Feuilles d`exercices capes

Ω = {1,2,3,4}Ω (P(Ω)) =
24(Ω) = n(P(Ω)) = 2n
n, p ∈N
n
p=n n−p n+ 1 p+ 1=n p+n p+ 1.
A B C E
(A∪B)c=Ac∩Bc(A∩B)c=Ac∪Bc.
A∩(B∪C) = (A∩B)∪(A∩C)A∪(B∩C) = (A∪B)∩(A∪C).
(A∪B)∩(A∪Bc) (A∪B)∩(Ac∪B)∩(Ac∪Bc).
A B C
A B C
A B C
A B C
A B C
Ω
A B
C
A B C Ac∩B Ac∩BcA∩C

[0, T ]
Ω
4
N
N
(Ω,F,P) Ω
n+ 1 A1, ..., An, A A1∩A2∩... ∩An⊂A
P[A]≥
n
X
i=1
P[Ai]−(n−1).
k
k≥1/2
n

pk=P( ).
pkk∈1,2,3
p
a= 95%
b= 1%
p= 0.5%
a b p
a b p

n
n
a b c ax2+bx +c= 0
[i|j]
0 6 [i|j] [j|i]
6 1/3
6
4
6
4
6
Ω
A
BΩ
A B
A B
A1
2
B p 1−p
A B
An
A

an=P[An] (an, n ≥1)
a1=1
2
an+1 =1
2an+ (1 −p)(1 −an)
n≥1
an=p−1
2n−11 2−1−p
3
2−p+1−p
3
2−p
(an, n ≥1)
n n
Xn
Xn
n
NbNnN=Nn+Nb
r X
X
Ω
n∈N?An
An
n∈N?Bnn B
BnB
{1, ..., n}p
41
2
p
0≤x < 1r∈N?
1
(1 −x)r=∞
X
k=0 k+r−1
r−1xk
p
1−p
r
X r
P[X=k]k∈N
r
E[X]