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A set of lecture notes from rice university's elec 303 course, taught by farinaz koushanfar in fall'08. The notes cover the topic of conditional probability, including definitions, examples, and applications. Students will learn about the multiplication rule, total probability theorem, and bayes rule, as well as how to calculate conditional probabilities using these theorems.
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ELEC 303, Koushanfar, Fall’
Farinaz Koushanfar ECE Dept., Rice University Aug 28, 2008
ELEC 303, Koushanfar, Fall’
ELEC 303, Koushanfar, Fall’
ELEC 303, Koushanfar, Fall’
ELEC 303, Koushanfar, Fall’
P(A|B) = P(A∩B) / P(B)
P(A∩B) = P(B).P(A|B) = P(A).P(B|A)
Ω A B
ELEC 303, Koushanfar, Fall’
ELEC 303, Koushanfar, Fall’
Slide courtesy of Prof. Dahleh, MIT
ELEC 303, Koushanfar, Fall’
P(A)=0.
P(Ac)=0.
Multiplication rule: Assuming that all of the conditioning events have positive probability,
... ( | )
( ) ( ) ( | ) ( | )... (^11)
1 1 2 1 3 1 2 n ni i
ni i PA A
P A PA PA A PA A A =^ −
= = ∩ I
I
ELEC 303, Koushanfar, Fall’
P(B) = P(A 1 ∩B) + P(A 2 ∩B) + P(A 3 ∩B) = P(A 1 )P(B|A 1 )+P(A 2 )P(B|A 2 )+P(A 3 )P(B|A 3 )
A 1 A 2 A 3
B
B
B
Bc
Bc
Bc
A 1 ∩B
A 2 ∩B
A 3 ∩B (^) Figure courtesy of Bertsekas&Tsitsiklis, Introduction to Probability, 2008
ELEC 303, Koushanfar, Fall’
Example courtesy of Prof. Dahleh, MIT
ELEC 303, Koushanfar, Fall’
Example courtesy of Prof. Dahleh, MIT
ELEC 303, Koushanfar, Fall’
{Absent and decision is present}
Example courtesy of Prof. Dahleh, MIT
ELEC 303, Koushanfar, Fall’
Example courtesy of Prof. Dahleh, MIT
ELEC 303, Koushanfar, Fall’
Example courtesy of Prof. Dahleh, MIT
ELEC 303, Koushanfar, Fall’
Let A 1 ,A 2 ,…,An be disjoint events that form a partition of the sample space, and assume that P(Ai) > 0, for all i. Then, for any event B such that P(B)>0, we have
( ) ( | ) ... ( ) ( | )
( ) ( | ) ( )
( | ) ( ) ( | ) 1 1 n n
i i i i i P A P B A P A PB A
P A P B A P B
P A B P A P B A
= =
