Econ 3190
Fall 2011
HW #1 Solution
1. Denote event in (a), (b), (c) and (d) as event
E.
(a)
P
(
E
) =
P
(
A
∪
B
) =
P
(
A
) +
P
(
B
)

P
(
A
∩
B
)
(b)
P
(
E
) =
P
(
A
∪
B
)

P
(
A
∩
B
) =
P
(
A
) +
P
(
B
)

2
P
(
A
∩
B
)
(c)
P
(
E
) =
P
(
A
∪
B
) =
P
(
A
) +
P
(
B
)

P
(
A
∩
B
)
(d)
P
(
E
) = 1

P
(
A
∩
B
)
2. Following De Morgan’s Laws
(a)
A
∪
(
A
c
∩
B
) = (
A
∪
A
c
)
∩
(
A
∪
B
) =
S
∩
(
A
∪
B
) = (
A
∪
B
)
(b) (
A
∩
B
)
∪
(
A
c
∩
B
) = (
A
∪
A
c
)
∩
B
=
B
3.
(a) Not necessarily.
We know
A
∩
B
=
φ,
but can we conclude
A
c
∩
B
c
=
φ
as well?
We have to consider 2 cases:
i.
A
and
B
are collectively exhaustive, that is to say
A
∪
B
=
S.
From above,
A
c
∩
B
c
= (
A
∪
B
)
c
=
S
c
=
φ.
As such, if
A
and
B
are collectively exhaustive and
A
and
B
are mutually exclusive,
A
c
and
B
c
are mutually exclusive as well.
ii.
A
and
B
are
NOT
collectively exhaustive, that is to say
A
∪
B
⊂
S.
And
A
c
∩
B
c
= (
A
∪
B
)
c
6
=
S
c
=
˙
φ
.
Hence, in this case,
A
c
and
B
c
are
NOT
mutually exclusive.
(b) If
A
and
B
are collectively exhaustive and
A
and
B
are mutually
exclusive,
A
c
and
B
c
are mutually exclusive as well.
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 '07
 HONG
 Probability, Cooperative, Af ternoon

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