p. 31
Conditional Probability
•
Q
: Should the following probabilities be different?
Event =
王建民本季戰績至少獲得
6
次勝投
Event = rain tomorrow
P
=?? if no information about where you are staying
P
=?? if you are staying in a desert
P
=?? if a typhoon will hit the place you stay tomorrow
•
Q
: What causes the differences?
For an event,
new information
(i.e., some other event has
occurred) could change its probability
We call the altered probability a conditional probability
• Mathematical Definition: If
A
and
B
are two events in a sample
space
and
P
(
A
)>0, then
is called the
conditional probability
of
B
given
A
.
P
(
B

A
)
≡
P
(
A
∩
B
)
P
(
A
)
P
=?? in the beginning
of the season
P
=?? in the middle
of the season
P
=??
now
p. 32
In the classical model,
P
(
A
) = #
A
/#
and
P
(
A
Å
B)=#(
A
Å
B
)/#
Example: A family is known to have 2 children,
at least one of
whom is a girl
.
Q
: Probability that the other is a boy=??
= {
bb
,
bg
,
gb
,
gg
}
A
={
bg
,
gb
,
gg
} and
B
={
bb
,
bg
,
gb
}
P
(
B

A
) = #(
A
Å
B
)/#
A
= 2/3
Note: #
is reduced to #
A
.
⇒
P
(
B

A
)=
#(
A
∩
B
)
/
#
Ω
#
A/
#
Ω
=
#(
A
∩
B
)
#
A
.
In effect by conditioning,
Ω
→
A,
B
→
B
∩
A.
and, for an arbitrary event
B
in
to occur when
A
has occurred, we need that both
A
and
B
occur
together, i.e.,
we are restricting the sample space from
to
A
,
i.e.,
The division by
P
(
A
) in the definition above rescales
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 Fall '11
 ShaoWeiCheng
 Conditional Probability, Probability, Probability theory, NR white balls

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