Étude d`un paradoxe Introduction 1 Étude du jeu A

A
p1= 0,495 1 −p1
B
p2= 0,095
p3= 0,745
A
Snn
S0= 0 Xn
n Xn
{−1,1}
Xnn≥1
Sn
E[Xn]E[Sn]

E1
E2 E3
1−q2
q1
1−q1
q3
q2
1−q3
aij EiEj
(aij )i,j≤N
a12 =q1a21 = 1 −q2a11 = 0
M=
0q11−q1
1−q20q2
q31−q30
.
0< qi<1i= 1,2,3
M2
bij M2
i j
M
M λ1
λ2
∃P∈GL(3,C), M =P
1 0 0
0λ10
0 0 λ2
P−1.

(Mn)n∈N
M∞
M∞k
k
(Mn)n∈N
B
XnB
E0E1
s= 3k+ 1 E2
s= 3k+ 2
Cn{0,1,2}
n
C0= 0
p2p3
M
P(C1= 0) P(C1= 1) P(C1= 2)
g0=E[X2|C1= 0] g1=E[X2|C1= 1]
g2=E[X2|C1= 2] X2
E[X2]
nE[Xn]
Mn−1·G G 
g0
g1
g2
E[Sn]

(Mn)n∈N
limn→∞ ESn
n
A1
2
B1
2
B