1.017/1.010 Class 6
Conditional Probability and Bayes Theorem
Conditional Probability
If two events
A
and
B
are not independent we can gain information about
P
(
A
)
if
we know that an event in
B
has occurred.
This is reflected in
conditional
probability
of
A
given
B
, written as
P
(
AB
)
:
)
(
)
(
)

(
B
P
AB
P
B
A
P
=
The
unconditional
probability
P
(
A
)
is often called the
a priori
probability while
the
conditional
probability
P
(
A

B
)
is often called the
a posteriori
probability.
Note that conditioning may take place in either direction:
P
(
AB
) =
P
(
A

B
)
P
(
B
) =
P
(
B

A
)
P
(
A
)
Conditional probabilities are valid probability measures that satisfy all the
fundamental axioms.
If
A
and
B
are independent:
P
(
AB
)
= P
(
A
)
Example:
A
= {Algae bloom occurs }
B
= {Daily average water temperature above 25 deg. C)
Obtain probabilities from long record of daily algae and temperature
observations:
Suppose
P
(
A
)
= 0.01
,
P
(
B
)
= 0.15
,
P
(
A, B
)
= 0.005
Then:
033
.
0
15
.
0
005
.
0
)
(
)
(
)

(
=
=
=
B
P
AB
P
B
A
P
Probability of a bloom increases significantly if we know that temperature is
above 25 deg. C.
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 Spring '05
 GeorgeKocur
 Conditional Probability, Probability, Probability theory, Bayes Theorem

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