Baye’s Theorem:
Partition of a Set
A family of sets A1, A2, …, An is said to form a partition of a set A if
A1,A2,...An are non empty sets
Ai 
Aj (i
A=A1
23.....
An
Bayes' theorem.
Theorem is a theorem of probability theory originally stated by the Reverend Thomas Bayes.
Let A1, A2, ... , An be a set of mutually exclusive events that together form the sample space S. Let B be any event from the same sample space, such that P(B) > 0. Then,
|
P( Ak | B ) = |
P( Ak ∩ B ) P( A1 ∩ B ) + P( A2 ∩ B ) + . . . + P( An ∩ B ) |
Note: Invoking the fact that P( Ak ∩ B ) = P( Ak )P( B | Ak ), Baye's theorem can also be expressed as
|
P( Ak | B ) = |
P( Ak ) P( B | Ak ) P( A1 ) P( B | A1 ) + P( A2 ) P( B | A2 ) + . . . + P( An ) P( B | An ) |
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