1
Probability Homework Solution #3
2.48
(a) The results follow directly from the definition of conditional probability.
[
]
[

]
[
]
P A
B
P A B
P B
∩
=
If
,
then
[
]
0 by Corollary 3 and thus
[

]
0.
[
]
If
,
then
, and thus [

]
.
[
]
[
]
If
,
then
, and thus [

]
1.
[
]
A
B
P A
B
P A B
P A
A
B
A
B
A
P A B
P B
P B
A
B
A
B
B
P A B
P B
∩
= ∅
∩
=
=
⊂
∩
=
=
⊃
∩
=
=
=
(b)
[
]
If
[

]
[
],
then multiplying both sides by
[
], we have:
[
]
[
]
[
] [
]
[
]
[
] [
]
We then also have that
[

]
[
].
[
]
[
]
We conclude that if
[

]
[
], then
and
tend t
P A
B
P A B
P A
P B
P B
P A
B
P A P B
P A
B
P A P B
P B
A
P B
P A
P A
P A B
P A
B
A
∩
=
>
∩
>
∩
=
>
=
>
o occur jointly.
2.49
(i)
[
]
[

]
for
[
]
0
[
]
0
[
],
0
[
]
0
[

]
,
[
]
[
]
[

]
1
Therefore,
0
[

]
1
P A
B
P A B
P B
P B
P A
B
P B
P A B
A
B
B
P A
B
P B
P A B
P A B
∩
=
>
≤
∩
<
⇒
≤
∩
⊂
∩
≤
⇒
≤
≤
≤
(ii)
[
]
[
]
[

]
1
[
]
[
]
P S
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document