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Grade 12 - Exp.
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4
3
2
1
Grade 12 - Comp.
/ Discrete laws
60 exercises (including 59 corrected)
a
Reminders: probabilities
(3 exercices)
Reminders: probability law
(1 exercice)
Reminders: binomial distribution
(4 exercices)
Non-symmetrical tree
(2 exercices)
Conditional probability and binomial distribution
(5 exercices)
Independent repetition of identical experiments
(2 exercices)
Independent repetition of identical experiments and random variables
(2 exercices)
Expectations, variances and independent repetition of identical experiments
(3 exercices)
Independent succession of random experiments
(5 exercices)
Random variables and repetitions of independent experiments
(2 exercices)
Random variables, expectations, standard deviations and repetitions
(5 exercices)
Random variables, expectations, standard deviations and conditional probabilities
(5 exercices)
Bernoulli test
(5 exercices)
Binomial law
(16 exercices)
image PNG : /home/_math/_exercice/d7/7168/metapost/casio.png
1.
R
eminders
:
pr
ob
abilities
E.7117
The
table
b
elo
w
pro
vides
an
o
v
erview
of
daily
newspap
er
readership
in
F
rance,
based
on
a
sample
of
800
p
eople
surv
ey
ed
in
2005
.
Ev
ery
da
y
or
almost
ev
ery
da
y
Once
or
t
wice
a
w
eek
Only-
during
certain
p
erio
ds
Rarely
Nev
er
T
otal
F
armers
op
erators
1
10
2
8
79
100
Craftsmen,
shopk
eep
ers
,
business
leaders
11
11
5
7
66
100
Managers
17
16
10
18
39
100
In
termediate
professions
Media
8
15
7
15
55
100
Emplo
y
ees
6
7
4
9
74
100
W
ork
ers
(including
agricultural
w
ork
ers)
4
5
3
5
83
100
Retired
6
7
2
6
79
100
Others
inactiv
e
5
9
4
9
73
100
T
otal
in
w
orkforce
58
80
37
77
548
800
P
ercen
tages
of
total
7.25
%
10
%
4.625
%
In
this
exercise,
the
results
will
b
e
giv
en
in
decimal
form
and
rounded
to
the
nearest
0.001
.
P
art
A.
1
The
last
ro
w
of
the
table
b
elo
w
sho
ws
the
share
of
eac
h
category
in
relation
to
the
total
sample.
Calculate
the
missing
v
alues
in
this
last
ro
w.
2
Giv
e
the
probabilit
y
that
a
p
erson
c
hosen
at
random
from
among
the
executiv
es
nev
er
reads.
P
art
B.
A
p
erson
is
c
hosen
at
random
from
this
sample
of
800
p
eople.
In
this
section,
w
e
note
the
follo
wing
ev
en
ts
:
J
Ev
en
t
:
ˇ
The
p
erson
c
hosen
nev
er
reads
ı
;
O
Ev
en
t
:
ˇ
The
p
erson
c
hosen
is
a
lab
orer
ı.
1
Calculate
the
probabilities
of
ev
en
ts
J
and
O
2
Calculate
the
probabilit
y
of
ev
en
t
J
∩
O
.
3
Calculate
the
probabilit
y
of
ev
en
t
J
∪
O
E.7118
Let
Ω
;
P
b
e
a
probabilized
space
where
A
and
B
are
t
w
o
ev
en
ts
of
Ω
suc
h
that
:
P
(
A
)
=
0.36
;
P
(
B
)
=
0.27
1
What
can
w
e
sa
y
ab
out
A
and
B
if
P
A
∪
B
=
0.63
?
2
Assume
:
P
A
∪
B
=
0.5
:
a
What
can
w
e
sa
y
ab
out
A
and
B
?
b
What
is
the
probabilit
y
of
an
ev
en
t
sim
ultaneously
re-
alizing
the
ev
en
ts
A
and
B
?
E.7119
The
w
orks
council
of
a
P
arisian
compan
y
w
an
ts
to
organize
a
w
eek
end
trip
to
the
pro
vinces.
A
surv
ey
is
conducted
among
the
1
200
emplo
y
ees
of
this
com-
pan
y
to
find
out
their
c
hoice
of
transp
ortation.
(The
only
me
ans
of
tr
ansp
ortation
offer
e
d
ar
e
tr
ain,
plane,
or
bus.)
.
The
results
of
the
surv
ey
of
compan
y
emplo
y
ees
are
listed
in
the
follo
wing
table
:
T
rain
Plane
Coac
h
T
otal
W
omen
468
196
56
720
Men
150
266
64
480
T
otal
618
462
120
1200
An
emplo
y
ee
of
this
compan
y
is
in
terview
ed
at
random
(it
is
assume
d
that
al
l
employe
es
have
the
same
chanc
e
of
b
eing
interviewe
d)
.
F
the
ev
en
t
:
ˇ
the
emplo
y
ee
is
a
w
oman
ı
;
T
the
ev
en
t
:
ˇ
the
emplo
y
ee
c
ho
oses
the
train
ı.
1
Calculate
the
probabilities
P
(
F
)
,
P
(
T
)
then
determine
the
probabilit
y
that
the
emplo
y
ee
will
not
c
ho
ose
the
train
(the
r
esults
wil
l
b
e
given
in
de
cimal
form)
2
a
Determine
the
probabilit
y
of
the
ev
en
t
F
∩
T
.
b
Deduce
the
probabilit
y
of
the
ev
en
t
F
∪
T
.
3
By
c
ho
osing
an
emplo
y
ee
at
random
from
among
the
em-
plo
y
ees
who
did
not
c
ho
ose
the
train,
what
is
the
proba-
bilit
y
that
this
emplo
y
ee
is
a
w
oman?
(The
r
esult
wil
l
b
e
r
ounde
d
to
thr
e
e
de
cimal
plac
es)
2.
R
eminders
:
pr
ob
ability
law
E.7120
Let
X
b
e
a
random
v
ariable
whose
prob-
abilit
y
distribution
is
giv
en
b
elo
w
:
x
i
1
2
3
4
5
6
P
X
=
x
i
0
;
05
0.12
0.15
0.23
0.17
a
1
Determine
the
v
alue
of
a
so
that
the
table
b
elo
w
repre-
sen
ts
a
probabilit
y
distribution.
2
Determine
the
follo
wing
probabilities
:
a
P
X
3
b
P
X
<
5
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7117
sacados/7117
chapExoCorrec/7118
sacados/7118
chapExoCorrec/7119
sacados/7119
chapExoCorrec/7120
sacados/7120
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
V
3
F
3
V
2
V
3
F
3
F
2
V
1
V
3
F
3
V
2
V
3
F
3
F
2
F
1
3.
R
eminders
:
binomial
distribution
E.7155
Here
are
the
c
hoice
trees
asso
ciated
with
rep
eating
a
Bernoulli
test
3
and
4
times
resp
ectiv
ely
:
1
F
or
the
threefold
rep
etition
of
Bernoulli’s
test,
complete
the
table
b
elo
w
:
Num
b
er
of
succès
0
1
2
3
Num
b
er
of
issues
2
F
or
the
four-fold
rep
etition
of
Bernoulli’s
test,
complete
the
table
b
elo
w
:
Num
b
er
of
succès
0
1
2
3
4
Num
b
er
of
issues
E.7121
A
MCQ
(multiple-choic
e
questionnair
e)
is
giv
en
to
studen
ts
:
it
con
tains
three
questions,
and
for
eac
h
question,
four
answ
ers
are
pro
vided,
only
one
of
whic
h
is
cor-
rect.
W
e
w
an
t
to
study
the
success
rate
for
this
m
ultiple-c
hoice
questionnaire
if
studen
ts
answ
er
completely
at
random
;
w
e
therefore
assume
that
the
answ
ers
giv
en
to
eac
h
question
are
indep
enden
t
of
eac
h
other.
W
e
note
:
F
i
:
ˇ
The
answ
er
giv
en
to
question
i
is
false
ı
;
V
i
:
ˇ
The
answ
er
giv
en
to
question
i
is
true
ı
;
1
Complete
the
w
eigh
ted
tree
b
elo
w
:
2
Let
X
b
e
the
random
v
ariable
that
asso
ciates
eac
h
ele-
men
tary
ev
en
t
with
the
n
um
b
er
of
correct
answ
ers
ob-
tained
in
the
m
ultiple-c
hoice
test.
a
Justify
that
the
random
v
ariable
X
follo
ws
a
binomial
distribution
with
parameters
3
and
0.25
.
b
In
order
to
obtain
the
probabilit
y
distribution
of
the
random
v
ariable
X
,
complete
the
table
b
elo
w
:
k
0
1
2
3
P
X
=
k
c
Calculate
the
exp
ected
v
alue
of
the
random
v
ariable
X
.
E.7168
Reminder
:
for
a
random
v
ariable
X
follo
wing
a
binomial
distribution
with
parameters
n
and
p
,
the
probabilit
y
that
the
random
v
ariable
tak
es
the
v
alue
k
is
:
P
X
=
k
=
n
k
·
p
k
·
1
−
p
n
−
k
Note
:
the
binomial
co
efficien
t
n
k
allo
ws
us
to
kno
w
the
n
um
b
er
of
paths
in
the
decision
tree
result-
ing
in
k
successes
for
n
rep
etitions.
Ab
o
v
e
is
a
screenshot
of
a
calculator
for
calculating
the
bi-
nomial
co
efficien
t
5
3
.
1
Using
the
calculator,
determine
the
binomial
co
efficien
ts
b
elo
w
:
a
5
3
b
4
0
c
4
2
d
7
5
2
Consider
the
random
v
ariable
X
follo
wing
the
binomial
distribution
with
parameter
B
12
;
0
;
3
.
Determine
the
follo
wing
probabilities
rounded
to
10
−
4
:
a
P
X
=
3
a
P
X
=
7
3
Consider
the
random
v
ariable
X
follo
wing
the
binomial
distribution
with
parameter
B
8
;
0
;
4
.
Determine
the
follo
wing
probabilities
using
the
complemen
tary
distri-
bution,
rounded
to
10
−
4
:
a
P
X
1
a
P
X
7
E.7169
Results
are
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
dra
wn
suc-
cessiv
ely
,
at
random
and
indep
enden
tly
of
eac
h
other.
The
random
v
ariable
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?
W
e’ll
round
the
v
alue
to
the
nearest
10
−
4
.
4.
Non-symmetric
al
tr
e
e
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7155
sacados/7155
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
chapExoCorrec/7121
sacados/7121
V
3
F
3
V
2
V
3
F
3
F
2
V
1
V
3
F
3
V
2
V
3
F
3
F
2
F
1
chapExoCorrec/7168
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chapExoCorrec/7169
sacados/7169
:
:
:
:::
A
G
:
:
:
:
:
:
A
:
:
:
A
G
A
L
E
0
;
55
0
;
37
A
L
G
E
E.7133
A
compan
y
w
as
in
terested
in
the
probabilit
y
of
one
of
its
emplo
y
ees,
c
hosen
at
random,
b
eing
absen
t
during
a
giv
en
w
eek
of
win
ter
2014
.
The
probabilit
y
of
an
emplo
y
ee
ha
ving
the
flu
in
a
giv
en
w
eek
w
as
estimated
to
b
e
0.07
.
If
the
emplo
y
ee
has
the
flu,
then
he
or
she
is
absen
t.
If
the
emplo
y
ee
do
es
not
ha
v
e
the
flu
that
w
eek,
the
probabil-
it
y
that
he
or
she
will
b
e
absen
t
is
estimated
at
0.04
.
An
emplo
y
ee
of
the
compan
y
is
c
hosen
at
random
and
the
follo
wing
ev
en
ts
are
considered
:
G
:
the
emplo
y
ee
has
the
flu
in
a
giv
en
w
eek
;
A
:
the
emplo
y
ee
is
absen
t
in
a
giv
en
w
eek.
1
Repro
duce
and
complete
the
tree,
indicating
the
proba-
bilities
of
eac
h
branc
h.
2
Sho
w
that
the
probabilit
y
p
(
A
)
of
the
ev
en
t
A
is
equal
to
0.1072
.
E.7216
In
Jan
uary
2015
,
the
director
of
a
con
temp
orary
art
m
useum
commissions
a
surv
ey
on
visitor
habits.
The
m
useum
has
a
w
ebsite.
T
o
purc
hase
a
tic
k
et,
in
terested
individuals
can
go
to
the
m
useum
tic
k
et
office
or
order
a
tic
k
et
online.
Three
t
yp
es
of
visits
are
a
v
ailable
:
Individual
visits
without
an
audio
guide.
Individual
visits
with
an
audio
guide.
Group
visits
for
at
least
10
p
eople.
In
this
case,
a
single
tic
k
et
is
issued
for
the
group.
The
w
ebsite
only
allo
ws
y
ou
to
purc
hase
individual
tic
k
ets
with
or
without
an
audio
guide.
F
or
group
visits,
y
ou
m
ust
go
to
the
m
useum
tic
k
et
office.
Ov
er
the
course
of
the
y
ear
2015
,
the
surv
ey
rev
ealed
that
:
55
%
of
admission
tic
k
ets
w
ere
purc
hased
at
the
m
useum
tic
k
et
office
;
of
the
tic
k
ets
purc
hased
at
the
m
useum
tic
k
et
office,
51
%
w
ere
for
individual
visits
without
audio
guide
ren
tal,
and
37
%
w
ere
for
visits
with
audio
guide
ren
tal
;
70
%
of
the
tic
k
ets
purc
hased
online
corresp
ond
to
indi-
vidual
visits
without
audio
guide
ren
tal.
W
e
randomly
select
a
m
useum
tic
k
et
purc
hased
on
2015
W
e
consider
the
follo
wing
ev
en
ts
:
E
:
ˇ
the
tic
k
et
w
as
purc
hased
online
ı
;
A
:
ˇ
the
tic
k
et
is
for
an
individual
visit
with
audio
guide
ren
tal
ı
;
L
:
ˇ
the
tic
k
et
corresp
onds
to
an
individual
visit
without
audio
guide
ren
tal
ı
;
G
:
ˇ
the
tic
k
et
corresp
onds
to
a
group
visit
ı
Remem
b
er
that
if
E
and
F
are
t
w
o
ev
en
ts,
p
(
E
)
denotes
the
probabilit
y
of
ev
en
t
E
and
p
F
(
E
)
denotes
the
probabilit
y
of
ev
en
t
E
giv
en
that
ev
en
t
F
has
o
ccurred.
W
e
denote
E
as
the
opp
osite
ev
en
t
of
E
.
1
Cop
y
and
complete
the
follo
wing
w
eigh
ted
tree
represen
t-
ing
the
situation
describ
ed
in
the
statemen
t
:
2
Sho
w
that
the
probabilit
y
that
the
tic
k
et
w
as
purc
hased
online
and
corresp
onds
to
an
individual
visit
with
audio
guide
ren
tal
is
equal
to
0.135
.
3
Sho
w
that
:
p
(
A
)
=
0.3385
4
The
tic
k
et
c
hosen
corresp
onds
to
an
individual
visit
with
audio
guide
ren
tal.
What
is
the
probabilit
y
that
this
tic
k
et
w
as
purc
hased
at
the
m
useum
tic
k
et
office?
The
result
will
b
e
rounded
to
the
nearest
thousandth
5.
Conditional
pr
ob
ability
and
binomial
distribution
E.7136
A
large
compan
y
has
just
com-
pleted
its
recruitmen
t
campaign,
whic
h
to
ok
place
in
t
w
o
stages
:
first
stage
:
review
of
the
candidate’s
application
;
second
stage
:
in
terview
for
recruitmen
t
purp
oses.
The
recruitmen
t
pro
cess
implemen
ted
b
y
the
compan
y
is
as
follo
ws
:
if
the
application
is
deemed
to
b
e
of
go
o
d
qualit
y
,
the
candidate
is
in
vited
to
an
in
terview
with
the
h
uman
re-
sources
director
;
if
the
application
is
not
deemed
to
b
e
of
go
o
d
qualit
y
,
the
candidate
undergo
es
tests
and
is
then
in
vited
to
an
in
terview
with
the
compan
y
director.
In
b
oth
cases,
at
the
end
of
the
in
terview,
the
candidate
is
either
hired
or
not.
By
c
ho
osing
a
candidate
at
random,
w
e
construct
a
random
exp
erimen
t
whose
probabilit
y
tree
is
represen
ted
b
elo
w
with
some
of
the
data
retriev
ed
from
the
publication
:
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7133
sacados/7133
:
:
:
:::
A
G
:
:
:
:
:
:
A
:
:
:
A
G
chapExoCorrec/7216
sacados/7216
A
L
E
0
;
55
0
;
37
A
L
G
E
chapExoCorrec/7136
sacados/7136
0
;
3
:
:
:
R
P
D
R
0
;
24
:
:
:
R
D
0
;
7
:
:
:
R
P
D
R
:::
:
:
:
R
D
0
;
95
0
;
7
S
0
;
3
S
R
0
;
05
0
;
2
S
0
;
8
S
R
where
D
is
the
ev
en
t
ˇ
the
candidate
has
a
file
that
is
consid-
ered
to
b
e
of
go
o
d
qualit
y
ı
ı
and
R
is
the
ev
en
t
"
the
candidate
is
hired
b
y
the
compan
y
ı.
1
a
Determine
the
probabilit
y
P
D
R
.
b
F
urthermore,
w
e
kno
w
that
38
%
candidates
w
ere
hired.
Deduce
the
probabilit
y
P
D
R
.
c
Complete
the
probabilit
y
tree
2
T
en
p
eople
apply
for
a
job
at
the
compan
y
.
Their
ap-
plications
are
review
ed
indep
enden
tly
of
eac
h
other.
Let
X
b
e
the
random
v
ariable
giving
the
n
um
b
er
of
p
eople
recruited
from
among
the
10
applican
ts.
a
Justify
that
X
follo
ws
a
binomial
distribution
with
pa-
rameters
n
=
10
and
p
=
0.38
.
b
Calculate
the
probabilit
y
that
at
least
one
of
the
ten
p
eople
will
b
e
hired.
Giv
e
the
exact
v
alue,
then
a
v
alue
rounded
to
10
−
3
.
E.7139
A
surv
ey
w
as
conducted
among
studen
ts
enrolled
in
a
high
sc
ho
ol
lunc
h
program.
The
results
w
ere
used
to
construct
the
probabilit
y
tree
b
elo
w
:
A
studen
t
is
c
hosen
at
random
from
among
those
enrolled
in
the
lunc
h
program.
W
e
note
the
follo
wing
ev
en
ts
:
R
the
ev
en
t
:
ˇ
the
studen
t
regularly
eats
in
the
cafete-
ria
ı
;
S
the
ev
en
t
:
ˇ
the
studen
t
is
satisfied
ı.
W
e
note
R
and
S
as
the
opp
osite
ev
en
ts
of
R
and
S
.
1
a
Sho
w
that
the
probabilit
y
of
ev
en
t
S
is
equal
to
0.675
.
b
Kno
wing
that
the
studen
t
is
not
satisfied
with
the
qual-
it
y
of
the
meals,
calculate
the
probabilit
y
that
he
eats
regularly
in
the
cafeteria.
Giv
e
the
result
rounded
to
10
−
3
.
2
F
our
studen
ts
are
randomly
selected
from
among
those
enrolled
in
the
half-b
oard
program
and
questioned
suc-
cessiv
ely
and
indep
enden
tly
.
Let
X
b
e
the
random
v
ariable
equal
to
the
n
um
b
er
of
studen
ts
who
sa
y
they
are
satisfied
with
the
qualit
y
of
the
meals.
Since
the
n
um
b
er
of
studen
ts
is
large
enough,
w
e
consider
that
X
follo
ws
a
binomial
distribution.
The
results
will
b
e
rounded
to
three
decimal
places.
a
Sp
ecify
the
parameters
of
this
binomial
distribution
b
Calculate
the
probabilit
y
of
ev
en
t
A
:
ˇ
all
four
studen
ts
are
satisfied
with
the
qualit
y
of
the
meals
ı.
c
Describ
e
ev
en
t
A
in
one
sen
tence
and
calculate
its
prob-
abilit
y
.
h
ttps:/
/c
hingmath.fr
0
;
3
:
:
:
R
P
D
R
0
;
24
:
:
:
R
D
0
;
7
:
:
:
R
P
D
R
:::
:
:
:
R
D
chapExoCorrec/7139
sacados/7139
0
;
95
0
;
7
S
0
;
3
S
R
0
;
05
0
;
2
S
0
;
8
S
R
0
;
65
0
;
3
E
0
;
7
E
H
0
;
35
0
;
6
E
0
;
4
E
F
E.7217
A
mobile
phone
op
erator
is
orga-
nizing
a
telephone
mark
eting
campaign
to
offer
its
customers
a
new
subscription
plan.
Some
of
the
p
eople
con
tacted
listen
to
the
explanation,
while
others
hang
up
immediately
(or
say
they
ar
e
not
inter
este
d)
.
A
p
erson
is
c
hosen
at
random
from
the
customer
database.
Eac
h
p
erson
has
the
same
probabilit
y
of
b
eing
c
hosen.
W
e
denote
H
the
ev
en
t
ˇ
the
p
erson
c
hosen
is
a
man
ı,
F
the
ev
en
t
ˇ
the
p
erson
c
hosen
is
a
w
oman
ı,
E
the
ev
en
t
"
the
p
erson
c
hosen
listens
to
the
salesp
erson’s
explanations
ı
and
E
the
opp
osite
ev
en
t
of
E
.
Reminder
of
notation
:
If
A
and
B
are
t
w
o
giv
en
ev
en
ts,
P
(
A
)
denotes
the
probabil-
it
y
that
ev
en
t
A
will
o
ccur
and
P
B
A
denotes
the
probabil-
it
y
of
ev
en
t
A
o
ccurring,
giv
en
that
ev
en
t
B
has
o
ccurred.
P
art
A
The
study
has
established
the
follo
wing
probabilit
y
tree
:
F
or
eac
h
statemen
t,
only
one
of
the
answ
ers
is
correct.
Cop
y
the
n
um
b
er
of
the
answ
er
and
the
v
alue
c
hosen
on
to
y
our
answ
er
sheet.
No
justification
is
required
1
The
probabilit
y
that
the
p
erson
c
hosen
will
listen
to
the
salesp
erson’s
explanations
is
equal
to
:
a
0
;
195
b
0
;
21
c
0
;
405
d
0
;
595
2
The
salesp
erson
addresses
a
p
erson
who
listens
to
them.
The
probabilit
y
,
rounded
to
t
w
o
decimal
places,
that
this
p
erson
is
a
man
is
equal
to
:
a
0
;
48
b
0
;
52
c
0
;
76
d
0
;
24
P
art
B
The
surv
eys
conducted
during
these
first
few
da
ys
also
sho
w
that
12
%
of
those
in
terview
ed
subscrib
e
to
this
new
pac
k
age.
Eac
h
emplo
y
ee
of
the
op
erator
mak
es
60
calls
p
er
da
y
.
W
e
assume
that
the
file
is
large
enough
for
the
c
hoices
to
b
e
considered
indep
enden
t
and
made
under
iden
tical
conditions.
Let
X
b
e
the
random
v
ariable
that
coun
ts
the
n
um
b
er
of
sub-
scriptions
made
b
y
a
giv
en
emplo
y
ee
on
a
giv
en
da
y
.
1
Justify
that
the
random
v
ariable
X
follo
ws
a
binomial
distribution,
the
parameters
of
whic
h
will
b
e
giv
en.
2
Determine
the
probabilit
y
that
the
emplo
y
ee
will
obtain
5
subscriptions
on
a
giv
en
da
y
.
(The
r
esult
wil
l
b
e
r
ounde
d
to
two
de
cimal
plac
es.)
3
Determine
the
probabilit
y
that
the
emplo
y
ee
will
obtain
at
least
one
subscription
on
a
giv
en
da
y
.
(The
value
wil
l
b
e
r
ounde
d
to
the
ne
ar
est
ten-thousandth)
.
E.7312
A
surv
ey
w
as
carried
out
among
studen
ts
at
a
high
sc
ho
ol
to
find
out
their
views
on
the
length
of
the
lunc
h
break
and
sc
ho
ol
rh
ythms.
The
surv
ey
rev
ealed
that
56.75
%
of
the
sc
ho
ol’s
studen
ts
w
ere
in
fa
v
or
of
spread-
ing
classes
out
o
v
er
the
sc
ho
ol
y
ear.
F
our
studen
ts
tak
en
at
random
from
among
the
sc
ho
ol’s
pupils
are
in
terview
ed
successiv
ely
and
indep
enden
tly
.
Let
X
b
e
the
random
v
ariable
that
giv
es
the
n
um
b
er
of
studen
ts
in
fa
v
or
of
a
more
staggered
distribution
of
classes
o
v
er
the
sc
ho
ol
y
ear.
As
the
n
um
b
er
of
studen
ts
is
sufficien
tly
large,
X
is
considered
to
follo
w
a
binomial
distribution.
1
Sp
ecify
the
parameters
of
this
binomial
distribution.
Results
will
b
e
rounded
to
the
nearest
10
−
4
.
2
Calculate
the
probabilit
y
that
none
of
the
four
studen
ts
surv
ey
ed
is
in
fa
v
or
of
spreading
classes
out
more
o
v
er
the
sc
ho
ol
y
ear.
3
Calculate
the
probabilit
y
that
exactly
t
w
o
studen
ts
are
in
fa
v
or
of
a
more
spread-out
course
distribution
o
v
er
the
sc
ho
ol
y
ear.
E.7137
A
large
compan
y
has
just
com-
pleted
its
recruitmen
t
campaign,
whic
h
to
ok
place
in
t
w
o
stages
:
first
stage
:
review
of
the
candidate’s
application
;
second
stage
:
in
terview
for
recruitmen
t
purp
oses.
The
recruitmen
t
pro
cess
implemen
ted
b
y
the
compan
y
is
as
follo
ws
:
if
the
application
is
deemed
to
b
e
of
go
o
d
qualit
y
,
the
candidate
is
in
vited
to
an
in
terview
with
the
h
uman
re-
sources
director
;
if
the
application
is
not
deemed
to
b
e
of
go
o
d
qualit
y
,
the
candidate
undergo
es
tests
and
is
then
in
vited
to
an
in
terview
with
the
compan
y
director.
In
b
oth
cases,
at
the
end
of
the
in
terview,
the
candidate
is
either
hired
or
not.
A
t
the
end
of
this
recruitmen
t
campaign,
the
compan
y
pub-
lishes
the
follo
wing
results
:
30
%
of
the
candidates
had
applications
that
w
ere
deemed
to
b
e
of
go
o
d
qualit
y
;
20
%
candidates
whose
applications
w
ere
not
considered
to
b
e
of
go
o
d
qualit
y
w
ere
recruited
;
38
%
candidates
w
ere
recruited.
1
W
e
tak
e
a
random
candidate
and
note
:
D
the
ev
en
t
ˇ
the
candidate
has
a
file
that
is
considered
to
b
e
of
go
o
d
qualit
y
ı
;
R
the
ev
en
t
ˇ
the
candidate
is
hired
b
y
the
compan
y
ı.
a
Represen
t
this
situation
using
a
w
eigh
ted
tree
b
Calculate
the
probabilit
y
that
the
candidate
do
es
not
ha
v
e
a
go
o
d
application
and
is
not
hired
b
y
the
com-
pan
y
.
c
Sho
w
that
the
probabilit
y
of
ev
en
t
D
∩
R
is
equal
to
0.24
.
d
Deduce
the
probabilit
y
that
a
candidate
will
b
e
hired
giv
en
that
their
application
is
considered
to
b
e
of
go
o
d
qualit
y
.
Complete
the
w
eigh
ted
tree
created
in
ques-
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7217
sacados/7217
0
;
65
0
;
3
E
0
;
7
E
H
0
;
35
0
;
6
E
0
;
4
E
F
chapExoCorrec/7312
sacados/7312
chapExoCorrec/7137
sacados/7137
Extrait de Centres Etrangers Juin 2014
Evènements
élémentaires
NN
N
NN
N
NB
B
N
NN
N
NN
N
NB
B
N
BN
N
BN
N
BB
B
B
1
e
r
tirage
2
e
tirage
3
X
V
2
F
2
F
2
F
V
2
V
1
F
1
F
1
F
F
2
V
1
F
1
F
1
F
F
2
V
1
F
1
F
1
F
F
V
2
V
1
F
1
F
1
F
V
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
F
2
V
1
F
1
F
1
F
V
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
F
2
V
1
F
1
F
1
F
V
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
F
tion
a
.
2
T
en
p
eople
apply
for
a
job
at
the
compan
y
.
Their
ap-
plications
are
review
ed
indep
enden
tly
of
eac
h
other.
Let
X
b
e
the
random
v
ariable
giving
the
n
um
b
er
of
p
eople
hired
among
the
10
applican
ts.
a
Justify
that
X
follo
ws
a
binomial
distribution
with
pa-
rameters
n
=
10
and
p
=
0.38
.
b
Calculate
the
probabilit
y
that
at
least
one
of
the
ten
p
eople
will
b
e
hired.
Giv
e
the
exact
v
alue,
then
a
v
alue
rounded
to
10
−
3
.
6.
Indep
endent
r
ep
etition
of
identic
al
exp
eriments
E.7683
Consider
a
random
exp
erimen
t
with
t
w
o
outcomes
:
S
success
and
E
failure.
These
t
w
o
outcomes
are
equiprobable.
W
e
rep
eat
this
exp
erimen
t
4
times
indep
enden
tly
.
The
c
hoice
tree
b
elo
w
is
used
to
describ
e
this
rep
etition
:
1
Ho
w
man
y
differen
t
outcomes
do
es
this
random
exp
eri-
men
t
ha
v
e?
2
Determine
the
probabilit
y
of
obtaining
4
success.
3
a
Ho
w
man
y
outcomes
represen
t
3
success?
b
Determine
the
probabilit
y
of
obtaining
3
success
in
this
random
exp
erimen
t.
E.7423
An
urn
con
tains
t
w
o
blac
k
balls
and
one
white
ball
;
the
game
is
pla
y
ed
with
the
ball
b
eing
re-
turned
:
i.e.,
once
the
ball
has
b
een
dra
wn,
it
is
returned
to
the
urn
b
efore
the
next
dra
w.
Here’s
a
decision
tree
based
on
the
dra
wing
of
t
w
o
balls
:
1
T
aking
in
to
accoun
t
the
order
in
whic
h
the
balls
are
dra
wn,
what
is
the
p
ossible
n
um
b
er
of
differen
t
dra
ws?
2
Determine
the
probabilit
y
of
the
follo
wing
ev
en
ts
:
a
A
:
ˇ
The
first
ball
dra
wn
is
blanc
he
ı.
b
B
:
ˇ
The
t
w
o
balls
dra
wn
are
différen
tes
ı
colors.
c
C
:
ˇ
The
second
ball
is
a
noire
ı
ball.
3
Giv
e
the
probabilities
of
the
follo
wing
ev
en
ts
:
a
A
∩
B
b
B
c
C
7.
Indep
endent
r
ep
etition
of
identic
al
exp
eriments
and
r
andom
variables
E.7386
Consider
a
m
ultiple-c
hoice
question-
naire
consisting
of
3
questions
eac
h
offering
4
answ
ers,
only
one
of
whic
h
is
correct.
A
random
exp
erimen
t
is
created
b
y
asking
participan
ts
to
randomly
answ
er
the
questions
prop
osed
in
the
form.
This
situation
is
represen
ted
b
y
the
c
hoice
tree
b
elo
w
:
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7683
sacados/7683
chapExoCorrec/7423
sacados/7423
Evènements
élémentaires
NN
N
NN
N
NB
B
N
NN
N
NN
N
NB
B
N
BN
N
BN
N
BB
B
B
1
e
r
tirage
2
e
tirage
chapExoCorrec/7386
sacados/7386
3
X
V
2
F
2
F
2
F
V
2
V
1
F
1
F
1
F
F
2
V
1
F
1
F
1
F
F
2
V
1
F
1
F
1
F
F
V
2
V
1
F
1
F
1
F
V
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
F
2
V
1
F
1
F
1
F
V
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
F
2
V
1
F
1
F
1
F
V
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
F
Evènements
élémentaires
N
NNN
N
NNN
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NNB
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NNN
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NBN
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NBN
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NNN
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NNB
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NBN
N
NBN
B
NBB
B
N
N
BNN
N
BNN
B
BNB
N
N
BNN
N
BNN
B
BNB
N
N
BBN
N
BBN
B
BBB
B
B
V
3
F
3
V
2
V
3
F
3
F
2
V
1
V
3
F
3
V
2
V
3
F
3
F
2
F
1
F
or
eac
h
elemen
tary
ev
en
t,
the
random
v
ariable
X
is
asso
ci-
ated
with
the
n
um
b
er
of
correct
answ
ers.
Giv
e
the
probabilit
y
distribution
of
the
random
v
ariable
X
.
E.5169
There
are
three
balls
in
an
urn
:
t
w
o
blac
k
balls
and
one
white
ball.
Three
balls
are
dra
wn
succes-
siv
ely
from
this
urn,
with
replacemen
t.
The
decision
tree
b
elo
w
illustrates
all
the
elemen
tary
ev
en
ts
of
this
random
exp
erimen
t
:
Eac
h
dra
w
is
asso
ciated
with
a
pa
y
off
as
follo
ws
:
0
e
if
no
blac
k
balls
are
dra
wn
;
1
e
if
only
one
blac
k
ball
is
dra
wn
;
2
e
if
t
w
o
blac
k
balls
are
dra
wn
;
5
e
if
all
three
balls
are
blac
k.
W
e
consider
the
random
v
ariable
X
whic
h
asso
ciates
eac
h
dra
w
with
the
corresp
onding
prize.
Complete
the
table
b
elo
w
giving
the
distribution
of
the
ran-
dom
v
ariable
X
:
k
0
1
2
5
P
X
=
k
8.
Exp
e
ctations,
varianc
es
and
indep
endent
r
ep
etition
of
identic
al
exp
eriments
E.5190
A
QCM
(multiple-choic
e
question-
nair
e)
is
prop
osed
to
studen
ts
:
it
com-
prises
three
questions
and
for
eac
h
of
these
questions,
four
answ
ers
are
pro-
p
osed,
of
whic
h
only
one
is
correct.
W
e
wish
to
study
the
p
ercen
tage
of
success
in
this
MCQ
if
the
studen
ts
answ
er
it
completely
at
random
;
w
e
then
assume
that
the
answ
ers
giv
en
to
eac
h
of
the
questions
are
indep
enden
t
of
eac
h
other.
W
e
note
:
F
i
:
ˇ
The
answ
er
pro
vided
to
question
i
is
fausse
ı
;
V
i
:
ˇ
The
answ
er
pro
vided
to
the
question
i
is
vraie
ı
;
1
Complete
the
w
eigh
ted
tree
sho
wn
ab
o
v
e.
2
W
e
note
X
the
random
v
ariable
coun
ting
the
n
um
b
er
of
correct
answ
ers
pro
vided
to
the
MCQ.
a
Determine
the
probabilit
y
distribution
of
the
random
v
ariable
X
.
b
Calculate
the
exp
ectation
of
the
random
v
ariable
X
.
E.5913
A
compan
y
pro
duces
electronic
comp
onen
ts
b
efore
êb
eing
as-
signed
to
sales.
Eac
h
comp
onen
t
undergo
es
t
w
o
indep
enden
t
qualit
y
tests
:
5
%
of
the
comp
onen
ts
fail
the
first
test
;
3
%
of
the
comp
onen
ts
fail
the
second
test.
A
t
the
end
of
the
pro
duction
line,
an
electronic
comp
onen
t
is
c
hosen
at
random.î
1
Determine
the
probabilit
y
that
the
electronic
comp
onen
t
failed
b
oth
tests.
0.0012
0.0013
0.0014
0.0015
2
Determine
the
probabilit
y
that
the
electronic
comp
onen
t
failed
only
one
of
the
t
w
o
tests.
0.0666
0.077
0.0776
0.078
3
The
coût
of
pro
duction
is
calculated
as
follo
ws
:
eac
h
comp
onen
t
coûte
100
e
and
for
eac
h
failed
test,
the
re-
pair
coûte
10
e
.
Let
X
b
e
the
random
v
ariable
that
asso
ciates
eac
h
ran-
domly
selected
comp
onen
t
with
its
pro
duction
cost
coût.
What
is
the
exp
ected
v
alue
of
the
random
v
ariable
X
?
100.8
101.8
102.8
103.8
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5169
sacados/5169
Evènements
élémentaires
N
NNN
N
NNN
B
NNB
N
N
NNN
N
NNN
B
NNB
N
N
NBN
N
NBN
B
NBB
B
N
N
NNN
N
NNN
B
NNB
N
N
NNN
N
NNN
B
NNB
N
N
NBN
N
NBN
B
NBB
B
N
N
BNN
N
BNN
B
BNB
N
N
BNN
N
BNN
B
BNB
N
N
BBN
N
BBN
B
BBB
B
B
chapExoCorrec/5190
sacados/5190
V
3
F
3
V
2
V
3
F
3
F
2
V
1
V
3
F
3
V
2
V
3
F
3
F
2
F
1
chapExoCorrec/5913
sacados/5913
Urne A
Urne B
1
2
3
P
(
C
P
(
P
P
P
(
P
P
C
P
(
C
P
(
P
P
P
(
P
P
C
:
:
:
:
:
:
R
:
:
:
V
B
:
:
:
:
:
:
R
:
:
:
V
N
E.5914
A
compan
y
man
ufactures
electronic
comp
onen
ts.
During
pro-
duction,
eac
h
comp
onen
t
undergo
es
t
w
o
qualit
y
tests.
If,
during
one
of
the
tests,ôthe
test
is
negativ
e,
the
comp
onen
t
is
repaired
and
returned
to
the
pro
duction
line.
6
%
of
comp
onen
ts
fail
the
first
test.
;
2
%
of
comp
onen
ts
fail
the
second
test.
A
t
the
end
of
the
pro
duction
line,
an
electronic
comp
onen
t
is
c
hosen
at
random.î
1
Determine
the
probabilit
y
that
the
electronic
comp
onen
t
failed
b
oth
tests.
0.0012
0.0013
0.0014
0.0015
2
Determine
the
probabilit
y
that
the
electronic
comp
onen
t
failed
only
one
of
the
t
w
o
tests.
0.0666
0.077
0.0776
0.078
3
The
coût
of
pro
duction
is
calculated
as
follo
ws
:
eac
h
comp
onen
t
coûte
100
e
and
for
eac
h
failed
test,
the
re-
pair
coûte
10
e
.
Let
X
b
e
the
random
v
ariable
that
asso
ciates
eac
h
ran-
domly
selected
comp
onen
t
with
its
pro
duction
cost
coût.
What
is
the
exp
ected
v
alue
of
the
random
v
ariable
X
?
100.8
101.8
102.8
103.8
9.
Indep
endent
suc
c
ession
of
r
andom
exp
eriments
E.5915
Consider
t
w
o
urns
A
and
B
con
tain-
ing
four
and
three
ob
jects
resp
ectiv
ely
as
sho
wn
b
elo
w
:
The
game
consists
of
dra
wing
an
ob
ject
from
urn
A
then
from
urn
B
:
1
Ho
w
man
y
differen
t
pairs
of
ob
jects
can
b
e
obtained
from
the
t
w
o
dra
ws?
2
Consider
the
follo
wing
t
w
o
ev
en
ts
:
C
:
ˇ
The
pair
of
ob
jects
includes
a
carré
ı
P
:
ˇ
the
pair
of
ob
jects
includes
a
p
en
tagone
ı
Determine
the
follo
wing
probabilities
:
a
P
(
C
∩
P
)
b
P
(
C
∩
P
)
c
P
(
C
∩
P
)
d
P
(
C
∩
P
)
3
a
Determine
the
follo
wing
t
w
o
probabilities
:
P
(
C
)
;
P
(
P
)
b
Cop
y
and
complete
the
tree
b
elo
w
with
the
probabilities
sho
wn
:
c
Can
w
e
find
the
results
of
ques-
tion
2
using
the
probabilit
y
tree
from
the
previous
question.
E.5191
T
w
o
urns
A
and
B
are
a
v
ailable,
eac
h
con
taining
balls
indistinguishable
b
y
touc
h.
Here
is
the
comp
osition
of
the
urns
:
Urn
A
:
three
blac
k
balls
and
t
w
o
white
balls
;
Urn
B
:
fiv
e
red
balls
and
t
w
o
green
balls.
One
ball
is
dra
wn
successiv
ely
from
eac
h
of
the
urns.
Consider
the
follo
wing
ev
en
ts
:
B
:
ˇ
the
ball
dra
wn
is
blanc
he
ı
;
N
:
ˇ
the
ball
dra
wn
is
noire
ı
;
R
:
ˇ
the
ball
dra
wn
is
rouge
ı
;
V
:
ˇ
the
ball
dra
wn
is
v
erte
ı.
1
Cop
y
and
complete
the
probabilit
y
tree
b
elo
w
:
2
Determine
the
v
alue
of
the
follo
wing
probabilities
:
a
P
(
B
∩
R
)
b
P
(
B
∩
V
)
c
P
(
N
∩
R
)
3
a
Giv
e
the
v
alue
of
:
P
(
B
∩
R
)+
P
(
N
∩
R
)
.
b
What
do
w
e
notice?
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5914
sacados/5914
chapExoCorrec/5915
sacados/5915
Urne A
Urne B
1
2
3
P
(
C
P
(
P
P
P
(
P
P
C
P
(
C
P
(
P
P
P
(
P
P
C
chapExoCorrec/5191
sacados/5191
:
:
:
:
:
:
R
:
:
:
V
B
:
:
:
:
:
:
R
:
:
:
V
N
C
B
A
A
B
A
C
A
C
A
B
A
E.5168
A
small
restauran
t
offers
three
main
courses
and
t
w
o
desserts
on
its
men
u.
Here
is
a
description
of
the
men
u
:
Spaghetti
.
.
.
.
.
.
.
6
e
Beef
fillet
.
.
.
.
.
.
.
7
e
Rib
steak
.
.
.
.
.
.
.
8
e
F
ruit
salad
.
.
.
.
.
.
.
.
2
e
Custard
.
.
.
.
.
.
.
.
.
.
3
e
Eac
h
customer
en
tering
the
restauran
t
orders
exactly
one
main
course
and
one
dessert.
1
T
aking
a
random
customer
lea
ving
the
restauran
t,
sp
ec-
ify
what
their
bill
migh
t
b
e.
2
Assuming
that
all
com
binations
main
course
=
dessert
ha
v
e
the
same
probabilit
y
of
b
eing
c
hosen
b
y
a
customer.
a
Ho
w
man
y
com
binations
can
b
e
created
from
this
men
u?
b
What
is
the
probabilit
y
that
a
customer
paid
8
e
?
11
e
?
c
Sho
w
that
the
probabilit
y
of
ha
ving
a
bill
for
10
e
is
1
3
.
d
Complete
the
table
b
elo
w
:
In
v
oice
amoun
t
8
9
10
11
Probabilit
y
E.5350
A
tourist
tra
v
els
for
t
w
o
da
ys
to
the
P
acific
coast
of
Mexico
in
the
lo
calit
y
of
ˇ
F
aro
de
Bucerias
ı.
On
the
w
a
y
to
the
b
eac
h,
t
w
o
paths
are
offered
to
the
b
eac
hes
of
ˇ
Maruata
ı
and
ˇ
Pla
y
a
V
en
tura
ı.
On
the
first
da
y
,
the
tourist
c
ho
oses
one
of
the
t
w
o
b
eac
hes
at
random.
On
the
second
da
y
,
he
will
c
hange
b
eac
hes
with
probabilit
y
3
4
.
What
is
the
probabilit
y
that
the
tourist
has
b
een
to
the
b
eac
h
ˇ
Pla
y
a
V
en
tura
ı
at
least
once?
E.6633
An
elephan
t
mo
v
es
to
three
p
oin
ts
in
its
territory
.
It
starts
at
p
oin
t
A
and
mak
es
three
mo
v
es
:
Either,
he
turns
clo
c
kwise
with
2
c
hances
out
of
three
other-
wise,
he
mo
v
es
coun
terclo
c
kwise.
It
is
assumed
that
eac
h
of
his
mo
v
es
is
indep
enden
t
of
the
previous
ones.
What
is
the
probabilit
y
that,
at
the
end
of
these
mo
v
es,
he
will
arriv
e
at
the
p
oin
t
B
?
10.
R
andom
variables
and
r
ep
etitions
of
indep
endent
exp
eriments
E.133
lebanon
-
2004
-
c
hoice
-
7
p
oin
ts
In
an
animal
exp
erimen
t,
a
rat
is
placed
at
the
start
of
a
run
and
ask
ed
to
c
ho
ose
one
of
3
exit
do
ors
:
if
it
b
orro
ws
the
do
or
A
,
it
go
es
out
if
he
b
orro
ws
one
of
the
do
ors
B
or
C
,
he
is
brough
t
bac
k
to
the
start,
and
this
un
til
he
c
ho
oses
the
do
or
A
.
W
e’l
l
give
the
r
esults
as
irr
e
ducible
fr
actions
P
art
A
:
It
is
assumed
that
the
rat
has
no
memory
:
it
c
ho
oses
a
do
or
at
random
and
can
b
orro
w
the
same
do
or
sev
eral
times
in
a
ro
w.
Eac
h
do
or
therefore
has
the
same
probabilit
y
of
b
eing
c
hosen.
1
What
is
the
probabilit
y
that
it
will
come
out
on
the
first
try?
2
What
is
the
probabilit
y
that
he
only
comes
out
on
the
second
try
(the
first
b
eing
misse
d
and
the
se
c
ond
b
eing
suc
c
essful)
?
3
What
is
the
probabilit
y
that
it
only
comes
out
on
the
fourth
try?
P
art
B
:
It
is
assumed
that
the
rat
has
a
p
erfect
memory
:
at
eac
h
step,
it
randomly
c
ho
oses
one
of
the
do
ors
it
has
nev
er
b
een
through.
1
On
the
app
endix
b
elo
w
(r
eturn
with
c
opy)
,
complete
the
tree
with
the
w
eigh
tings.
2
X
denotes
the
n
um
b
er
of
trials
it
tak
es
to
exit.
What
v
alues
can
the
n
um
b
er
X
tak
e?
3
Complete
the
table
b
elo
w
:
V
alue
of
X
1
2
3
Probabilit
y
1
3
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5168
sacados/5168
chapExoCorrec/5350
sacados/5350
chapExoCorrec/6633
sacados/6633
C
B
A
sacados/133
Liban - 2004 - au choix - 7 points
A
B
A
C
A
C
A
B
A
V
3
F
3
V
2
V
3
F
3
F
2
V
1
V
3
F
3
V
2
V
3
F
3
F
2
F
1
-1
-2
2
1
1
2
E.4797
A
balanced
die
has
letters
written
on
its
faces
:
t
w
o
A
,
t
w
o
B
and
t
w
o
C
.
The
random
exp
erimen
t
consists
of
rolling
the
die
three
times
and
noting,
eac
h
time,
the
letter
of
the
hidden
face.
Th
us,
at
eac
h
output
of
the
random
exp
erimen
t,
a
three-letter
w
ord
is
constructed.
1
Describ
e
the
27
mots
that
can
b
e
obtained
in
this
random
exp
erimen
t.
It
is
assumed
that
eac
h
of
these
w
ords
has
the
same
probabil-
it
y
of
exit.
2
Giv
e
the
probabilit
y
of
the
follo
wing
ev
en
ts
:
a
A
:
ˇ
Le
w
ord
obtained
con
tains
exactly
once
the
letter
B
ı
;
b
B
:
ˇ
The
w
ord
obtained
con
tains
exactly
t
wice
the
let-
ter
B
ı
;
c
C
:
ˇ
The
w
ord
con
tains
the
same
letter
in
the
first
and
third
place
ı.
3
Based
on
the
construction
of
these
w
ords,
w
e
ha
v
e
cre-
ated
a
new
game
with
the
follo
wing
rules
:
The
pla
y
er
wins
2
e
if
the
w
ord
con
tains
the
letter
only
once
B
;
the
pla
y
er
wins
5
e
if
the
w
ord
con
tains
exactly
t
w
o
instances
of
the
letter
B
.
The
pla
y
er
wins
10
e
if
the
w
ord
is
ˇ
BBB
ı.
W
e
note
ˇ
X
=
k
ı
the
ev
en
t
ˇ
the
pla
y
er
wins
k
e
ı.
Com-
plete
the
table
b
elo
w
:
k
0
2
5
10
P
(
X
=
k
)
11.
R
andom
variables,
exp
e
ctations,
standar
d
deviations
and
r
ep
etitions
E.4811
A
QCM
(multiple-choic
e
questionnair
e)
is
prop
osed
to
studen
ts
:
it
comprises
three
questions
and
four
answ
ers
are
prop
osed,
only
one
of
whic
h
is
correct.
W
e
wish
to
study
the
p
ercen
tage
of
success
in
this
MCQ
if
the
studen
ts
answ
er
it
completely
at
random
;
w
e
then
assume
that
the
answ
ers
giv
en
to
eac
h
of
the
questions
are
indep
en-
den
t
of
eac
h
other.
W
e
note
:
F
i
:
ˇ
The
answ
er
pro
vided
to
question
i
is
fausse
ı
;
V
i
:
ˇ
The
answ
er
pro
vided
to
the
question
i
is
vraie
ı
;
1
Complete
the
probabilit
y
tree
b
elo
w
:
2
Note
X
the
random
v
ariable
coun
ting
the
n
um
b
er
of
cor-
rect
answ
ers
pro
vided
to
the
MCQ.
a
Determine
the
probabilit
y
distribution
of
the
random
v
ariable
X
.
b
Calculate
the
exp
ectation
of
the
random
v
ariable
X
.
E.7501
An
urn
con
tains
three
balls
indistin-
guishable
to
the
touc
h
b
earing
the
n
um-
b
ers
−
2
,
−
1
and
2
.
Consider
the
random
exp
erimen
t
of
suc-
cessiv
ely
dra
wing
a
ball
from
the
urn
three
times
and
putting
it
bac
k
eac
h
time.
Consider
the
random
v
ariable
X
whic
h
asso
ciates
with
eac
h
exp
erimen
t,
the
sum
of
the
n
um
b
ers
obtained
on
the
balls
dra
wn.
Determine
the
exp
ectation
of
the
random
v
ariable
X
An
y
trace
of
researc
h
or
initiativ
e,
ho
w
ev
er
incomplete,
will
b
e
tak
en
in
to
accoun
t
in
the
assessmen
t.
E.7562
An
urn
con
tains
three
balls
indistin-
guishable
to
the
touc
h
b
earing
the
n
um-
b
ers
1
,
1
and
2
.
Consider
the
random
exp
erimen
t
of
suc-
cessiv
ely
dra
wing
a
ball
from
the
urn
three
times
and
putting
it
bac
k
eac
h
time.
Consider
the
random
v
ariable
X
whic
h
asso
ciates
with
eac
h
exp
erimen
t,
the
sum
of
the
n
um
b
ers
obtained
on
the
balls
dra
wn.
Determine
the
exp
ectation
of
the
random
v
ariable
X
An
y
trace
of
researc
h
or
initiativ
e,
ho
w
ev
er
incomplete,
will
b
e
tak
en
in
to
accoun
t
in
the
assessmen
t.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4797
sacados/4797
chapExoCorrec/4811
sacados/4811
V
3
F
3
V
2
V
3
F
3
F
2
V
1
V
3
F
3
V
2
V
3
F
3
F
2
F
1
chapExoCorrec/7501
sacados/7501
-1
-2
2
chapExoCorrec/7562
sacados/7562
1
1
2
E.5196
A
c
ho
colate
factory
pro
duces
b
o
xes
of
c
ho
colates
eac
h
y
ear,
including
50
%
with
milk
c
ho
colate,
30
%
with
dark
c
ho
colate,
and
20
%
with
white
c
ho
colate.
70
%
of
the
b
o
xes
con
tain
plain
c
ho
colates,
while
the
others
con
tain
c
ho
colates
filled
with
caramel.
These
prop
ortions
are
indep
enden
t
of
the
t
yp
e
of
c
ho
colate
used
to
mak
e
the
b
o
x.
Consider
the
follo
wing
ev
en
ts
:
L
:
ˇ
milk
c
ho
colate
is
used
ı
;
N
:
ˇ
dark
c
ho
colate
is
used
ı
;
B
:
ˇ
white
c
ho
colate
is
used
ı
;
N
a
:
ˇ
the
c
ho
colates
are
plain
ı
;
C
:
ˇ
the
c
ho
colates
are
filled
with
caramel
ı
;
All
results
will
b
e
giv
en
in
decimal
form.
1
Dra
w
the
w
eigh
ted
tree
asso
ciated
with
this
situation.
2
A
b
o
x
is
c
hosen
at
random
from
the
factory
output.
De-
termine
the
probabilities
of
the
follo
wing
ev
en
ts
:
a
ˇ
the
b
o
x
con
tains
dark
and
natures
ı
c
ho
colates.
b
ˇ
the
b
o
x
con
tains
dark
or
plain
c
ho
colates
ı
3
The
compan
y
sets
b
o
x
prices
as
follo
ws
:
le
base
price
for
a
b
o
x
of
c
ho
colate
is
9
e
;
if
the
c
ho
colate
used
is
dark
c
ho
colate
then
the
price
is
increased
b
y
4
e
;
if
the
c
ho
colate
used
is
white
c
ho
colate
then
the
price
is
increased
b
y
2
e
;
if
the
c
ho
colates
are
caramel-filled,
the
price
p
er
b
o
x
increases
b
y
2
e
.
The
random
v
ariable
X
asso
ciates
a
b
o
x
pro
duced
b
y
the
factory
with
its
b
elly
price.
a
Dra
w
the
table
represen
ting
the
probabilit
y
distribu-
tion
asso
ciated
with
the
random
v
ariable
X
.
b
Determine
the
exp
ectation
of
the
random
v
ariable
X
round
to
the
nearest
ten
th.
E.6631
A
n
ursery
offers
three
t
yp
es
of
trees
:
acacias,
plane
trees,
and
oaks.
Eac
h
of
these
trees
can
b
e
pur-
c
hased
in
differen
t
sizes
:
either
as
a
ˇ
y
oung
sapling
ı
(
0.75
meters)
,
or
as
an
ˇ
mature
tree
ı
(
2
meters)
.
A
t
the
âsapling
stage,
acacias,
plane
trees,
and
oaks
are
w
orth
50
e
,
65
e
and
80
e
resp
ectiv
ely
.
If
the
customer
w
an
ts
to
buy
the
adult
form,
they
m
ust
add
15
e
.
In
his
end-of-y
ear
review,
he
notes
that
40
%
of
the
trees
sold
are
oaks
and
that
acacias
and
plane
trees
share
the
remaining
sales
equally
.
He
also
notices
that,
regardless
of
the
t
yp
e
of
tree,
a
quarter
of
sales
are
alw
a
ys
made
on
"
adult
ı.
The
random
exp
erimen
t
considered
consists
of
randomly
dra
wing
an
in
v
oice
from
the
financial
y
ear
2015
.
The
follo
wing
ev
en
ts
are
considered
:
A
:
ˇ
The
tree
purc
hased
is
a
acacia
ı
P
:
ˇ
The
tree
purc
hased
is
a
platane
ı
C
:
ˇ
The
tree
purc
hased
is
a
c
hêne
ı
J
:
ˇ
The
tree
is
a
sapling
ı
1
Dra
w
a
w
eigh
ted
tree
represen
ting
this
situation
2
Consider
the
random
v
ariable
X
asso
ciating
the
in
v
oice
dra
wn
with
its
amoun
t.
a
Dra
w
the
table
of
the
probabilit
y
distribution
of
the
random
v
ariable
X
.
b
Calculer
the
exp
ectation
of
the
random
v
ariable
X
.
c
Donner
the
standard
deviation
of
X
to
the
nearest
ten
th.
12.
R
andom
variables,
exp
e
ctations,
standar
d
deviations
and
c
onditional
pr
ob
a-
bilities
E.4890
A
c
ho
colate
factory
pro
duces
b
o
xes
of
c
ho
colate
throughout
the
y
ear,
of
whic
h
50
%
are
made
with
milk
c
ho
colate,
30
%
with
dark
c
ho
colate,
and
20
%
with
white
c
ho
colate.
70
%
of
the
b
o
xes
con
tain
plain
c
ho
colates,
while
the
others
con
tain
caramel-filled
c
ho
colates.
These
prop
ortions
are
inde-
p
enden
t
of
the
t
yp
e
of
c
ho
colate
used
to
mak
e
the
b
o
x.
Consider
the
follo
wing
ev
en
ts
:
L
:
ˇ
milk
c
ho
colate
is
used
ı
;
N
:
ˇ
dark
c
ho
colate
is
used
ı
;
B
:
ˇ
white
c
ho
colate
is
used
ı
;
N
a
:
ˇ
the
c
ho
colates
are
plain
ı
;
C
:
ˇ
the
c
ho
colates
are
filled
with
caramel
ı
;
All
results
will
b
e
giv
en
in
decimal
form.
1
Dra
w
the
probabilit
y
tree
asso
ciated
with
this
situation.
2
A
b
o
x
is
c
hosen
at
random
from
the
factory
output.
De-
termine
the
probabilities
of
the
follo
wing
ev
en
ts
:
a
ˇ
The
b
o
x
con
tains
dark
and
plain
c
ho
colates
ı
b
ˇ
The
b
o
x
con
tains
dark
or
plain
c
ho
colates
ı
3
The
compan
y
sets
the
prices
of
the
b
o
xes
as
follo
ws
:
the
base
price
of
a
b
o
x
of
c
ho
colates
is
9
e
;
if
dark
c
ho
colate
is
used,
the
price
is
increased
b
y
4
e
;
if
white
c
ho
colate
is
used,
the
price
is
increased
b
y
2
e
;
if
the
c
ho
colates
are
filled
with
caramel,
the
price
of
the
b
o
x
increases
b
y
2
e
.
The
random
v
ariable
X
asso
ciates
eac
h
b
o
x
pro
duced
b
y
the
factory
with
its
selling
price.
a
Dra
w
a
table
represen
ting
the
probabilit
y
distribution
asso
ciated
with
the
random
v
ariable
X
.
b
Determine
the
exp
ected
v
alue
of
the
random
v
ariable
X
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5196
sacados/5196
chapExoCorrec/6631
sacados/6631
chapExoCorrec/4890
sacados/4890
C
C
M
C
C
T
C
C
P
E.7561
A
n
ursery
offers
three
t
yp
es
of
trees
:
acacias,
plane
trees,
and
oaks.
Eac
h
of
these
trees
can
b
e
purc
hased
in
differen
t
sizes
:
either
as
a
ˇ
y
oung
sapling
ı
(
0.75
meters)
,
or
as
an
ˇ
mature
tree
ı
(
2
meters)
.
A
t
the
âsapling
stage,
acacias,
plane
trees,
and
oaks
are
w
orth
50
e
,
65
e
and
80
e
resp
ectiv
ely
.
If
the
customer
w
an
ts
to
buy
the
adult
form,
they
m
ust
add
15
e
.
During
his
end-of-y
ear
review,
he
notes
that
40
%
of
the
trees
sold
are
oaks
and
that
acacias
and
plane
trees
share
the
re-
maining
sales
equally
.
He
also
notices
that
regardless
of
the
t
yp
e
of
tree,
a
quarter
of
sales
are
alw
a
ys
for
"
adult
ı.
The
random
exp
erimen
t
considered
consists
of
randomly
dra
wing
an
in
v
oice
from
the
fiscal
y
ear
2015
.
Consider
the
follo
wing
ev
en
ts
:
A
:
ˇ
The
tree
purc
hased
is
an
acacia
ı
P
:
ˇ
The
tree
purc
hased
is
a
plane
tree
ı
C
:
ˇ
The
tree
purc
hased
is
an
oak
tree
ı
J
:
ˇ
The
tree
is
a
y
oung
sapling
ı
1
a
Dra
w
a
w
eigh
ted
tree
represen
ting
this
situation
b
Giv
e
the
probabilit
y
of
the
ev
en
t
C
∩
J
2
W
e
consider
the
random
v
ariable
X
asso
ciating
the
in-
v
oice
dra
wn
with
its
amoun
t.
a
Dra
w
up
the
probabilit
y
distribution
table
for
the
ran-
dom
v
ariable
X
.
b
Calculate
the
exp
ected
v
alue
of
the
random
v
ariable
X
.
E.4822
A
restauran
t
offers
t
w
o
t
yp
es
of
dessert
on
its
men
u
:
an
assortmen
t
of
macarons,
c
hosen
b
y
50
%
customers
;
a
slice
of
tarte
tatin,
c
hosen
b
y
30
%
customers.
the
rest
of
the
customers
do
not
order
an
y
dessert.
No
customer
orders
more
than
one
dessert.
In
addition,
80
%
customers
ha
v
e
coffee,
regardless
of
whether
or
not
they
ha
v
e
dessert.
A
customer
at
this
restauran
t
is
randomly
surv
ey
ed.
W
e
note
p
the
probabilit
y
asso
ciated
with
this
random
exp
erimen
t.
W
e
note
:
M
the
ev
en
t
:
ˇ
The
customer
orders
an
assortmen
t
of
macarons
ı
;
T
the
ev
en
t
:
ˇ
The
customer
tak
es
a
slice
of
tarte
tatin
ı
;
P
ev
en
t
:
ˇ
The
customer
do
es
not
tak
e
an
y
dessert
ı
;
C
ev
en
t
:
ˇ
The
customer
has
a
coffee
ı
and
C
the
opp
osite
ev
en
t
of
C
.
1
Complete
the
probabilit
y
tree
b
elo
w
:
2
Determine
the
probabilit
y
that
a
customer
c
hose
the
mac-
aron
assortmen
t
and
had
a
coffee.
3
An
assortmen
t
of
macarons
is
sold
6
e
,
a
slice
of
tarte
tatin
is
sold
7
e
,
and
a
coffee
is
sold
2
e
.
Eac
h
customer
orders
one
dish
(and
only
one)
at
a
single
price
of
18
e
,
and
do
es
not
order
more
than
one
dessert
or
more
than
one
coffee.
W
e
denote
X
the
random
v
ariable
asso
ciating
eac
h
cus-
tomer
with
the
price
of
their
bill.
a
Determine
the
v
alues
tak
en
b
y
the
random
v
ariable
X
.
b
Dra
w
up
a
table
represen
ting
the
probabilit
y
distribu-
tion
of
the
random
v
ariable
X
.
c
Determine
the
exp
ected
v
alue
of
the
random
v
ariable
X
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7561
sacados/7561
chapExoCorrec/4822
sacados/4822
C
C
M
C
C
T
C
C
P
F
G
M
C
F
G
M
C
R
R
S
p
R
R
S
n
R
R
S
r
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E.4823
A
red
fruit
pro
ducer
offers
rasp-
b
erries,
redcurran
ts
and
blueb
erries
for
direct
sale.
The
customer
can
buy
either
tra
ys
of
fruit
for
tasting,
or
tra
ys
of
fruit
for
jam.
The
gro
w
er
has
noticed
that,
among
his
customers,
9
sur
10
purc
hase
a
tra
y
of
fruit
for
jam.
Regardless
of
the
t
yp
e
of
tra
y
purc
hased,
in
50
%
des
cases
the
customer
c
ho
oses
blueb
erry
for
fruit,
30
%
raspb
erries
in
the
other
cases,
redcurran
t
is
c
hosen.
Note
:
C
the
ev
en
t
:
ˇ
le
customer
buys
a
punnet
of
fruit
at
con-
fiture
ı
;
F
l
the
ev
en
t
:
ˇ
the
customer
requests
raspb
erries
ı
;
G
the
ev
en
t
:
ˇ
le
customer
requests
groseilles
ı
;
M
l’ev
en
t
:
ˇ
le
customer
requests
m
yrtilles
ı
;
It
is
assumed
that
the
fruit
c
hosen
do
es
not
dep
end
on
the
t
yp
e
of
tra
y
purc
hased
and
that
eac
h
customer
buys
only
one
tra
y
.
1
Complete
the
probabilit
y
tree
b
elo
w
:
2
Determine
the
probabilit
y
of
C
∩
F
.
3
The
pro
ducer
sets
the
prices
of
his
tra
ys
as
follo
ws
:
The
base
price
of
a
punnet
of
fruit
for
jam
is
5
euros
and
that
of
a
punnet
of
fruit
for
eating
is
3
euros
;
If
the
selected
tra
y
con
tains
raspb
erries,
he
adds
1
eu-
ros
to
the
price
of
the
tra
y
;
If
the
selected
tra
y
con
tains
blueb
erries,
he
adds
2
eu-
ros
to
the
price
of
the
tra
y
;
If
the
selected
punnet
con
tains
redcurran
ts,
the
base
price
remains
unc
hanged.
W
e
note
X
the
random
v
ariable
asso
ciating
eac
h
cus-
tomer
with
the
price
of
the
punnet
purc
hased.
a
What
are
the
v
alues
tak
en
b
y
the
random
v
ariable
X
?
b
Dra
w
up
a
table
represen
ting
the
probabilit
y
distribu-
tion
of
X
.
c
Determine
the
exp
ected
v
alue
of
the
random
v
ariable
X
.
E.4837
A
sp
orts
shop
offers
do
wnhill
skis,
sno
wb
oards,
and
cross-coun
try
skis
for
ren
t.
Its
ren
tal
equipmen
t
consists
of
60
%
do
wnhill
skis,
with
the
remainder
divided
equally
b
et
w
een
sno
wb
oards
and
cross-
coun
try
skis.
After
eac
h
da
y
of
ren
tal,
the
equipmen
t
is
c
hec
k
ed
and
re-
paired
if
necessary
.
Regardless
of
the
t
yp
e
of
equipmen
t
ren
ted,
30
%
of
the
equipmen
t
requires
repair.
Eac
h
piece
of
ren
ted
equipmen
t
is
listed
on
a
form
that
trac
ks
its
status.
A
form
is
selected
at
random.
The
follo
wing
ev
en
ts
are
considered
:
S
p
:
ˇ
The
card
is
for
a
pair
of
do
wnhill
skis
ı
;
S
n
:
ˇ
The
record
is
for
a
sno
wb
oard
ı
;
S
r
:
ˇ
The
record
is
for
a
pair
of
touring
skis
ı
;
R
:
ˇ
The
equipmen
t
needs
repair
ı
;
R
is
its
opp
osite
ev
en
t.
All
results
for
the
first
four
questions
will
b
e
rounded
to
10
−
3
.
1
Cop
y
and
complete
the
probabilit
y
tree
b
elo
w
:
2
a
Calculate
the
probabilit
y
that
the
card
dra
wn
con-
cerns
a
pair
of
piste
skis
not
in
need
of
repair.
b
Calculate
P
S
p
∪
R
:
the
probabilit
y
that
the
card
dra
wn
concerns
a
pair
of
piste
skis
or
equipmen
t
not
requiring
repair.
3
The
cost
of
ren
ting
do
wnhill
skis
or
a
sno
wb
oard
is
20
e
,
that
of
a
pair
of
touring
skis
is
15
e
.
In
the
ev
en
t
of
repair,
an
additional
cost
of
15
e
est
will
b
e
c
harged.
Consider
the
random
v
ariable
X
whic
h
asso
ciates
the
as-
so
ciated
billing
amoun
t
with
a
plug.
a
Dra
w
a
table
represen
ting
the
probabilit
y
distribution
of
the
random
v
ariable
X
.
b
Determine
the
exp
ectation
of
the
random
v
ariable
X
.
13.
Bernoul
li
test
E.7827
Consider
an
exp
erimen
t
with
t
w
o
p
ossible
outcomes
:
one
called
ˇ
suc-
cess
ı
and
denoted
S
with
probabil-
it
y
0.4
;
the
other
called
ı
failure
ı
and
denoted
E
.
W
e
decide
to
rep
eat
this
same
ex-
p
erimen
t
three
times.
W
e
obtain
the
probabilit
y
tree
sho
wn
opp
o-
site.
W
e
assume
that
these
rep
etitions
are
indep
enden
t
of
eac
h
other.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4823
sacados/4823
F
G
M
C
F
G
M
C
chapExoCorrec/4837
sacados/4837
R
R
S
p
R
R
S
n
R
R
S
r
chapExoCorrec/7827
sacados/7827
E
S
E
S
E
S
E
S
E
S
E
S
E
S
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
1
Complete
this
probabilit
y
tree?
2
a
Ho
w
man
y
paths
ha
v
e
3
successes?
b
Giv
e
the
probabilit
y
of
obtaining
three
successes
at
the
end
of
this
random
exp
erimen
t.
3
a
Ho
w
man
y
paths
ha
v
e
0
successes?
b
Giv
e
the
probabilit
y
of
obtaining
no
successes
at
the
end
of
this
random
exp
erimen
t.
4
a
Ho
w
man
y
paths
ha
v
e
2
successes?
b
Giv
e
the
probabilit
y
of
obtaining
exactly
t
w
o
successes
at
the
end
of
this
random
exp
erimen
t?
E.4887
Here
are
the
c
hoice
trees
asso
ciated
with
rep
eating
a
Bernoulli
test
3
and
4
times
resp
ectiv
ely
:
1
F
or
the
threefold
rep
etition
of
Bernoulli’s
test,
complete
the
table
b
elo
w
:
Num
b
er
of
succès
0
1
2
3
Num
b
er
of
issues
2
a
F
or
the
fourfold
rep
etition
of
Bernoulli’s
test,
com-
plete
the
table
b
elo
w
:
Num
b
er
of
succès
0
1
2
3
4
Num
b
er
of
issues
b
Is
there
a
metho
d
for
obtaining
the
second
table
from
the
first?
E.4886
In
a
game
arising
from
a
random
exp
er-
imen
t,
only
t
w
o
outcomes
are
considered
:
success
(
S
)
at
this
game
and
failure
(
E
)
.
The
probabilit
y
of
success
is
0.15
.
P
art
A
:
successiv
e,
indep
enden
t
rep
etition
of
4
games
Here
is
the
c
hoice
tree
corresp
onding
to
this
rep
etition
of
ex-
p
erimen
ts
;
1
Determine
the
probabilit
y
of
the
ev
en
t
A
:
ˇ
The
pla
y
er
w
on
exactly
3
fois
ı
2
Determine
the
probabilit
y
of
the
ev
en
t
B
:
ˇ
The
pla
y
er
w
on
exactly
2
fois
ı
P
art
B
:
successiv
e
and
indep
enden
t
rep
etition
of
5
games
Here’s
the
c
hoice
tree
corresp
onding
to
this
game
:
3
Determine
the
probabilit
y
of
the
ev
en
t
C
:
ˇ
The
pla
y
er
w
on
exactly
3
fois
ı.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4887
sacados/4887
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
chapExoCorrec/4886
sacados/4886
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
P
F
P
F
P
F
P
F
P
F
P
F
P
F
P
F
P
F
P
F
P
F
P
F
P
F
P
F
P
F
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
E.389
1
A
pla
y
er
flips
an
un
balanced
coin
with
a
probabilit
y
of
0
;
63
of
landing
on
heads.
What
is
the
probabilit
y
of
landing
on
tails?
W
e
w
an
t
to
examine
certain
outcomes
resulting
from
four
tosses
of
this
coin.
W
e
assume
that
these
tosses
are
indep
en-
den
t
of
one
another.
Note
:
The
elemen
tary
ev
en
ts
P
-
P
-
F
-
P
and
P
-
F
-
P
-
P
are
differen
t
outcomes
of
this
random
exp
erimen
t,
but
eac
h
of
them
results
in
the
same
n
um
b
er
of
heads
and
tails.
2
a
Ho
w
man
y
outcomes
corresp
ond
to
getting
3
heads
and
1
tails?
b
What
is
the
probabilit
y
of
an
outcome
con
taining
3
heads
and
1
tails?
c
Use
this
to
determine
the
probabilit
y
of
getting
3
heads
during
this
exp
erimen
t.
3
a
Whic
h
of
the
follo
wing
calculations
represen
ts
the
probabilit
y
of
a
roll
with
exactly
2
heads
:
(1
−
0
;
63)
2
;
0
;
63
3
×
(1
−
0
;
63)
0
;
63
2
×
(1
−
0
;
63)
2
;
0
;
63
×
(1
−
0
;
63)
3
b
Determine
the
probabilit
y
of
getting
2
heads
during
this
roll.
E.4911
1
Consider
the
c
hoice
tree
b
elo
w
from
the
fourfold
rep
eti-
tion
of
a
Bernoulli
trial
:
Determine
the
binomial
co
efficien
ts
b
elo
w
:
a
4
2
b
4
3
2
Using
the
calculator,
determine
the
v
alue
of
the
follo
wing
binomial
co
efficien
ts
:
a
5
3
a
12
5
a
8
6
a
7
2
14.
Binomial
law
E.7830
A
pla
y
er
has
a
balanced
cubic
die
whose
faces
are
n
um
b
ered
from
1
to
6
.
On
eac
h
thro
w,
he
wins
if
he
gets
2
,
3
,
4
,
5
or
6
;
he
loses
if
he
gets
1
.
A
game
consists
of
5
successiv
e,
indep
enden
t
thro
ws
of
the
die.
Determine
the
exact
probabilit
y
that
the
pla
y
er
will
lose
3
times
during
a
game,
then
its
v
alue
rounded
to
the
ten
th.
E.7831
Consider
a
random
v
ariable
X
follo
wing
the
binomial
distribution
with
parameters
n
=
15
and
p
=
0.63
.
1
Using
a
calculator,
determine
the
follo
wing
binomial
co-
efficien
ts
:
a
15
13
b
15
14
c
15
15
2
Using
a
calculator,
giv
e
the
v
alue
rounded
to
the
nearest
10
−
4
of
the
follo
wing
probabilities
:
a
P
X
=
13
b
P
X
=
14
c
P
X
=
15
3
Deduce
the
v
alue,
rounded
to
the
nearest
10
−
4
,
of
the
probabilit
y
of
ev
en
t
X
12
.
E.4938
Let
X
b
e
a
random
v
ariable
follo
wing
a
binomial
distribution
with
parameters
n
=
35
and
p
=
0.34
.
Determine
the
exact
v
alue
and
then
the
v
alue
rounded
to
10
−
5
of
eac
h
of
the
follo
wing
probabilities
:
a
P
X
=
5
b
P
X
=
10
c
P
X
=
25
E.4926
Consider
a
random
v
ariable
X
follo
wing
the
binomial
distribution
with
parameters
n
=
15
and
p
=
0.63
.
W
e
will
giv
e
the
exact
probabilit
y
of
eac
h
of
the
probabilities
requested.
1
Determine
the
follo
wing
probabilities
:
a
P
X
=
0
b
P
X
=
1
c
P
X
=
5
2
Giv
e
the
probabilit
y
of
ev
en
t
X
14
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/389
sacados/389
P
F
P
F
P
F
P
F
P
F
P
F
P
F
P
F
P
F
P
F
P
F
P
F
P
F
P
F
P
F
chapExoCorrec/4911
sacados/4911
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
S
E
chapExoCorrec/7830
sacados/7830
chapExoCorrec/7831
sacados/7831
chapExoCorrec/4938
sacados/4938
chapExoCorrec/4926
sacados/4926
0
1
2
3
4
5
0,1
0,2
0,3
0,4
3
X
V
2
F
2
F
2
F
V
2
V
1
F
1
F
1
F
F
2
V
1
F
1
F
1
F
F
2
V
1
F
1
F
1
F
F
V
2
V
1
F
1
F
1
F
V
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
F
2
V
1
F
1
F
1
F
V
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
F
2
V
1
F
1
F
1
F
V
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
F
1
4
1
4
1
4
3
X
V
3
4
2
F
V
3
4
1
4
2
V
3
4
1
F
F
V
3
4
1
4
1
4
2
V
3
4
1
F
V
3
4
1
4
1
V
3
4
0
F
F
F
E.4928
A
random
v
ariable
X
is
assumed
to
fol-
lo
w
a
binomial
distribution
with
parameters
n
=
5
and
p
=
0.6
.
1
Using
a
calculator,
dra
w
up
a
table
represen
ting
the
la
w
of
the
v
ariable
X
with
probabilities
rounded
to
the
near-
est
thousandth.
2
Deduce
the
follo
wing
probabilities
rounded
to
the
h
un-
dredth
:
a
P
X
1
b
P
X
>
1
E.4930
1
Consider
the
random
v
ariable
X
follo
wing
a
binomial
dis-
tribution
with
parameters
n
=
5
and
p
=
0.5
.
a
Using
a
calculator
and
rounding
v
alues
to
10
−
4
,
dra
w
up
a
table
sho
wing
the
probabilit
y
distribution
of
the
random
v
ariable
X
.
b
Dra
w
the
bar
c
hart
represen
ting
the
la
w
of
the
random
v
ariable
X
.
2
Consider
the
random
v
ariable
X
follo
wing
a
binomial
dis-
tribution
with
parameters
n
=
5
and
p
=
0.37
.
a
Using
the
calculator
and
rounding
v
alues
to
10
−
4
,
dra
w
up
a
table
sho
wing
the
probabilit
y
distribution
of
the
random
v
ariable
X
.
b
Dra
w
the
bar
c
hart
represen
ting
the
la
w
of
the
random
v
ariable
X
.
E.7385
Consider
a
m
ultiple-c
hoice
questionnaire
consisting
of
3
questions
eac
h
offering
4
answ
ers,
only
one
of
whic
h
is
correct.
W
e
create
a
random
exp
erimen
t
b
y
asking
participan
ts
to
ran-
domly
answ
er
the
questions
prop
osed
in
the
form.
This
situation
is
represen
ted
b
y
the
c
hoice
tree
b
elo
w
:
F
or
eac
h
elemen
tary
ev
en
t,
w
e
asso
ciate
the
random
v
ariable
X
with
the
n
um
b
er
of
correct
answ
ers.
1
Giv
e
the
probabilit
y
la
w
of
the
random
v
ariable
X
.
F
or
eac
h
question
in
the
M.C.Q.,
the
probabilit
y
of
a
correct
answ
er
is
1
4
and
a
wrong
answ
er
is
3
4
.
The
c
hoice
tree
is
simplified
b
y
the
probabilit
y
tree
b
elo
w
:
2
a
Using
this
probabilit
y
tree,
what
calculation
finds
the
v
alues
of
the
probabilities
:
P
X
=
0
;
P
X
=
3
b
What
metho
d
can
b
e
one
apply
to
quic
kly
obtain
the
v
alues
of
P
X
=
1
and
P
X
=
2
?
E.7487
T
en
p
eople
apply
for
a
job
in
the
compan
y
.
Their
applications
are
studied
indep
enden
tly
of
eac
h
other.
W
e
denote
b
y
X
the
random
v
ariable
giving
the
n
um
b
er
of
p
eople
recruited
among
the
10
p
eople.
W
e
kno
w
38
%
of
the
applican
ts
ha
v
e
b
een
recruited.
1
Justify
that
X
follo
ws
a
binomial
distribution
with
pa-
rameters
n
=
10
and
p
=
0.38
.
2
Calculate
the
probabilit
y
that
at
least
one
of
the
ten
p
eo-
ple
will
b
e
recruited.
The
exact
v
alue
and
then
a
v
alue
of
the
result
rounded
to
10
−
3
will
b
e
giv
en.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4928
sacados/4928
chapExoCorrec/4930
sacados/4930
0
1
2
3
4
5
0,1
0,2
0,3
0,4
0
1
2
3
4
5
0,1
0,2
0,3
0,4
chapExoCorrec/7385
sacados/7385
3
X
V
2
F
2
F
2
F
V
2
V
1
F
1
F
1
F
F
2
V
1
F
1
F
1
F
F
2
V
1
F
1
F
1
F
F
V
2
V
1
F
1
F
1
F
V
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
F
2
V
1
F
1
F
1
F
V
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
F
2
V
1
F
1
F
1
F
V
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
1
V
0
F
0
F
0
F
F
F
1
4
1
4
1
4
3
X
V
3
4
2
F
V
3
4
1
4
2
V
3
4
1
F
F
V
3
4
1
4
1
4
2
V
3
4
1
F
V
3
4
1
4
1
V
3
4
0
F
F
F
chapExoCorrec/7487
sacados/7487
Extrait Antilles-Guyanes Juin 2014
Représentation 1
Représentation 2
E.7829
Let
X
follo
w
a
binomial
distribution
with
parameters
15
and
0.35
.
C’est-à-dire
:
X
∼
B
(
15
;
0.35
)
Determine
the
exact
v
alue,
then
the
v
alue
rounded
to
the
thousandth
of
the
follo
wing
probabilities
:
a
P
X
=
5
b
P
X
=
7
c
P
X
=
9
E.7640
By
c
ho
osing
a
pupil
at
random
from
among
those
registered
for
half-b
oard,
w
e
kno
w
that
the
probabilit
y
of
the
c
hosen
pupil
b
eing
satisfied
with
half-b
oard
is
0.675
.
Eac
h
time,
a
group
of
4
studen
ts
is
in
terview
ed.
W
e
denote
X
the
random
v
ariable
equal
to
the
n
um
b
er
of
stu-
den
ts
declaring
themselv
es
satisfied
with
the
qualit
y
of
the
meals.
As
the
n
um
b
er
of
studen
ts
is
sufficien
tly
large,
X
is
considered
to
follo
w
a
binomial
distribution.
Results
will
b
e
rounded
to
the
nearest
thousandth.
1
Sp
ecify
the
parameters
of
this
binomial
distribution.
2
Calculate
the
probabilit
y
of
the
ev
en
t
A
:
ˇ
the
four
stu-
den
ts
are
satisfied
with
the
qualit
y
of
repas
ı.
3
Describ
e
in
a
sen
tence
the
ev
en
t
A
and
calculate
its
prob-
abilit
y
.
E.4888
Consider
an
urn
con
taining
6
red
balls
and
4
blue
balls
indistinguishable
b
y
touc
h.
1
One
ball
is
dra
wn
at
random
from
this
urn.
What
is
the
probabilit
y
of
getting
a
red
ball?
W
e
decide
to
dra
w
three
balls
in
succession.
Eac
h
time,
the
ball
is
put
bac
k
in
to
the
urn
(the
dr
aw
is
said
to
b
e
with
de-
livery)
.
2
Consider
the
Bernoulli
sc
heme
resulting
from
this
rep
eti-
tion,
where
success
is
asso
ciated
with
the
ev
en
t
ˇ
ha
ving
dra
wn
a
ball
rouge
ı
a
What
are
the
parameters
of
this
Bernoulli
sc
heme?
b
Construct
the
tree
represen
ting
this
situation.
c
Ho
w
man
y
elemen
tary
ev
en
ts
realize
the
ev
en
t
:
ˇ
ha
ving
dra
wn
2
balls
rouges
ı
E.4909
In
this
exercise,
all
results
will
b
e
rounded
to
the
nearest
10
−
3
.
In
a
cit
y
,
a
surv
ey
sho
ws
that
87
%
of
the
inhabitan
ts
o
wn
a
cell
phone.
Three
inhabitan
ts
are
c
hosen
successiv
ely
,
at
random
and
in-
dep
enden
tly
.
Note
X
the
n
um
b
er
of
p
eople
selected
who
o
wn
a
cell
phone.
1
What
v
alues
can
the
random
v
ariable
X
tak
e?
The
random
v
ariable
X
is
assumed
to
follo
w
a
binomial
dis-
tribution
with
parameters
3
and
0.87
:
2
Giv
e,
in
a
table,
the
probabilit
y
distribution
of
the
ran-
dom
v
ariable
X
?
3
What
is
the
probabilit
y
that
at
least
t
w
o
of
his
three
p
eople
o
wn
a
cell
phone?
E.4939
Let
X
b
e
a
random
v
ariable
follo
wing
a
binomial
distribution
with
parameters
n
=
35
and
p
=
0.35
.
Using
the
calculator,
determine
the
follo
wing
probabilities
rounded
to
the
nearest
thousandth
:
a
P
X
5
b
P
X
10
c
P
X
20
E.4937
Of
the
t
w
o
represen
tations
b
elo
w
and
without
justification,
giv
e
the
one
that
represen
ts
the
la
w
of
a
random
v
ariable
X
follo
wing
the
binomial
la
w
with
param-
eters
n
=
10
and
p
=
0.3
:
E.4929
A
random
v
ariable
X
is
assumed
to
follo
w
a
binomial
distribution
with
parameters
n
=
8
and
p
=
0.37
.
Determine
the
probabilit
y
v
alue
P
3
X
7
rounded
to
the
nearest
h
undredth.
E.4927
A
v
accine
is
b
eing
tested
on
a
p
opula-
tion
of
100
individuals.
30
of
them
react
to
this
v
accine
with
high
fev
ers.
Eac
h
test
phase
is
carried
out
on
a
group
of
5
in-
dividuals
c
hosen
at
random
and
indep
enden
tly
b
et
w
een
eac
h
test.
Let
us
note
X
the
random
v
ariable
that
asso
ciates
with
eac
h
test
phase
the
n
um
b
er
of
individuals
who
had
a
reaction
with
high
fev
ers.
1
Justify
that
the
random
v
ariable
X
follo
ws
a
binomial
distribution
whose
parameter
v
alues
will
b
e
sp
ecified.
2
Determine
the
probabilit
y
that
2
individuals
reacted
to
the
v
accine
with
fev
er
o
v
er
a
test
phase.
3
Ov
er
a
test
phase,
what
is
the
probabilit
y
that
at
least
4
individuals
ha
v
e
reacted
with
fev
er.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7829
sacados/7829
chapExoCorrec/7640
sacados/7640
chapExoCorrec/4888
sacados/4888
chapExoCorrec/4909
sacados/4909
chapExoCorrec/4939
sacados/4939
chapExoCorrec/4937
sacados/4937
Représentation 1
Représentation 2
chapExoCorrec/4929
sacados/4929
chapExoCorrec/4927
sacados/4927
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