MATHF105 Probabilités I. Fiche d`exercices 1.

Aσ A,B ∈ A A∩B∈ A A\B∈ A
Aσ A1,A2, . . . ∈ A Ti≥1Ai∈ A
BσR[a,b]∈ B a < b
x∈ B x∈R(a,b)∈ B [a,∞)∈ B
A,B ⊂ΩσAA,B ∈ A
Aσ A1,A2, . . . ∈ A
A= lim sup
n
An:= \
N≥1[ n≥N
An.
A∈ A A
(Ω,A,P )A, B ∈ A A⊂B
P(A)≤P(B)
(Ω,A,P )A1,A2, . . . ∈ A
P [
i≥1
Ai!≤X
i≥1
P(Ai).
(Ω,A,P )A1,A2, . . . ∈ A
A1⊇A2⊇ · · · Tn≥1An=A
P(An)→P(A).
(Ω,A,P )A1,A2, A3∈ A P(A1∪A2) =
P(A1) + P(A2)−P(A1∩A2)
P(A1∪A2∪A3)
=P(A1) + P(A2) + P(A3)−P(A1∩A2)−P(A1∩A3)−P(A2∩A3)
+P(A1∩A2∩A3).
(Ω,A,P )A1, . . . , An∈ A
P(A1∪ · · · ∪ An) =
n
X
k=1
(−1)k+1 X
I⊂{1,...,n}
|I|=k
P \
i∈I
Ai!
.
(Ω,A,P )A1, . . . , An∈ A
P(A1∪ · · · ∪ An)≥
n
X
k=1
P(Ai)−X
1≤i<j≤n
P(Ai∩Aj).