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1
Conditional Probability
[]
[]
PA B
PAB
PB
=
A
B
S
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Example. Toss 2 dice and take the sum.
: odd toss,
{3, 5, 7, 9, 11}
: {2,3}
[
]
[{3}]
2 / 36
2
[]
[ ]
[{2,3}]
1/ 36
2 / 36
3
A
B
PA B
P
PAB
PB
P
=
=
=
=
+
Homework.
A couple has two kids and at least one is a boy. What is the probability that both are boys?
Ans. 1/3.
3
Bayes Rule
[]
[]
PA B
PB A
PA
PB
=
Notice that
[
] [][] [][]
PA B PAB PB PB A PA
=
=
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Partitioning of the Sample Space
The events
12
,
,,
K
AA
A
partition the sample space
S
iff
ij
A
A
for all i and j i
j
φ
=
≠
and
1
K
j
j
AS
=
=
.
Notice that in this case,
1
1
[
][
]
1
K
K
jj
j
j
P
APAP
S
=
=
=
=
=
∑
1
A
2
A
3
A
5
Conditional Probability after Partitioning
We want to find
[ ].
PB
Suppose the event
B
depends on some other events
.
j
A
Find the events that partition the sample space
S
.
123
1
2
3
,
,
are disjoint and
AAA
A A A S
=
.
1
23
Note that
,
,
are disjoint
BA
.
11
2
2
3
3
Therefore
[
]
[
][
]
[  ][ ] [  ][ ] [  ][ ]
PB A
PB A PA
=
++
=
1
A
2
A
3
A
B
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Example. Defect
A store gets ½ of its items from Factory 1, ¼ from Factory 2, and ¼ from Factory 3.
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This note was uploaded on 11/23/2010 for the course EE EE528 taught by Professor Majungsoo during the Spring '10 term at Korea Advanced Institute of Science and Technology.
 Spring '10
 Majungsoo

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