Problem 1.2.1 Solution
(a) An outcome specifies whether the fax is high
(
h
)
, medium
(
m
)
,orlow
(
l
)
speed, and whether
the fax has two
(
t
)
pages or four
(
f
)
pages. The sample space is
S
= {
ht
,
hf
,
mt
,
mf
,
lt
,
lf
}
.
(1)
(b) The event that the fax is medium speed is
A
1
={
,
}
.
(c) The event that a fax has two pages is
A
2
,
,
}
.
(d) The event that a fax is either high speed or low speed is
A
3
,
,
,
}
.
(e) Since
A
1
∩
A
2
}
and is not empty,
A
1
,
A
2
, and
A
3
are not mutually exclusive.
(f) Since
A
1
∪
A
2
∪
A
3
= {
,
,
,
,
,
} =
S
,
(2)
the collection
A
1
,
A
2
,
A
3
is collectively exhaustive.
Problem 1.2.2 Solution
(a) The sample space of the experiment is
S
= {
aaa
,
aaf
,
afa
,
faa
,
ffa
,
faf
,
aff
,
fff
}
.
(1)
(b) The event that the circuit from
Z
fails is
Z
F
= {
,
,
,
}
.
(2)
The event that the circuit from
X
is acceptable is
X
A
= {
,
,
,
}
.
(3)
(c) Since
Z
F
∩
X
A
,
} 6=
φ
,
Z
F
and
X
A
are not mutually exclusive.
(d) Since
Z
F
∪
X
A
,
,
,
,
,
S
,
Z
F
and
X
A
are not collectively
exhaustive.
(e) The event that more than one circuit is acceptable is
C
= {
,
,
,
}
.
(4)
The event that at least two circuits fail is
D
= {
,
,
,
}
.
(5)
(f) Inspection shows that
C
∩
D
=
φ
so
C
and
D
are mutually exclusive.
(g) Since
C
∪
D
=
S
,
C
and
D
are collectively exhaustive.
3