Probabilités... sans les variables aléatoires

(Ω,T)
B A

[[1, n]]
N
N Z Q
P(N)R
{0,1}N{0,1} P(N)
E1, ..., En
n
[
k=1
Ek
n∈NEn[
n∈N
En
E F E ×F
[
k∈N
Ek\
k∈N
Ek
[
n∈N∗
[1/n, 3−1/n]\
n∈N∗
[1/n, 3−1/n]
[
n∈N∗
]−1/n, 1+1/n[\
n∈N∗
]−1/n, 1+1/n[
n∈NEn+1 ⊂En[
n∈N
En\
n∈N
En
n∈NEn⊂En+1 \
k∈N
Ek[
k∈N
Ek
Ω = {P, F }
Ω = {1,2, ..., 6}26
{R>3},{R62},{R},{R= 1 R= 3}.

Ω = {(1,1); (1,2); ...; (6,6)}= [[1,6]]2236 {x=y} {x
} {x+y64} {(x, y) = (2,3)}
0
1
Ω Ω P(Ω) [0,1]
P(Ω) = 1
A, B ∈ P(Ω) P(A∪B) = P(A) + P(B)
X⊂ΩP(X) = |X|
|Ω|
Ω
Ω
P P({x})x∈Ω Ω
PΩ
A⊂ΩP(A)=1−P(A)
A, B ⊂ΩP(A∪B) = P(A) + P(B)−P(A∩B)
A⊂B⊂ΩP(A)6P(B)
{0,1}N
1 2
E1, ..., En
P n
[
k=1
Ek!=
n
X
k=1
P(Ek)
ΩP(Ω)
Ω
Ω

σ
ΩσΩTΩ
∅∈T
T
T
Ω
T={∅,Ω}
T=P(Ω)
A⊂ΩT={∅, A, A, Ω}A
i∈I Ai∈ T i, j ∈I Ai∩Aj=∅[
i∈I
Ai= Ω
(Ai)i∈I
ΩT
T
(Ω,T)
(Ω,T,P)TΩP
T[0,1]
P(Ω) = 1
(An)n∈NPP(An)
P [
n∈N
An!=
∞
X
n=0
P(An)
σ
P N
[
n=1
An!6
N
X
n=0
P(An)
P [
n∈N
An!6
∞
X
n=0
P(An).
(Ω,T,P)
P(∅) = 0
A∈ T P(A)=1−P(A)
A, B ∈ T A⊂BP(A)6P(B)
Ω Ω = {x0, x1, ..., xn, ...} T =P(Ω)
σPP({xn}) 1 A⊂ΩTP
x∈A
P({x})
P(A) Ω
Ω = {xn|n∈N}Ppn1
ΩT=P(Ω) A⊂ΩP(A) = P
xn∈A
pn

(An)n∈NE=[
k∈N
AkT
AnE F =\
k∈N
Bk
(Bn)n∈N
Ann
Ann13
Ann
Bnn
1/3
Bnn
Bnn
P(An)P(E)P(Bn)P(F)
(Ω,T,P)
(An)n∈N
P(An)−→
n→+∞
P [
k∈N
Ak!
(Bn)n∈N
P(Bn)−→
n→+∞
P \
k∈N
Bk!
P [
k∈N
Ak!= lim
n→+∞
P(An)P \
k∈N
Bk!= lim
n→+∞
P(Bn)
C0=A0n∈N∗Cn=An\An−1E Cn
P(E) =
∞
X
n=0
P(Cn) = lim
n→∞
n
X
k=0
P(Ck).
n
P
k=0
P(Ck) = Pn ∪
k=0
Ck=P(An)
PFFP
AnElim An=E
P(lim An) = lim P(An),