4.1.
PROBABILITY SPACES
79
20. Prove Proposition 1.2.6 (7).
21. Let (Ω
,
F
,
P
) be a probability space and
A
∈ F
such that
P
(
A
)
>
0.
Show that (Ω
,
F
, μ
) is a probability space, where
μ
(
B
) :=
P
(
B

A
)
.
22. Let (Ω
,
F
,
P
) be a probability space. Show that if
A, B
∈ F
are inde
pendent this implies
(a)
A
and
B
c
are independent,
(b)
A
c
and
B
c
are independent.
23. Let (Ω
,
F
,
P
) be the model of rolling two dice, i.e. Ω =
{
(
k, l
) : 1
≤
k, l
≤
6
}
,
F
= 2
Ω
, and
P
((
k, l
)) =
1
36
for all (
k, l
)
∈
Ω
.
Assume
A
:=
{
(
k, l
) :
l
= 1
,
2 or 5
}
,
B
:=
{
(
k, l
) :
l
= 4
,
5 or 6
}
,
C
:=
{
(
k, l
) :
k
+
l
= 9
}
.
Show that
P
(
A
∩
B
∩
C
) =
P
(
A
)
P
(
B
)
P
(
C
)
.
Are
A, B, C
independent ?
24.
(a) Let Ω :=
{
1
, ...,
6
}
,
F
:= 2
Ω
and
P
(
B
) :=
1
6
card(
B
). We define
A
:=
{
1
,
4
}
and
B
:=
{
2
,
5
}
. Are
A
and
B
independent?
(b) Let Ω :=
{
(
k, l
) :
k, l
= 1
, ...,
6
}
,
F
:= 2
Ω
and
P
(
B
) :=
1
36
card(
B
).
We define
A
:=
{
(
k, l
) :
k
= 1 or
k
= 4
}
and
B
:=
{
(
k, l
) :
l
= 2 or
l
= 5
}
.
Are
A
and
B
independent?
25. In a certain community 60 % of the families own their own car, 30 %
own their own home, and 20% own both (their own car and their own
home).
If a family is randomly chosen, what is the probability that
this family owns a car or a house but not both?
26. Prove Bayes’ formula: Proposition 1.2.15.
27. Suppose we have 10 coins which are such that if the
i
th one is flipped
then heads will appear with probability
i
10
, i
= 1
,
2
, . . .
10
.
One chooses
randomly one of the coins. What is the conditionally probability that
one has chosen the fifth coin given it had shown heads after flipping?
Hint:
Use Bayes’ formula.