a
Middle
School
Grade 6
Grade 7
Grade 8
Grade 9
Ressource
Out Middle
High
School
Grade 10
Grade 11
Grade 11 - STMG
Grade 12 - STMG
Grade 12
Grade 12 - Comp.
Grade 12 - Exp.
Ressource
Out High
4
3
2
1
Outside the high school program
/ Bernoulli and binomial distribution
23 exercises (including 20 corrected)
a
Confidence and fluctuation intervals
(9 exercices)
Old annuals (pre-2011 program)
(1 exercice)
Truncated geometric law
(1 exercice)
Binomial law: fluctuation interval
(6 exercices)
image PNG : /home/_math/_exercice/d7/7851/metapost/a.png
image PNG : /home/_math/_exercice/d7/7857/metapost/a.png
1.
Confidenc
e
and
fluctuation
intervals
E.5435
Consider
a
random
v
ariable
fol-
lo
wing
a
binomial
distribution
with
parameters
9
and
0.3
:
X
∼
B
9
;
0.3
The
table
b
elo
w
pro
vides
information
on
the
probabilit
y
dis-
tribution
of
the
random
v
ariable
X
:
k
0
1
2
3
4
5
6
7
8
9
P
(
X
k
)
0.04
0.196
0.463
0.73
0.901
0.975
0.996
1.0
1.0
1.0
Justify
that
the
random
v
ariable
X
has
at
least
95
%
c
hance
of
taking
its
v
alues
in
the
in
terv
al
0
;
5
.
E.5434
Consider
a
random
v
ariable
follo
w-
ing
a
binomial
distribution
with
parameters
9
and
0.5
:
X
∼
B
9
;
0.5
The
table
b
elo
w
pro
vides
information
on
the
probabilit
y
dis-
tribution
of
the
random
v
ariable
X
:
k
0
1
2
3
4
5
6
7
8
9
P
(
X
k
)
0.002
0.02
0.09
0.254
0.5
0.746
0.91
0.98
0.998
1.0
1
F
or
whic
h
v
alues
of
k
do
w
e
ha
v
e
:
P
X
k
>
0.025
?
2
F
or
whic
h
v
alues
of
k
do
w
e
ha
v
e
:
P
X
k
0.975
?
3
Justify
that
the
random
v
ariable
X
has
at
least
95
%
c
hance
of
taking
its
v
alues
in
the
in
terv
al
2
;
7
.
E.5436
Consider
a
random
v
ariable
X
follo
w-
ing
a
binomial
distribution
with
parameters
30
and
0.32
:
X
∼
B
30
;
0.32
k
0
1
2
3
4
5
6
7
8
P
(
X
k
)
0
0
0.001
0.005
0.018
0
;
049
0.11
0.208
0.341
k
9
10
11
12
13
14
15
16
17
P
(
X
k
)
0.494
0.645
0.774
0
;
871
0.934
0.97
0.988
0.995
0.999
k
18
19
20
21
22
23
24
25
26
27
28
29
30
P
(
X
k
)
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Determine
the
smallest
in
tegers
a
and
b
suc
h
that
:
P
X
a
)
>
0.025
;
P
X
b
0.975
2
Justify
that
:
P
a
X
b
0.95
3
Noting
F
=
X
30
the
random
v
ariable
represen
ting
the
ran-
dom
frequency
of
success.
Justify
that
:
P
a
30
F
b
30
0.95
E.5445
Consider
the
random
v
ariable
X
follo
w-
ing
the
binomial
distribution
with
parameters
20
and
0.34
:
X
∼
B
20
;
0.34
The
questions
will
b
e
answ
ered
without
justification
and
us-
ing
the
calculator.
1
Determine
the
t
w
o
smallest
in
tegers
a
and
b
v
erifying
the
inequalities
:
P
(
X
a
)
>
0.025
;
P
(
X
b
)
0.975
2
Determine
the
fluctuation
in
terv
al
J
at
the
threshold
of
95
%
.
E.5438
Consider
a
random
v
ariable
X
follo
wing
a
binomial
distribution
with
parameters
100
and
0.81
.
k
71
72
73
74
75
76
77
78
79
P
(
X
k
)
0.010
0.018
0.032
0.053
0.084
0.127
0.185
0.257
0.343
k
80
81
82
83
84
85
86
87
88
P
(
X
k
)
0.439
0.540
0.640
0.733
0.813
0.877
0.924
0.957
0.977
Determine
the
fluctuation
in
terv
al
at
95
%
.
E.5490
Consider
a
random
v
ariable
X
follo
w-
ing
a
binomial
distribution
with
parameters
100
and
0.35
(
X
∼
B
(
100
;
0.35
)
)
.
The
follo
wing
questions
are
to
b
e
answ
ered
using
a
calculator,
and
the
results
should
b
e
giv
en
to
three
decimal
places
where
necessary
:
1
Determine
the
v
alue
of
the
follo
wing
probabilities
:
a
P
X
=
43
b
P
X
38
c
P
X
31
2
a
Determine
the
v
alue
of
the
smallest
in
tegers
a
and
b
that
satisfy
the
follo
wing
t
w
o
conditions
:
P
X
a
>
0.025
;
P
X
b
0.975
b
Giv
e
the
fluctuation
in
terv
al
at
95
%
of
the
frequency
asso
ciated
with
the
random
v
ariable
X
.
E.5446
One
compan
y
claims
that
80
%
of
its
customers
are
satisfied
with
its
pro
ducts.
1
A
consumer
asso
ciation
wishes
to
v
erify
this
claim
and
commissions
a
study
of
50
customers
of
this
compan
y
.
a
Determine
the
fluctuation
in
terv
al
at
the
threshold
of
95
%
.
b
The
study
obtains
a
satisfaction
rate
of
71
%
.
A
ccord-
ing
to
this
study
,
what
can
w
e
sa
y
ab
out
the
compan
y’s
affirmation
at
the
threshold
of
5
%
?
2
The
asso
ciation
renews
its
study
,
this
time
fo
cusing
on
100
customers.
This
new
study
still
obtains
a
satisfaction
rate
of
71
%
.
What
can
w
e
sa
y
ab
out
the
compan
y’s
assertion
at
the
threshold
of
5
%
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5435
sacados/5435
chapExoCorrec/5434
sacados/5434
chapExoCorrec/5436
sacados/5436
chapExoCorrec/5445
sacados/5445
chapExoCorrec/5438
sacados/5438
chapExoCorrec/5490
sacados/5490
chapExoCorrec/5446
sacados/5446
E.5449
Consider
a
game
using
t
w
o
urns
con
tain-
ing
balls
of
indistinguishable
red
and
blue
colors.
The
dra
ws
in
the
t
w
o
urns
are
assumed
to
b
e
indep
enden
t
of
eac
h
other.
Here
is
the
comp
osition
of
eac
h
of
the
urns
:
The
A
est
urn
comp
osed
of
three
red
balls
and
t
w
o
blue
balls
;
The
B
urn
is
made
up
of
12
red
balls
and
28
b
oules
blue
balls.
Eac
h
participan
t
m
ust
remo
v
e
one
ball
from
the
A
urn,
then
one
ball
from
the
B
urn.
1
a
Construire
a
probabilit
y
tree
represen
ting
this
game.
b
Sho
w
that
the
probabilit
y
of
getting
at
least
one
red
ball
has
v
alue
18
25
2
W
e
rep
eat
20
times
this
game
indep
enden
tly
.
Consider
the
random
v
ariable
X
whic
h
coun
ts
the
n
um
b
er
of
games
in
whic
h
at
least
one
red
ball
has
b
een
dra
wn.
a
What
is
the
probabilit
y
la
w
follo
w
ed
b
y
the
random
v
ariable
X
?
b
Determine
the
probabilit
y
of
P
X
18
.
Then,
giv
e
the
v
alue
rounded
to
the
nearest
thousandth
of
this
probabilit
y
.
3
a
Determine
the
fluctuation
in
terv
al
at
95
%
of
the
bi-
nomial
distribution
of
a
random
v
ariable
Y
follo
wing
a
binomial
distribution
with
parameters
100
and
18
25
.
b
A
statistical
study
w
as
conducted
on
the
rep
etition
100
times
of
this
game.
The
frequency
of
o
ccurrence
of
the
ev
en
t
ˇ
at
least
one
red
ball
w
as
obtained
ı
w
as
0.55
.
Should
this
study
b
e
rejected
with
a
risk
of
5
%
?
E.7851
A
farm
sorts
its
tomato
es
according
to
size
(c
ali-
br
e)
.
It
informs
its
suppliers
that
40
%
of
its
toma-
to
es
are
of
a
calibre
greater
than
or
equal
to
6
(a
diameter
gr
e
ater
than
47
mm
)
.
Visiting
the
farm,
a
supplier
tak
es
30
tomato
es
and
observ
es
that,
among
them,
10
tomato
es
are
of
a
size
greater
than
or
equal
to
6
.
Opp
osite
is
the
cum
ulativ
e
distribution
function
of
a
binomial
random
v
ariable
with
parameters
30
and
0.4
.
The
follo
wing
results
can
b
e
extracted
from
it
:
P
X
10
≈
0.2915
;
P
X
20
≈
0.9991
1
a
Determine
the
v
alue
of
the
smallest
in
teger
a
that
satisfies
the
condition
:
P
X
a
>
0.025
b
Determine
the
v
alue
of
the
smallest
in
teger
b
that
sat-
isfies
the
condition
:
P
X
b
0.975
c
Deduce
the
fluctuation
in
terv
al
at
the
threshold
of
95
%
of
the
observ
ed
frequency
for
the
random
v
ariable
X
.
(The
limits
wil
l
b
e
r
ounde
d
to
10
−
3
)
2
What
can
b
e
said
ab
out
the
supplier’s
observ
ation?
2.
Old
annuals
(pr
e-2011
pr
o
gr
am)
E.3127
A
pla
y
er
has
an
urn
con
taining
3
red
balls,
4
white
balls
and
n
green
balls
(
0
n
10
)
.
The
balls
are
indistinguishable
to
the
touc
h.
1
The
pla
y
er
randomly
dra
ws
a
ball
from
the
urn.
Calcu-
late
the
probabilit
y
of
eac
h
of
the
follo
wing
ev
en
ts
:
a
R
:
ˇ
The
ball
dra
wn
is
rouge
ı
;
b
B
:
ˇ
the
ball
dra
wn
is
blanc
he
ı
;
c
V
:
ˇ
the
ball
dra
wn
is
v
erte
ı.
2
The
pla
y
er
decides
to
pla
y
a
game.
This
tak
es
place
as
sho
wn
b
elo
w.
The
pla
y
er
dra
ws
a
ball
from
the
urn
:
If
it’s
red,
he
wins
16
F
;
If
it’s
white,
he
loses
12
F
;
If
it’s
green,
he
puts
the
ball
bac
k
in
the
urn,
then
dra
ws
a
ball
from
the
urn
;
If
this
ball
is
red,
he
wins
8
F
;
If
this
ball
is
white,
he
loses
2
F
;
If
this
ball
is
green,
he
neither
loses
nor
gains
an
y-
thing.
The
dra
ws
are
equiprobable
and
t
w
o
successiv
e
dra
ws
are
indep
enden
t.
A
t
the
b
eginning
of
the
game,
the
pla
y
er
o
wns
12
F
.
Let
X
b
e
the
random
v
ariable
that
tak
es
as
its
v
alue
the
sum
the
pla
y
er
o
wns
at
the
end
of
the
game
(one
dr
aw
or
two
dr
aws
as
the
c
ase
may
b
e)
.
a
Determine
the
v
alues
tak
en
b
y
X
.
b
Determine
the
probabilit
y
distribution
of
X
.
c
Sho
w
that
the
mathematical
exp
ectation
of
X
has
the
v
alue
:
E
(
X
)
=
12+16
·
n
(
n
+7)
2
.
3
Consider
the
function
f
defined
on
the
in
terv
al
0
;
10
b
y
f
(
x
)
=
x
(
x
+7)
2
.
Study
the
v
ariations
of
f
.
4
Deduce
the
v
alue
of
n
for
whic
h
the
mathematical
exp
ec-
tation
X
is
maxim
um.
Calculate
this
maxim
um
v
alue
(the
r
esult
wil
l
b
e
given
as
an
irr
e
ducible
fr
action)
.
3.
T
runc
ate
d
ge
ometric
law
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5449
sacados/5449
chapExoCorrec/7851
sacados/7851
chapExoCorrec/3127
sacados/3127
Sportifs de haut-niveau Octobre 1998 4 points
E.7422
Consider
the
follo
wing
random
exp
eri-
men
t
:
thro
w
a
die
5
times
successiv
ely
(and
indep
endently)
and
note
the
thro
w
n
um
b
er
where
the
face
6
first
app
eared.
If
the
6
face
has
not
app
eared
in
an
y
of
these
5
thro
ws
then
0
is
noted.
1
Construct
the
probabilit
y
tree
asso
ciated
with
this
ran-
dom
exp
erimen
t.
2
Consider
the
random
v
ariable
that
asso
ciates
the
n
um
b
er
denoted
with
eac
h
trial
of
the
random
exp
erimen
t.
Determine
the
probabilit
y
distribution
of
the
random
v
ariable
X
.
4.
Binomial
law
:
fluctuation
interval
E.4910
Results
will
b
e
rounded
to
the
nearest
10
−
3
.
In
a
w
atc
h
factory
,
when
the
pro
duction
mac
hines
are
prop-
erly
adjusted,
4
%
of
the
w
atc
hes
pro
duced
are
defectiv
e.
T
o
c
hec
k
the
state
of
pro
duction,
10
w
atc
hes
are
tak
en
suc-
cessiv
ely
,
at
random
and
indep
enden
tly
of
eac
h
other.
The
random
v
ariable
X
asso
ciates
the
n
um
b
er
of
defectiv
e
w
atc
hes
with
this
dra
w.
1
Justify
that
the
random
v
ariable
X
follo
ws
a
binomial
distribution.
Sp
ecify
its
parameters.
2
What
is
the
probabilit
y
that
at
least
1
w
atc
h
is
defectiv
e?
The
exact
v
alue
and
the
v
alue
rounded
to
the
nearest
thousandth
will
b
e
giv
en.
3
An
engineer
is
testing
the
pro
duction
line.
After
sev
eral
batc
h
tests
of
10
w
atc
hes,
he
obtains
81
%
of
batc
hes
with
at
least
1
defectiv
e
w
atc
hes.
What
can
w
e
sa
y
ab
out
the
state
of
the
pro
duction
line?
E.7858
Consider
a
random
v
ariable
follo
w-
ing
a
binomial
distribution
with
parameters
9
and
0.3
:
X
∼
B
9
;
0.3
The
table
b
elo
w
giv
es
information
on
the
probabilit
y
distri-
bution
of
the
random
v
ariable
X
:
k
0
1
2
3
4
5
6
7
8
9
P
(
X
k
)
0
;
04
0.196
0.463
0.73
0.901
0.975
0.996
1.0
1.0
1.0
Justify
that
the
random
v
ariable
X
has
at
least
a
95
%
c
hance
of
taking
its
v
alues
in
the
in
terv
al
0
;
5
.
E.7859
Consider
a
random
v
ariable
follo
w-
ing
a
binomial
distribution
with
parameters
9
and
0.5
:
X
∼
B
9
;
0.5
The
t
w
o
tables
b
elo
w
giv
e
information
ab
out
the
probabilit
y
distribution
of
the
random
v
ariable
X
:
k
0
1
2
3
4
5
6
7
8
9
P
(
X
k
)
0
;
002
0.02
0.09
0.254
0.5
0.746
0.91
0.98
0.998
1.0
1
F
or
what
v
alues
of
k
do
w
e
ha
v
e
:
P
X
k
>
0.025
?
2
F
or
what
v
alues
of
k
do
w
e
ha
v
e
:
P
X
k
0.975
?
3
Justify
that
the
random
v
ariable
X
at
least
95
%
c
hance
of
taking
its
v
alues
in
the
in
terv
al
2
;
7
.
E.7857
A
farm
grades
its
tomato
es
according
to
their
size
(c
alibr
e)
.
It
informs
its
suppliers
that
40
%
of
its
tomato
es
are
of
a
calibre
greater
than
or
equal
to
6
(a
diameter
gr
e
ater
than
47
mm
)
.
Visiting
the
farm,
a
supplier
tak
es
30
tomato
es
and
observ
es
that,
among
them,
10
tomato
es
are
of
a
size
greater
than
or
equal
to
6
.
Opp
osite
is
the
cum
ulativ
e
distribution
function
of
a
binomial
random
v
ariable
with
parameters
30
and
0.4
.
The
follo
wing
results
can
b
e
extracted
from
it
:
P
X
10
≈
0.2915
;
P
X
20
≈
0.9991
1
a
Determine
the
v
alue
of
the
smallest
in
teger
a
that
satisfies
the
condition
:
P
X
a
>
0.025
b
Determine
the
v
alue
of
the
smallest
in
teger
b
that
sat-
isfies
the
condition
:
P
X
b
0.975
c
Deduce
the
fluctuation
in
terv
al
at
the
threshold
of
95
%
of
the
observ
ed
frequency
for
the
random
v
ariable
X
.
(The
limits
wil
l
b
e
r
ounde
d
to
10
−
3
)
2
What
can
b
e
said
ab
out
the
supplier’s
observ
ation?
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7422
sacados/7422
chapExoCorrec/4910
sacados/4910
chapExoCorrec/7858
sacados/7858
chapExoCorrec/7859
sacados/7859
chapExoCorrec/7857
sacados/7857
E.141
A
t
a
funfair,
a
lottery
is
held
ev
ery
hour.
Eac
h
time,
thirt
y
tic
k
ets
are
sold,
of
whic
h
ten
are
win-
ners
(it
is
assume
d
that
al
l
tickets
have
the
same
pr
ob
ability
of
b
eing
b
ought.)
.
F
or
eac
h
result,
giv
e
the
exact
v
alue
and
then
the
appro
ximate
v
alue
rounded
to
the
thousandth.
1
Luk
e
buys
a
tic
k
et.
What
is
the
probabilit
y
that
this
tic
k
et
is
a
winner?
2
Mark
en
ters
three
consecutiv
e
lotteries
for
whic
h
he
tak
es
a
tic
k
et
eac
h
time
(the
lotteries
ar
e
assume
d
to
b
e
inde-
p
endent)
.
What
is
the
probabilit
y
that
Marc
has
at
least
one
winning
tic
k
et?
3
P
eter
tak
es
part
in
a
lottery
,
buying
three
tic
k
ets
sim
ul-
taneously
.
a
What
is
the
probabilit
y
that
P
eter
do
es
not
ha
v
e
a
winning
tic
k
et?
b
What
is
the
probabilit
y
that
P
eter
has
at
least
one
winning
tic
k
et?
4
Who
is
more
lik
ely
to
ha
v
e
at
least
one
winning
tic
k
et,
P
eter
or
Mark?
5
The
adv
ert
announces
ˇ
Every
thir
d
ticket
is
a
winner!
Buy
thr
e
e
tickets!
ı.
This
text
suggests
that,
b
y
buying
three
tic
k
ets,
y
ou
are
sure
to
win.
What
can
w
e
sa
y
ab
out
this
prop
ert
y?
E.7861
Consider
a
game
using
t
w
o
urns
con
tain-
ing
red
and
blue
balls
that
are
indistinguishable
to
the
touc
h.
Assume
that
the
dra
ws
from
the
t
w
o
urns
are
indep
enden
t
of
eac
h
other.
Here
is
the
comp
osition
of
eac
h
urn
:
Urn
A
con
tains
three
red
balls
and
t
w
o
blue
balls
;
Urn
B
con
tains
12
red
balls
and
28
blue
balls.
Eac
h
participan
t
m
ust
dra
w
one
ball
from
urn
A
,
then
one
ball
from
urn
B
.
1
a
Construct
a
probabilit
y
tree
represen
ting
this
game.
b
Sho
w
that
the
probabilit
y
of
obtaining
at
least
one
red
ball
is
18
25
2
This
game
is
rep
eated
20
times
indep
enden
tly
.
W
e
ha
v
e
the
random
v
ariable
X
,
whic
h
coun
ts
the
n
um
b
er
of
games
in
whic
h
at
least
one
red
ball
w
as
dra
wn.
a
What
is
the
probabilit
y
distribution
follo
w
ed
b
y
the
random
v
ariable
X
?
b
Determine
the
probabilit
y
of
P
X
18
.
Then
giv
e
the
v
alue
of
this
probabilit
y
rounded
to
the
nearest
thou-
sandth.
3
a
Determine
the
fluctuation
in
terv
al
at
95
%
of
the
bi-
nomial
distribution
of
a
random
v
ariable
Y
follo
wing
a
binomial
distribution
with
parameters
100
and
18
25
.
b
A
statistical
study
w
as
conducted
on
the
rep
etition
of
this
game
100
times.
The
frequency
of
o
ccurrence
of
the
ev
en
t
ˇ
at
least
one
red
ball
w
as
obtained
ı
ı
w
as
0.55
.
Should
this
study
b
e
rejected
with
a
risk
of
5
%
?
5.
Unclassifie
d
financial
ye
ars
E.7114
Appro
ximate
results
should
b
e
rounded
to
the
thousandth.
A
compan
y
mass-pro
duces
USB
flash
driv
es
for
the
computer
industry
.
A
random
sample
of
100
k
eys
is
tak
en
from
the
da
y’s
pro-
duction
for
c
hec
king.
The
pro
duction
is
large
enough
for
this
sampling
to
b
e
considered
a
random
dra
w
of
100
k
eys.
The
probabilit
y
that
a
k
ey
U
S
B
tak
en
at
random
from
one
da
y’s
pro
duction
is
defectiv
e
is
0.015
.
Consider
the
random
v
ariable
X
whic
h,
to
an
y
sample
th
us
defined,
asso
ciates
the
n
um
b
er
of
defectiv
e
k
eys
in
that
sam-
ple.
1
Justify
that
the
random
v
ariable
X
follo
ws
a
binomial
distribution,
the
parameters
of
whic
h
w
e
will
determine.
2
Calculate
the
probabilities
p
X
=
0
and
p
X
=
1
.
3
Calculate
the
probabilit
y
that,
in
suc
h
a
sample,
at
most
t
w
o
k
eys
are
defectiv
e.
E.124
T
o
get
to
the
bus
stop
at
7
h
30
,
whic
h
tak
es
her
to
sc
ho
ol,
V
alerie
has
a
c
hoice
of
three
routes
:
A
,
B
,
or
C
.
The
probabilit
y
that
she
will
c
ho
ose
route
A
is
1
2
,
and
the
probabilit
y
that
she
will
c
ho
ose
route
B
is
1
3
.
The
probabilit
y
that
she
misses
the
bus
that
passes
at
7
h
30
,
kno
wing
that
she
has
c
hosen
route
A
,
is
3
10
,
and
the
probabil-
it
y
that
she
misses
the
bus
that
passes
at
7
h
30
,
kno
wing
that
she
has
c
hosen
route
B
is
2
5
,
the
probabilit
y
that
she
misses
the
bus
that
passes
at
7
h
30
,
kno
wing
that
she
has
c
hosen
route
C
,
is
1
2
.
1
In
this
section,
w
e
will
giv
e
the
results
in
the
form
of
irreducible
fractions.
a
Calculate
the
probabilit
y
that
V
alerie
c
ho
oses
route
C
.
b
Construct
a
probabilit
y
tree
sho
wing
the
v
arious
p
os-
sibilities.
Rep
ort
the
probabilities
giv
en
in
the
state-
men
t.
c
Calculate
the
probabilit
y
that
V
alerie
tak
es
the
bus
that
passes
at
7
h
30
and
that
she
has
c
hosen
route
B
.
d
Sho
w
that
the
probabilit
y
that
V
alerie
tak
es
the
bus
that
passes
at
7
h
30
is
19
30
.
e
Kno
wing
that
V
alerie
to
ok
the
bus
that
passes
at
7
h
30
,
calculate
the
probabilit
y
that
she
c
hose
route
C
h
ttps:/
/c
hingmath.fr
sacados/141
France - Juin 2001 - 4 points
chapExoCorrec/7861
sacados/7861
chapExoCorrec/7114
sacados/7114
chapExoCorrec/124
sacados/124
2
In
this
section,
results
will
b
e
rounded
to
the
nearest
10
−
3
.
V
alerie
tries
to
tak
e
the
bus
that
lea
v
es
at
7
h
30
four
da
ys
a
w
eek
under
the
same
conditions.
W
e
assume
that
for
V
alerie,
catc
hing
or
missing
the
bus
that
passes
at
7
h
30
on
a
giv
en
da
y
of
the
w
eek
is
indep
en-
den
t
of
catc
hing
or
missing
the
bus
that
passes
at
7
h
30
on
another
da
y
of
the
w
eek.
a
Calculate
the
probabilit
y
that
V
alerie
will
tak
e
the
bus
that
passes
at
7
h
30
four
times
during
the
w
eek.
b
Calculate
the
probabilit
y
that
she
will
tak
e
the
bus
that
passes
at
7
h
30
exactly
t
wice
during
the
w
eek.
Reminder
:
the
probabilit
y
of
E
giv
en
F
is
giv
en
b
y
:
P
F
(
E
)
=
P
(
E
∩
F
)
P
(
F
)
E.132
The
results
of
a
surv
ey
of
a
p
opu-
lation
of
whic
h
52%
of
the
p
eople
are
w
omen
and
48%
men,
sho
w
that
80%
of
w
omen
and
70%
of
men
pla
y
the
Lotto
at
least
once
a
mon
th.
1
One
individual
from
this
p
opulation
is
c
hosen
at
random.
All
c
hoices
are
equiprobable.
Note
:
H
:
the
ev
en
t
ˇ
The
chosen
individual
is
a
man.
ı
H
the
opp
osite
ev
en
t
of
H
,
i.e.
ˇ
The
chosen
individual
is
a
woman.
ı
L
the
ev
en
t
ˇ
The
individual
plays
the
L
otto
at
le
ast
onc
e
a
month.
ı
L
the
opp
osite
ev
en
t
of
L
,
i.e.
ˇ
The
individual
plays
the
L
otto
less
than
onc
e
a
month.
ı.
P
H
(
L
)
the
conditional
probabilit
y
of
the
ev
en
t
L
with
resp
ect
to
the
ev
en
t
H
.
A
pr
ob
ability
tr
e
e
c
an
b
e
r
epr
esente
d.
a
Calculate
the
probabilit
y
of
the
ev
en
t
H
∩
L
and
then
that
of
the
ev
en
t
H
∩
L
.
b
Sho
w
that
the
probabilit
y
of
L
is
equal
to
0.752.
c
Determine
P
L
(
H
)
,
probabilit
y
that
the
c
hosen
individ-
ual
is
a
man
kno
wing
that
he
pla
ys
Lotto
at
least
once
a
mon
th.
Giv
e
the
result
rounded
to
10
−
4
2
This
p
opulation
b
eing
sufficien
tly
large,
w
e
rep
eat
four
times,
indep
enden
tly
,
under
iden
tical
conditions
(or
that
c
an
b
e
c
onsider
e
d
as
such)
,
the
exp
erimen
t
in
the
first
question
ˇ
R
andomly
sele
ct
an
individual
fr
om
this
p
opu-
lation
ı.
a
Determine
the
probabilit
y
that
one
and
only
one
of
the
four
c
hosen
individuals
pla
ys
Lotto
at
least
once
a
mon
th,
with
the
others
pla
ying
less
than
once
a
mon
th.
Giv
e
the
result
rounded
to
10
−
4
.
b
Determine
the
probabilit
y
that
at
least
one
of
the
four
selected
individuals
pla
ys
Lotto
at
least
once
a
mon
th.
Giv
e
the
result
rounded
to
10
−
4
E.128
A
l
l
r
esults
must
b
e
given
as
irr
e-
ducible
fr
actions.
W
e
ha
v
e
a
c
hec
k
erb
oard
in
whic
h
eac
h
of
the
nine
squares
is
mark
ed
with
one
of
the
three
n
um
b
ers
1,
2,
or
3,
as
sho
wn
in
the
diagram
opp
osite
:
Three
indistinguishable
pa
wns
are
randomly
distributed
on
the
c
hec
k
erb
oard
(one
p
awn
p
er
squar
e)
and
w
e
call
S
the
sum
of
the
three
n
um
b
ers
mark
ed
on
the
three
squares
o
ccu-
pied
b
y
the
pa
wns.
All
distributions
are
equally
probable.
1
W
rite
P
ascal’s
triangle
up
to
the
ten
th
ro
w
and
deduce
9
3
2
Consider
the
follo
wing
ev
en
ts
E
,
F
,
and
G
:
E
:
ˇ
The
sum
S
is
e
qual
to
3
ı
;
F
:
ˇ
The
sum
S
is
e
qual
to
9
ı
;
G
:
ˇ
The
sum
S
is
e
qual
to
6
ı
a
Determine
the
probabilities
P
(
E
)
and
P
(
F
)
of
ev
en
ts
E
and
F
.
b
Sho
w
that
the
probabilit
y
of
ev
en
t
G
is
equal
to
1
3
.
3
Let
A
b
e
the
ev
en
t
ˇ
The
sum
is
divisible
by
3
ı
and
B
the
ev
en
t
ˇ
The
thr
e
e
p
awns
ar
e
aligne
d
in
a
c
olumn,
r
ow,
or
diagonal
ı.
a
Determine
the
probabilit
y
P
(
A
)
and
P
(
B
)
ev
en
ts
A
and
B
.
b
Calculate
the
probabilit
y
P
A
(
B
)
of
ev
en
t
B
giv
en
that
ev
en
t
A
has
o
ccurred.
c
Are
ev
en
ts
A
and
B
indep
enden
t?
h
ttps:/
/c
hingmath.fr
sacados/132
Japon - Juin 2002 - 7 points - obligatoire
sacados/128
Antilles - 2003 - au choix - 6 points
v
ert
.
.
.
.
.
.
.
.
.
rouge
ou
orange
.
.
.
rouge
ou
orange
.
.
.
v
ert
.
.
.
.
.
.
.
.
.
E.142
T
o
hire
staff,
a
compan
y
organizes
selection
tests.
Among
the
candidates
who
tak
e
the
tests,
there
are
60
%
men.
A
statistical
study
sho
ws
that
the
compan
y
hires
70
%
of
the
male
candidates
and
80
%
of
the
female
candidates.
Reminder
:
The
conditional
probabilit
y
of
A
giv
en
B
is
:
P
B
(
A
)
=
P
(
A
∩
B
)
P
(
B
)
P
art
A
A
t
the
end
of
the
tests,
one
p
erson
is
randomly
selected
from
among
all
the
candidates
and
in
terview
ed.
W
e
note
:
H
the
ev
en
t
ˇ
the
p
erson
is
a
man
ı.
F
the
ev
en
t
ˇ
the
p
erson
is
a
woman
ı.
E
the
ev
en
t
ˇ
the
p
erson
is
hir
e
d
ı.
E
the
complemen
tary
ev
en
t
(or
opp
osite)
of
E
.
1
a
What
is
the
probabilit
y
P
(
F
)
that
the
p
erson
in
ter-
view
ed
is
a
w
oman?
b
What
is
the
probabilit
y
that
the
p
erson
in
terview
ed
is
not
emplo
y
ed,
giv
en
that
she
is
a
w
oman?
2
Construct
a
probabilit
y
tree
illustrating
this
situation.
3
Calculate
the
probabilit
y
P
(
E
∩
F
)
that
the
resp
onden
t
is
a
w
oman
and
is
not
emplo
y
ed.
4
Sho
w
that
:
P
(
E
)
=
0
;
26
P
art
B
In
this
part,
the
results
will
b
e
rounded
to
the
nearest
thou-
sandth.
A
t
the
end
of
the
tests,
four
p
eople
are
in
terview
ed
at
random.
W
e
will
consider
that
these
four
c
hoices
are
indep
enden
t
in
pairs.
1
What
is
the
probabilit
y
that
none
of
the
four
p
eople
will
b
e
hired?
2
What
is
the
probabilit
y
that
at
least
one
of
the
four
p
eo-
ple
will
not
b
e
hired?
3
What
is
the
probabilit
y
that
exactly
t
w
o
p
eople
will
b
e
hired?
E.3726
Amélie
has
to
cross
the
main
street
of
a
village,
whic
h
is
lined
with
t
w
o
traffic
ligh
ts.
T
o
n
∈
1
;
2
,
w
e
note
E
n
the
ev
en
t
ˇ
Amélie
is
stopp
ed
b
y
the
n
e
red
ligh
t
or
orange
ı
and
E
n
the
opp
osite
ev
en
t.
The
orange
ligh
t
is
considered
a
red
ligh
t.
Let
p
n
b
e
the
probabilit
y
of
E
n
and
q
n
that
of
E
n
.
The
probabilit
y
of
the
first
traffic
ligh
t
b
eing
red
or
am
b
er
is
1
8
.
It
is
assumed
that
the
follo
wing
t
w
o
conditions
are
met
:
The
probabilit
y
of
the
second
traffic
ligh
t
b
eing
red
or
am
b
er,
if
the
first
traffic
ligh
t
is
red,
is
1
20
.
The
probabilit
y
that
the
second
traffic
ligh
t
is
red
or
or-
ange,
if
the
first
ligh
t
is
green,
is
equal
to
9
20
.
W
e’re
in
terested,
first
of
all,
in
the
first
traffic
ligh
ts.
1
Cop
y
and
complete
the
w
eigh
ted
tree
b
elo
w.
2
Note
X
the
random
v
ariable
equal
to
the
n
um
b
er
of
green
ligh
ts
among
these
t
w
o
traffic
ligh
ts.
Determine
the
prob-
abilit
y
distribution
of
X
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/142
sacados/142
Liban - Juin 2005 - 7 points
chapExoCorrec/3726
sacados/3726
v
ert
.
.
.
.
.
.
.
.
.
rouge
ou
orange
.
.
.
rouge
ou
orange
.
.
.
v
ert
.
.
.
.
.
.
.
.
.
ChingMath
by ChingLab
©
Middle School
Grade 6
Grade 7
Grade 8
Grade 9
High School
Grade 10
Grade 11
Grade 12
Grade 12 opt. Complémentaire
Grade 12 opt. Experte
Various
Contact me
F.A.Q
Other website
ChingWeb