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<MATH>P(A|B)=P(A|B \bigcap C)P(C|B)+P(A|B \bigcap C^c)P(C^c|B) </MATH> | <MATH>P(A|B)=P(A|B \bigcap C)P(C|B)+P(A|B \bigcap C^c)P(C^c|B) </MATH> | ||
| − | <MATH>P(A|B \bigcap C)P(C \bigcap B)/P(B) </MATH> | + | <MATH>P(A|B \bigcap C)P(B \bigcap C)/P(B) +P(A|B \bigcap C^c)P(B \bigcap C^c)/P(B)</MATH> |
Revision as of 19:08, 15 September 2008
The theorem of total probalility states that
$ P(A)=P(A|C)P(C)+P(A|C^c)P(C^c) $
$ P(A|B)=P(A|B \bigcap C)P(C|B)+P(A|B \bigcap C^c)P(C^c|B) $
$ P(A|B \bigcap C)P(B \bigcap C)/P(B) +P(A|B \bigcap C^c)P(B \bigcap C^c)/P(B) $
