1 Espaces probabilisés discrets
◦
Ω1={1,2,3,4,5,6}2 (i, j)∈Ω1
i j
Ω2={(i, j), i, j ∈ {1, . . . , 6}, i ≤j} (i, j)∈Ω2
i j
Ω3={2,3,4, . . . , 12} i∈Ω3 i
ABD
A= ,
B= ,
C= .
E (An)n∈N E m∈N
Sm=
n≥m
An, Tm=
n≥m
An.
(Tm) (Sm)
An⊃ An.
(Ac
n)=( An)c.
(Ac
n) = ( An)c.
P( nAn)≤ nP(An).
P nAn≥ nP(An).
p p′
p
n pn
1/2
(Ω,P)
pω=P({ω})
Ω (pω)ω∈Ω
ω∈Ω
pω= 1 .
P Ω
P({ω}) = pω∀ω∈Ω.
(E, P) {An}n≥1 B1=A1
n≥2Bn=An\An−1
An ∪i≥1Ai Bi
P+∞
n=1
An=
n→+∞P(An).
Ω = {1,2,3,4,5,6} i∈Ω
i
Ω = {a, b, c} a b
c
Ω = {ω1, ω2, ω3} ω1
ω2 ω3
Ω
ABC D
Ω
A=
B=
C= D=
Ω1={1,2, . . . , 10} Ω2=N Ω1
Ω2
◦
(E, P) {An}n≥1
P+∞
n=1
An≤
+∞
n=1
P(An).
+∞
n=1
P(An)<+∞=⇒P(
n
An) = 0 .
Remarque :
r
1/r2
N
p
A B
A B
P(A∩B) A B
A B N p
JojoTranquile
JojoTranquile
1/4
1/2
A= P(A)
♣ (Ω,P) A B
A B
P(B|A) = P(A)
P(A∪B) = P(A) + P(B)
P(A∪B) = P(A)P(B)
P(B|A) = P(A|B)
P(A|B) = P(A)
♣ (Ω,P) AB C
PC Ω PC(E) = P(E|C) E⊂Ω
A B
PC
A B C P
AB C P
AB C P
A B P
♣ (Ω,P) A B
P(A)≤1/2 P(B)≤1/2 P(A∩B)≤1/4
P(A)≥1/2 P(B)≥1/2 P(A∩B)≥1/4
P(A) = P(B) = 1 P(A∩B) = 1
♣ A B
A B
A B
A B
A B
♣ (Ω,P) A B
A∪B= Ω P(A)P(B) = 1
A∩B=∅ P(A)P(B) = 0
A∪B= Ω P(A) + P(B) = 1
A∪B= Ω P(A) + P(B)≥1
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