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For example, given a coin, are the two outcomes independent? | For example, given a coin, are the two outcomes independent? | ||
| − | <math> P( \lbrace C_1=H \rbrace \bigcap \lbrace C_2 =H \rbrace )</math> | + | <math> P( \lbrace C_1=H \rbrace \bigcap \lbrace C_2 =H \rbrace ) = 1/4</math> |
| − | <math> P( C_1=H ) \times P(C_2=H)</math> | + | <math> P( C_1=H ) \times P(C_2=H) = 1/2 \times 1/2 = 1/4</math> |
| + | |||
| + | Since the product of the two probabilities is equal to overall probability, the events are independent. | ||
[http://en.wikipedia.org/wiki/Help:Formula] | [http://en.wikipedia.org/wiki/Help:Formula] | ||
Revision as of 10:24, 8 September 2008
Independence
In Two Events
$ P(A \bigcap B) = P(A) \times P(B) $
For example, given a coin, are the two outcomes independent?
$ P( \lbrace C_1=H \rbrace \bigcap \lbrace C_2 =H \rbrace ) = 1/4 $
$ P( C_1=H ) \times P(C_2=H) = 1/2 \times 1/2 = 1/4 $
Since the product of the two probabilities is equal to overall probability, the events are independent.
In Multiple Events
$ \bigcap_i A_i = \prod_i P(A_i) $
For i $ \in $ S
Conditional Probability
A & B are conditionally independent given C if the following formula holds true.
$ P(A \bigcap B|C) = P(A|C) \times P(B|C) $
