Probabilités sur un univers ni ou dénombrable

ENE
ϕ:N→E
n7→ xn
E={xnn∈N}
N Z Z
N→Z
n7→ n
2n
n7→ −n+1
2n
E
Ω
Ω
Ω = {1,...,6}
Ω = {O, N}100
Ω = N∗
Ω = N

Ω
Ω
(i, j) 1 6i, j 66i j
card (Ω) = 6 ×6 = 36
Ω
(i, j)i6j
card (Ω) = 1 + 2 + 3 + 4 + 5 + 6 = 21
Ω
Ω
Ω = {1,2,3,4,5,6};A={2,4,6}
k∈Nk>15000
Ω = N;B= [[15000,+∞[[
C
Ω = {(P, P ),(P, F ),(F, P ),(F, F )};C={(P, F ),(F, P ),(F, F )}
∅
Ω
ω∈Ω Ω
A⊂Ω Ω
A⊂B A B A B
A∪B A B A B
A∩B A B A B
AcAΩA∗
A∩B=∅A B A B
∗A
ΩA1, . . . , An
Ai16i6n
n
[
i=1
Ai=A1∪A2∪ · · · ∪ An
Ai16i6n
n
\
i=1
Ai=A1∩A2∩ · · · ∩ An

(Ai)i∈N
[
i∈N
Ai=
+∞
[
i=1
Ai=Ai, i ∈N∗
\
i∈N
Ai=
+∞
\
i=1
Ai=Ai, i ∈N∗
A= ={2,4,6}B= ={3,6}
A∪B= ={2,3,4,6}
A∩B= ={6}
A B C
Fnn
n= 2k+ 1 F2k+1 k∈NA
F2k+1 k∈N
A=
+∞
[
k=0
F2k+1
B F2kk∈N∗
B=
+∞
[
k=1
F2k
C Fkk∈N∗
C=
+∞
\
n=1
Fn
Ω
ω
ω{1,2,3,4,5}6
ω{1,2,3,4,5}
C
C={1,2,3,4,5}N∗
Fn
Fn={1,2,3,4,5}n−1× {6}
Ω
Fn
i∈N∗
Ai={ }.

E1=
5
\
i=1
Ai, E2=
+∞
[
i=5
Ai, E3= 4
[
i=1
Ai!∩ +∞
\
i=5
Ai!, E4=\
i>4
Ai
Cn=Ti>nAi(Cn)n>1Cn
Cn+1 C=[
n>1
Cn
Ai
Bn={ }
B={ }
I⊂N{Ai, i ∈I}
Ω{Ai, i ∈I}
i6=j Ai∩Aj=∅
[
i∈I
Ai= Ω
A1={ } A2={ }
A, B C
Fn={n}
{Fn, n ∈N∗} ∪ C
ΩPP(Ω) [0,1]
P(Ω) = 1
(Ak)k∈N
P [
k∈N
Ak!=
+∞
X
k=0
P(Ak)
(Ω, P )P(Ω) P(A)
A∈ P(Ω) 0 1
Ω Ω

PΩ
P(∅)=0
A∩B=∅P(A∪B) = P(A) + P(B)
A1,· · · , An
P n
[
k=1
Ak!=
n
X
k=1
P(Ak)
A∈ P(Ω) P(A) = 1 −P(A)
A, B ∈ P(Ω) A⊂B P (A)6P(B)
A, B ∈ P(Ω) P(A∪B) = P(A) + P(B)−P(A∩B)
(Ak)k∈NAk=∅k∈N
P(∅) = P
[
k∈N
Ak
=
+∞
X
k=0
P(Ak) = P(A0) + P(A1) +
+∞
X
k=0
P(Ak)>P(A0) + P(A1)
P(∅>2P(∅)P(∅)60P(∅)=0
(Ak)k∈NA0=A A1=B Ak=∅k>2
P(A∪B) = P
[
k∈N
Ak
=
+∞
X
k=0
P(Ak) = P(A0) + P(A1) +
+∞
X
k=0
P(Ak) = P(A) + P(B) +
+∞
X
k=0
P(∅)
|{z}
=0
P(A∪B) = P(A) + P(B)
A∈ P(Ω) A A Ω
1 = P(Ω) = P(A∪A) = P(A) + P(A)
P(A)=1−P(A)
A, B ∈ P(Ω) A⊂B B =A∪(A∩B)
P(B) = P(A) + P(A∩B)
| {z }
>0
>P(A)
P(A)6P(B)
A∪B= (A∩B)∪(A∩B)∪(B∩A)
A= (A∩B)∪(A∩B)B= (A∩B)∪(B∩A)
P(A∪B) = P(A∩B) + P(A∩B) + P(B∩A)
=P(A)−P(A∩B)+P(A∩B) + P(B)−P(A∩B)
P(A∪B) = P(A) + P(B)−P(A∩B)
{An, n ∈N}
+∞
X
n=0
P(An) = P(Ω) = 1
n>1Fnn
P(Fn) = (5/6)n−1(1/6)
P(A)P(B)
C
A B P (A) = P(B)=0,75 P(A∩B)