Probabilités - Normalesup.org

2 12
4 5 1 + 4
2+3 3+2 4+1 2 1
6
36 2
36 5 36 9
6
6
1 6 2 6 36
10 6
6
65
69
×1
6
6
6
6k5
6k−1
×1
6

+∞
X
k=1
k×5
6k−1
×1
6=1 6×1
(1 −5
6)2= 6
6
1 2 3
R+
32
Ω
Ω
|Ω|=32
5Ω

A B Ω
A
A
1
•Ω
•
•Ω
•
•
Ω Ω
•A
•B
•C D
•E0E1E4
ΩT ⊂ P(Ω) Ω T
Ω
•Ω∈ T
• T A∈ T ¯ A∈ T
∀i∈NAi∈ T S
i∈N
Ai∈ T
ΩT=P(Ω)
Ω
Ω = R R+
(Ω,T) Ω T
Ω
(Ω,T)PT → [0; 1]

•P(Ω) = 1
•(Ai)i∈NTP(S
i∈N
Ai) = X
i∈N
P(Ai)
σ
(Ω,T, P )P
(Ω,T)
P(1) = P(2) = · · · =P(6) = 1
6σ
P(1) = 1
21 P(2) = 2 21 P(6) = 6 21
P(Ω) 1
P P (∅) = 0
P(∅) + P(∅) = P(∅∪∅) = P(∅)P(∅) = 0
A∈ T P(¯ A) = 1 −P(A)
A¯ A P (A) + P(¯ A) = P(A∪¯ A) =
P(Ω) = 1
A⊂B∈ T 2P(A)6P(B)
P(A)+P(B\A) =
P(B)P(B\A)>0 0 1 P(B)>
P(A)
(A, B)∈ T 2P(A∪B) = P(A) + P(B)−P(A∩B)
P(n ∪
i=1Ai) =
n
X
k=1
((−1)k−1X
16i1<i2<···<ik6n
P(Ai1∩Ai2· · · ∩ Aik))
σ P (A) = P(A∩B) + P(A\B)P(B) = P(A∩B) + P(B\A)
P(A∪B) = P(A\B) + P(B\A) + P(A∩B)
(Ai)i∈NX
i∈N
P(Ai) = 1 ∀B∈ T
P(B) = X
i∈N
P(B∩Ai)

Ai
P(∪
i∈NAi) = X
i∈N
P(Ai)AiΩ
1P(B)
B∩AiB
B
Ω
1
Ω
∀A∈ P(Ω) P(A) = (A)
(Ω)
n=|Ω|
1n
1
n
k1
nk
4
16 32
50.000 08
51
92
10 3
6 5 2 1
6
(Ω,T, P )A B P (A)6= 0
B A PA(B) = P(A∩B)
P(B)
T[0,1]
B7→ P(A∩B)
P(A)TA PA