Formule de Pointcarré : le sujet du problème (PDF

(Ω,A, p)
A1, A2, A3
p(A1∪A2∪A3) = p(A1)+p(A2)+p(A3)−p(A1∩A2)−p(A1∩A3)−p(A2∩A3)+p(A1∩A2∩A3)
n≥2 (Ai)1≤i≤n
A
p(
n
[
i=1
Ai) =
n
X
k=1
((−1)k−1X
J⊂I
|J|=k
p(\
j∈J
Aj))
|J|J I ={1, ..., n}
J I
ˆ
AJ=Tj∈JAjBJ=ˆ AJ∩(Tj∈JcAc
j) 1 ≤m≤n
Bm=[
J⊂I
|J|=m
BJ
B0=
n
\
j=1
Ac
j
J J0BJBJ0
m∈I ω ∈Bm⇔ω m Ai
S0= 1 1 ≤r≤n Sr=PJ⊂I
|J|=r
p(ˆ AJ)
A∈ A
p(B0∩A) = p(A) +
n
X
k=1
(−1)kX
J⊂I
|J|=k
p(\
j∈J
Aj∩A)
1≤m≤n
p(Bm) = Sm−(m+ 1
m)Sm+1 +... + (−1)n−m(n m)Sn
m
(Ω,A, p)
A B

A Bc
AcBc
(Ai)II
I1I2(Bi)I
Bi={Aisi i ∈I1
Ac
isi i ∈I2
(Bi)I
Ω
{1, ..., n}n
(Ω,P(Ω), p)p d n Ad=
{kd tels que k ∈Ωet kd ∈Ω}
Ad
p1< p2< ... < prn
(Api)1≤i≤r
ϕ(n)A n
n