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week 3
1
Conditional Probability
•
Idea – have performed a chance experiment but don’t know the outcome
(
ω
), but have some partial information (event
A
) about
ω
.
Question: given this partial information what’s the probability that the
outcome is in some event
B
?
•
Example:
Toss a coin 3 times. We are interested in event
B
that there are 2 or more
heads. The sample space has 8 equally likely outcomes.
The probability of the event
B
is …
Suppose we know that the first coin came up H. Let
A
be the event the first
outcome is H. Then
and
The conditional probability of
B
given
A
is
{}
TTT
TTH
THT
HTT
THH
HTH
HHT
HHH
,
,
,
,
,
,
,
=
Ω
( )
()
A
P
B
A
P
∩
=
=
8
4
8
3
4
3
{ }
HTT
HTH
HHT
HHH
A
,
,
,
=
{ }
HTH
HHT
HHH
B
A
,
,
=
∩
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View Full Documentweek 3
2
•
Given a probability space (
Ω
,
F
,
P
) and events
A, B
F
with P(
A
) > 0
The conditional probability of B given the information that
A
has occurred is
•
Example
:
We toss a die. What is the probability of observing the number 6 given that
the outcome is even?
•
Does this give rise to a valid probability measure?
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 Fall '10
 MCDUNNOUGH
 Conditional Probability, Probability

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