
P(A|Bi)×P(Bi).

P(B)∈]0,1[ (B, Bc)
A
P(A) = P(A|B)×P(B) + P(A|Bc)×P(Bc).
A B P (A)>0P(B)>0
P(A|B) = P(B|A)×P(A)
P(B).
P(A∩B) = P(A|B)×P(B) = P(B∩A)×P(A).
P(Ac)>0
P(B) = P(B|A)×P(A) + P(B|Ac)×P(Ac),
P(A|B) = P(B|A)×P(A)
P(B|A)×P(A) + P(B|Ac)×P(Ac).
B1B2B3
B1
B2
B3
B1
48
113
A B (Ω,F, P )
P P (A∩B) = P(A)×P(B)

A B C (Ω,F, P )
P
P(A∩B) = P(A)×P(B)P(B∩C) = P(B)×P(C)P(C∩A) = P(C)×P(A)
P(A∩B∩C) = P(A)×P(B)×P(C)
A B (A, Bc) (Ac, B)
(Ac, Bc)
P(A∩Bc) = P(B)−P(A∩B)
=P(B)−P(A)×P(B)
=P(B)[1 −P(A)]
=P(B)×P(Ac)
A B P (B) = 0 P(B) =
1
P(B)>0A B P (A|B) = P(A)