a
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School
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Ressource
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4
3
2
1
Outside the high school program
/ Sets, combinatorial analysis and probability
18 exercises (including 15 corrected)
a
Old annals - matching
(3 exercices)
TD1 L4
(5 exercices)
TD2 L4
(4 exercices)
TD3 L4
(3 exercices)
TD4 L4
(2 exercices)
Newton's binomial formula
(1 exercice)
ChingQuizz
:
2
exercises
a
v
ailable
for
Quizz
assessmen
t
:
1.
Old
annals
-
matching
E.3154
P
art
A
A
die
in
the
shap
e
of
a
regular
tetrahedron
is
pro
vided,
p
os-
sessing
one
blue
face,
t
w
o
red
faces
and
one
green
face
;
the
die
is
assumed
to
b
e
p
erfectly
balanced.
A
game
consists
in
making
t
w
o
successiv
e
and
indep
enden
t
thro
ws
of
this
die.
On
eac
h
thro
w,
the
color
of
the
hidden
face
is
noted.
Consider
the
follo
wing
ev
en
ts
:
E
is
the
ev
en
t
ˇ
at
the
end
of
a
game,
the
t
w
o
faces
noted
are
v
ertes
ı
;
F
is
the
ev
en
t
ˇ
at
the
end
of
a
game,
the
t
w
o
noted
faces
are
of
the
same
couleur
ı.
1
Calculate
the
probabilities
of
ev
en
ts
E
and
F
and
the
probabilit
y
of
E
kno
wing
F
.
2
T
en
iden
tical
and
indep
enden
t
games
are
pla
y
ed.
Calculate
the
probabilit
y
of
obtaining
at
least
t
wice
the
ev
en
t
F
during
these
ten
games
(an
appr
oximate
de
cimal
value
wil
l
b
e
given
to
the
ne
ar
est
10
−
3
)
P
art
B
W
e
w
an
t
to
kno
w
whether
the
die
used
can
b
e
considered
p
erfectly
balanced.
T
o
do
this,
w
e
n
um
b
er
the
four
faces
of
this
die
from
1
to
4,
then
w
e
roll,
this
die
160
times
noting
the
n
um
b
er
n
i
of
times
eac
h
face
is
hidden
;
w
e
obtain
the
follo
wing
results
:
face
i
1
2
3
4
effectif
n
i
30
48
46
32
W
e
note
f
i
the
frequency
relativ
e
to
face
n
i
and
d
2
obs
the
real
4
i
=1
f
i
−
1
4
2
.
W
e
then
sim
ulate
1000
times
the
exp
erimen
t
consisting
of
dra
wing
a
n
um
b
er
at
random
160
times
from
the
set
1
;
2
;
3
;
4
then,
for
eac
h
sim
ulation,
w
e
calculate
d
2
=
4
i
=1
F
i
−
1
4
2
,
where
F
i
is
the
frequency
of
o
ccurrence
of
the
n
um
b
er
i
.
The
9
ième
decile
of
the
statistical
series
of
1000
v
alues
of
d
2
is
equal
to
0.0098
.
In
view
of
the
exp
erimen
t
p
erformed
and
at
the
risk
of
10
%
,
can
the
die
b
e
considered
p
erfectly
balanced?
E.3199
1
A
t
a
sho
oting
range,
a
sho
oter
fires
successiv
e
shots
at
a
ballo
on
in
order
to
p
op
it.
With
eac
h
shot,
there
is
a
probabilit
y
of
0.2
of
p
opping
the
ballo
on.
The
sho
oter
stops
when
the
ballo
on
is
p
opp
ed.
The
successiv
e
shots
are
assumed
to
b
e
indep
enden
t.
a
What
is
the
probabilit
y
that
after
t
w
o
shots
the
bal-
lo
on
will
still
b
e
in
tact?
b
What
is
the
probabilit
y
that
t
w
o
shots
are
enough
to
puncture
the
ballo
on?
c
What
is
the
probabilit
y
p
n
that
n
shots
are
enough
to
puncture
the
ballo
on?
d
F
or
what
v
alues
of
n
do
w
e
ha
v
e
:
p
n
>
0.99
?
2
This
sho
oter
participates
in
the
follo
wing
game
:
First,
he
rolls
a
regular
tetrahedral
die
with
sides
n
um-
b
ered
from
1
to
4
(the
side
obtaine
d
with
such
a
die
is
the
hidden
side)
;
let
k
b
e
the
n
um
b
er
on
the
face
obtained.
The
sho
oter
then
go
es
to
the
sho
oting
range
and
is
al-
lo
w
ed
k
shots
to
p
op
the
ballo
on.
Sho
w
that,
if
the
die
is
w
ell
balanced,
the
probabilit
y
of
p
opping
the
ballo
on
is
equal
to
0.4096
(a
weighte
d
tr
e
e
c
an
b
e
use
d)
.
3
The
sho
oter
decides
to
test
the
tetrahedral
die
to
see
if
it
is
w
ell
balanced
or
if
it
is
loaded.
T
o
do
this,
he
rolls
the
die
200
times
and
obtains
the
follo
wing
table
:
F
ace
k
1
2
3
4
Num
b
er
of
outputs
on
the
fron
t
panel
k
58
49
52
41
a
Calculate
the
frequencies
of
outputs
f
k
observ
ed
for
eac
h
of
the
faces.
b
W
e
set
d
2
=
4
k
=1
f
k
−
1
4
2
.
Calculate
d
2
.
c
W
e
no
w
p
erform
1
000
sim
ulations
of
200
thro
ws
of
a
w
ell-balanced
tetrahedral
die
and
calculate
the
n
um-
b
er
d
2
for
eac
h
sim
ulation.
F
or
the
statistical
series
of
1
000
v
alues
of
d
2
,
w
e
obtain
the
follo
wing
results
:
Min.
D
1
Q
1
Médiane
Q
3
D
9
Max.
0.001
24
0.001
92
0.002
35
0,
002
81
0.003
45
0.004
52
0,
010
15
A
t
the
risk
of
10
%
,
can
w
e
consider
this
die
to
b
e
loaded?
h
ttps:/
/c
hingmath.fr
sacados/3154
France Septembre 2005 3 points
sacados/3199
France Juin 2006 5 points
R
1
R
2
R
3
R
4
B
0,7
0,3
E.3219
F
or
eac
h
of
the
three
questions,
full
marks
will
b
e
a
w
arded
if
the
answ
er
is
correctly
justified.
The
three
questions
are
indep
enden
t.
1
The
probabilit
y
of
an
individual
in
a
p
opulation
ha
ving
disease
M
is
equal
to
0.003
.
A
screening
test
for
this
disease
has
b
een
dev
elop
ed
;
with
this
test,
w
e
can
sa
y
that
:
if
a
p
erson
has
disease
M
,
the
test
is
p
ositiv
e
in
50
%
of
cases
;
the
test
is
p
ositiv
e
for
3
%
of
health
y
p
eople.
What
is
the
probabilit
y
,
to
the
nearest
0.01
,
of
ha
ving
disease
M
when
the
test
is
p
ositiv
e?
a
0.95
b
0.9
c
0.15
d
0.05
2
Consider
a
nail
b
oard
of
this
t
yp
e.
A
ball
B
is
thro
wn
from
the
top
of
the
b
oard
and
falls
in
to
one
of
four
con
tainers
lab
eled
R
1
,
R
2
,
R
3
,
and
R
4
.
A
t
eac
h
stage,
the
ball
has
a
probabilit
y
of
0.3
of
going
to
the
left
and
0.7
of
going
to
the
righ
t
(left
and
right
r
elative
to
the
observer)
.
Let
p
1
b
e
the
probabilit
y
that
the
ball
falls
in
to
bin
R
1
or
bin
R
3
and
p
2
b
e
the
probabilit
y
that
the
ball
falls
in
to
bin
R
2
or
bin
R
4
.
What
are
the
v
alues
of
p
1
and
p
2
?
a
p
1
=
p
2
=
0.5
b
p
1
=
0.216
and
p
2
=
0.784
c
p
1
=
0.468
and
p
2
=
0.532
d
p
1
=
0.468
and
p
2
=
0.432
3
The
first
1000
decimal
places
of
ı
are
giv
en
here
b
y
a
computer
:
1415926535
8979323846
2643383279
5028841971
6939937510
5820974944
5923078164
0628620899
8628034825
3421170679
8214808651
3233066470
9384460959
0582235725
3594085234
8111745028
4102701930
5211055596
4462294895
4930301964
4288109756
6593344612
8475648233
7867831652
7120190914
5648566923
4603486534
5432664825
3393607260
2491412737
2450700660
6315580574
8815209209
6282925409
1715364367
8925903600
1133053054
8820466525
3841469519
4151160943
3057270365
7595919530
9218611738
1932611793
1051185480
7446297996
2749567355
8857527240
9122793318
3011949129
8336733624
4065664308
6025394946
3952247371
9070217986
0943702770
5392171762
9317675238
4674818467
6691051320
0056812714
5263560827
7857753427
9778900917
3637178721
4684409012
2495343054
6549585371
0507922796
8925892354
2019956112
1290219608
6403441815
9813629774
7713099605
1870721134
9999998372
9780499510
5973173281
6096318599
0244594553
4690830264
2522300253
3446850352
6193110017
1010003137
8387528865
8753320830
1420617177
6691473035
9825349042
8755460731
1595620633
8235378759
3751957781
8577805321
7122600661
3001927876
6111959092
1642019894
By
grouping
these
decimal
places
in
to
v
alues
b
et
w
een
0
and
9
,
w
e
obtain
the
follo
wing
table
:
V
aleurs
0
1
2
3
4
Occurrences
93
116
102
102
94
V
aleurs
5
6
7
8
9
Occurrences
97
94
95
101
106
Using
a
table,
w
e
sim
ulated
1000
exp
erimen
ts
of
1000
random
dra
ws
of
a
n
um
b
er
b
et
w
een
0
and
9
.
F
or
eac
h
exp
erimen
t,
w
e
calculated
d
2
=
k
=9
k
=0
f
k
−
0.1
2
,
where
f
k
represen
ts
the
observ
ed
frequency
of
digit
k
for
the
exp
erimen
t.
W
e
then
obtained
a
statistical
series
for
whic
h
w
e
calcu-
lated
the
first
and
nin
th
deciles
(
d
1
and
d
9
),
the
first
and
third
quartiles
(
A
1
and
Q
3
)
and
the
median
(
M
e
)
:
d
1
=
000.422
;
Q
1
=
0.000
582
;
M
e
=
0.000
822
Q
3
=
0.00
1136
;
d
9
=
0.00145
By
calculating
d
2
on
the
first
1000
decimal
places
of
ı
,
w
e
obtain
:
a
0.000
456
b
0.004
56
c
0.000
314
4
A
statistician
who
disco
v
ers
the
table
and
do
es
not
kno
w
that
these
are
the
decimal
places
of
ı
h
yp
othesizes
that
the
series
is
the
result
of
indep
enden
t
random
dra
ws
fol-
lo
wing
an
equipartite
distribution.
He
tak
es
a
10
%
risk
of
rejecting
this
h
yp
othesis
when
it
is
true.
Do
es
he
ac-
cept
this
h
yp
othesis?
a
Y
es
b
No
c
He
cannot
conclude
2.
TD1
L4
E.8074
Consider
an
urn
in
whic
h
there
are
fiv
e
green,
four
blue
and
one
red
tok
en.
The
fol-
lo
wing
random
exp
erimen
t
consists
of
dra
wing
at
random
a
tok
en
from
the
urn
and
noting
its
color.
1
Describ
e
the
univ
erse
Ω
of
this
exp
erimen
t.
2
Are
the
outcomes
of
this
random
exp
erimen
t
equiproba-
h
ttps:/
/c
hingmath.fr
sacados/3219
Antilles-Guyane Septembre 2004 5 points
R
1
R
2
R
3
R
4
B
0,7
0,3
chapExoCorrec/8074
sacados/8074
ble?
3
Name
an
elemen
tary
ev
en
t
and
a
non-elemen
tary
ev
en
t
asso
ciated
with
this
exp
erimen
t.
E.8075
Let
A
,
B
and
C
b
e
three
ev
en
ts
in
a
univ
erse
Ω
.
T
ranslate
the
follo
wing
ev
en
ts
in
to
ensem
blistic
terms
(using
only
the
symb
ols
for
union,
inter-
se
ction
and
c
omplementary
p
assage,
as
wel
l
as
A
,
B
and
C
)
:
a
Only
B
comes
true.
b
A
and
C
come
true,
but
not
B
.
c
T
w
o
or
few
er
of
A
,
B
and
C
come
true.
E.8076
Let
b
e
a
univ
erse
Ω
and
let
b
e
three
ev
en
ts
A
,
B
and
C
of
Ω
.
T
ranslate
in
to
ensem
blis-
tic
terms
(using
only
the
symb
ols
for
union,
interse
ction
and
c
omplementary
p
assage,
as
wel
l
as
and)
the
follo
wing
ev
en
ts
:
1
Only
A
o
ccurs.
2
A
and
B
come
true,
but
not
C
.
3
all
three
ev
en
ts
come
true.
4
at
least
one
of
the
three
ev
en
ts
comes
true.
5
at
least
t
w
o
of
the
three
ev
en
ts
come
true.
6
none
come
true.
7
at
most
one
of
the
three
comes
true.
8
exactly
t
w
o
of
the
three
come
true.
E.8077
T
w
o
cards
are
dra
wn
sim
ul-
taneously
from
a
dec
k
of
32
cards.
Consider
the
follo
wing
ev
en
ts
:
A
:
ˇ
b
oth
cards
are
carreaux
ı
B
:
ˇ
there
is
a
king
and
a
sept
ı
C
:
ˇ
b
oth
cards
are
nom
bres
ı
What
do
the
follo
wing
ev
en
ts
represen
t?
a
B
b
A
∩
B
c
C
∩
B
d
A
∪
C
∩
B
E.8078
T
w
o
ev
en
ts
A
and
B
are
suc
h
that
:
P
A
=
0.35
;
P
B
=
0.4
;
P
A
∩
B
=
0.1
Calculer
P
A
∪
B
3.
TD2
L4
E.8079
A
bag
con
tains
6
balls,
indis-
tinguishable
b
y
touc
h,
n
um
b
ered
:
1
;
1
;
2
;
3
;
3
;
4
A
ball
is
dra
wn
at
random
and
its
n
um
b
er
noted.
W
e
name
X
the
random
v
ariable
whic
h,
to
a
dra
wn
ball
asso-
ciates
its
n
um
b
er
to
it.
Determine
the
probabilit
y
la
w
of
the
random
v
ariable
X
.
E.8080
Thirteen
p
eople
gather
at
a
table
to
ha
v
e
a
bite
to
eat.
In
ho
w
man
y
differen
t
w
a
ys
can
they
sit
on
the
thirteen
armc
hairs
if
:
1
these
armc
hairs
are
on
the
same
side
of
a
rectangular
table
(think
of
Da
Vinci’s
L
ast
Supp
er)
?
2
these
c
hairs
are
arranged
around
a
table
with
a
c
hairp
er-
son’s
seat
and
six
c
hairs
on
either
side
(think
of
futur
e
pr
ofessional
me
etings)
?
3
these
armc
hairs
are
arranged
regularly
around
a
round
table,
assuming
that
no
armc
hair
stands
out
from
the
others
(think
of
a
fe
ast
among
diehar
d
Gauls)
?
E.8081
A
dra
w
er
con
tains
5
distinct
pairs
of
blac
k
sho
es,
3
distinct
pairs
of
green
sho
es
and
2
distinct
pairs
of
red
sho
es.
W
e
c
ho
ose
2
sho
es
at
random.
1
Ho
w
man
y
p
ossible
dra
ws
are
there?
2
Ho
w
man
y
dra
ws
con
tain
t
w
o
sho
es
of
the
same
color?
3
Ho
w
man
y
dra
ws
con
tain
a
left
fo
ot
and
a
righ
t
fo
ot?
4
Ho
w
man
y
dra
ws
con
tain
a
left
fo
ot
and
a
righ
t
fo
ot
of
the
same
color?
E.8082
An
urn
con
tains
5
red
balls
and
3
blac
k
balls
indistinguishable
to
the
touc
h.
A
ball
is
tak
en
at
random,
its
color
observ
ed
and
then
returned
to
the
urn,
also
adding
a
ball
of
the
same
color
as
the
one
tak
en.
A
second
ball
is
then
tak
en
at
random
and
the
color
of
the
second
ball
observ
ed.
1
What
is
the
probabilit
y
of
dra
wing
a
red
ball
on
the
first
dra
w?
2
Kno
wing
that
a
red
ball
w
as
dra
wn
on
the
first
sampling,
what
is
the
probabilit
y
of
dra
wing
a
blac
k
ball
on
the
second
sampling?
3
Calculate
the
probabilit
y
of
dra
wing
a
red
ball
and
a
blac
k
ball
in
this
exp
erimen
t.
4
V
erify
that
the
probabilit
y
of
dra
wing
a
red
ball
on
the
first
dra
w
kno
wing
that
a
red
ball
w
as
dra
wn
on
the
sec-
ond
dra
w
is
equal
to
2
3
.
4.
TD3
L4
E.8083
Consider
a
random
v
ariable
X
that
follo
ws
the
binomial
distribution
with
parameters
20
and
0.4
.
Results
will
b
e
giv
en
to
the
nearest
0.001
.
1
Calculate
P
X
=
3
and
P
X
=
11
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8075
sacados/8075
chapExoCorrec/8076
sacados/8076
chapExoCorrec/8077
sacados/8077
chapExoCorrec/8078
sacados/8078
chapExoCorrec/8079
sacados/8079
chapExoCorrec/8080
sacados/8080
chapExoCorrec/8081
sacados/8081
chapExoCorrec/8082
sacados/8082
chapExoCorrec/8083
sacados/8083
0
2
4
6
8
10
12
14
16
18
20
C
2
C
1
C
3
2
Calculate
P
X
2
;
P
X
18
(r
ound
this
pr
ob
ability
to
the
ne
ar
est
0.000
001
)
.
E.8084
An
urn
con
tains
30
tok
ens,
of
whic
h
6
are
red.
Eac
h
da
y
a
p
erson
randomly
dra
ws
a
tok
en
from
the
urn
and
then
puts
it
bac
k
in.
Consider
a
p
ositiv
e
non-zero
in
teger
n
and
denote
X
the
random
v
ari-
able
corresp
onding
to
the
n
um
b
er
of
red
tok
ens
dra
wn
after
n
consecutiv
e
da
ys.
1
What
is
the
la
w
of
X
?
Justify
.
If
necessary
,
v
alues
in
the
follo
wing
questions
will
b
e
rounded
to
the
nearest
thousandth.
2
What
is
the
probabilit
y
that
on
10
consecutiv
e
da
ys,
the
p
erson
dra
ws
4
red
tok
ens?
at
least
1
red
tok
en?
3
What
m
ust
b
e
the
minim
um
n
um
b
er
of
consecutiv
e
da
ys
for
the
probabilit
y
of
no
red
tok
ens
dra
wn
to
b
e
less
than
0.001
?
E.8085
A
transp
ort
compan
y
wishes
to
optimize
con
trols
in
order
to
limit
fraud.
This
compan
y
carries
out
a
study
based
on
2
journeys
p
er
da
y
during
the
20
w
orking
da
ys
of
a
mon
th,
i.e.
a
total
of
40
journeys.
It
is
assumed
that
the
c
hec
ks
are
indep
enden
t
of
eac
h
other
and
that
the
probabilit
y
of
an
y
tra
v
eler
b
eing
c
hec
k
ed
is
equal
to
p
.
A
journey
costs
10
euros
and
in
the
ev
en
t
of
fraud
the
fine
is
100
euros
(we
only
p
ay
the
fine
and
not
the
journey
in
addi-
tion)
.
Theo
systematically
c
heats
on
the
40
journeys
studied.
W
e
denote
X
the
random
v
ariable
that
coun
ts
the
n
um
b
er
of
trips
where
Theo
w
as
c
hec
k
ed.
1
It
is
assumed
that
p
=
0.5
.
The
probabilit
y
that
Theo
will
b
e
c
hec
k
ed
at
most
2
times
is
:
a
0.2
b
0.97
c
7
×
10
−
10
d
2
×
10
−
1
2
Let
Z
b
e
the
random
v
ariable
giving
the
algebraic
gain
made
b
y
Theo
o
v
er
the
40
journeys.
Justify
that
Z
=
400
−
100
·
X
then
calculate
E
(
Z
)
.
3
The
v
alue
of
p
is
no
longer
kno
wn.
F
or
what
v
alues
of
p
is
systematic
fraud
fa
v
orable
to
Theo?
Justify
.
W
e
will
giv
e
the
set
of
v
alues
of
p
fulfilling
this
condition
in
the
form
of
an
in
terv
al
whose
b
ounds
will
b
e
rounded
to
the
nearest
h
undredth.
5.
TD4
L4
E.8086
T
en
solar
panels
are
installed
on
the
ro
of
of
a
house
lo
cated
in
a
region
with
regular
sun-
shine
and
pro
duce
electricit
y
.
W
e
denote
b
y
Y
the
random
v
ariable
whic
h,
on
eac
h
da
y
,
asso
ciates
the
electrical
pro
duc-
tion
supplied
b
y
these
10
panels...
expressed
in
k
W
h
.
The
v
ariable
Y
follo
ws
the
binomial
distribution
with
param-
eters
—
=
9
and
ff
=
3
.
1
What
is
the
probabilit
y
(to
within
10
−
2
)
that
daily
pro-
duction
will
b
e
b
et
w
een
6
and
12
k
W
h
?
2
Whic
h
of
the
three
probabilit
y
densit
y
functions
sho
wn
b
elo
w
can
b
e
the
la
w
of
Y
?
Justify
.
3
The
o
ccupan
ts
of
the
house
consume
on
a
v
erage
10
k
W
h
p
er
da
y
(excluding
he
ating
and
hot
water)
.
a
What
is
the
probabilit
y
(to
within
10
−
3
)
that
the
pan-
els’
daily
output
exceeds
a
v
erage
daily
consumption?
b
What
w
ould
this
family’s
a
v
erage
daily
consumption
ha
v
e
to
b
e,
in
k
W
h
,
for
this
probabilit
y
to
b
e
appro
x-
imately
90
%
?
W
e’ll
round
the
answ
er
to
the
ten
th.
E.8087
1
Let
X
b
e
a
random
v
ariable
follo
wing
the
normal
distri-
bution
N
20
;
25
,
i.e.
its
exp
ectation
equals
20
and
its
standard
deviation
equals
5
.
a
Giv
e
the
v
alue
of
the
n
um
b
ers
a
and
b
so
that
the
ran-
dom
v
ariable
Z
defined
b
y
:
Z
=
X
−
a
b
follo
ws
the
cen
tered
and
reduced
normal
distribution.
b
Calculate
to
the
nearest
10
−
3
:
P
X
28)
;
P
X
>
28)
;
P
X
=
28)
P
X
28)
;
P
12
<
x
<
28)
;
P
15
<
x
<
25)
c
Determine,
to
the
nearest
unit,
the
n
um
b
er
¸
suc
h
that
:
P
X
<
¸
)
=
0.99
d
Determine,
to
the
nearest
unit,
the
n
um
b
er
˛
suc
h
that
:
P
20
−
˛
<
X
<
20
+
˛
)
=
0.95
2
Let
Y
b
e
suc
h
that
Y
follo
ws
N
m
;
4
.
Calculate
m
to
the
nearest
0.1
so
that
:
P
Y
>
25
=
0.95
3
Let
T
b
e
suc
h
that
T
follo
ws
N
20
;
ff
2
.
Calculate
ff
to
the
nearest
unit
so
that
:
P
0
<
T
<
40
=
0.99
6.
Newton
’s
binomial
formula
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8084
sacados/8084
chapExoCorrec/8085
sacados/8085
chapExoCorrec/8086
sacados/8086
0
2
4
6
8
10
12
14
16
18
20
C
2
C
1
C
3
chapExoCorrec/8087
sacados/8087
E.8559
Let
n
b
e
a
non-zero
natural
n
um
b
er
:
1
Establish
equalit
y
:
n
k
=1
k
·
n
k
=
n
·
2
n
−
1
2
Deduce
the
equalit
y
:
n
k
=1
k
−
1
·
n
k
=
n
·
2
n
−
1
−
2
n
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8559
sacados/8559
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