polycopié - IMJ-PRG

Ω Ω P: Ω →[0,1]
X
x∈Ω
P(x)=1 P(x)x A
ΩP(A) = X
x∈A
P(x)A
Ω = { }
P P ( ) = P( ) = 1
2
ΩP
A⊂ΩP(A)≤1
ΩP
ΩP
A⊂Ω
Ω\A A ΩA
P(A)=1−P(A)
A B ΩA B
P(A∩B) = P(A)P(B)

A={1,2}B={1,3,5}
A B P (A) = 0,2P(B) =
0,5P(A∪B)
p1−p
n
A B ΩP(B)>0
A B PB(A)P(A|B)
P(A|B) = P(A∩B)
P(B).
A B P (A|B) = P(A)
A=B={2}
P(B|A)
A B P (A) = 0,2P(B) =
0,5P(A|B)
1/2 1/2
1/2
1/2 1/2
1/2

r d
v
R
P(r) = 9
64
P(d) = 15
28
P(R|d) = 3
7
P(R) = 55
196
P(r|R) = 12
77
S=NJ−(10 −NJ) = 2NJ−10,
NJN=S S = 0 N= 0
S < 0
S
P(S) = P(S) = 1
2.
3
4
1
2
A B PB(A) = P(A|B)
9−1PS(N= 8) = P({N= 8}|S) = 10
210
PS(N= 10) = P({N= 10}|S) = 1
20

P({N= 0}|S) =
1
2
P({N= 0}|S) =
0
PS(N= 10) =
P({N= 10}|S) = 310
410
P(N= 10) = 310 + 210
2×410
P{N=10}(S) =
P(S|{N= 10}) = 1
1 + 2
310
3
410
+1
210
A B P (A∩B) = P(A)P(B)
A B PB(A)P(A|B)
P(A|B) = P(A∩B)
P(B)
Ωn
P(x) = 1
n
x∈Ωn
p∈[0,1] pΩ = {0,1}
P(1) = p P (0) = 1 −p.
p=b
`b ` −b
n∈N∗p∈[0,1] Ω = {0, . . . , n}
P(k) = n
kpk(1 −p)n−k.
p=b
`n
b ` −b