Independence
In Two Events
Two events A and B are independent if the following formula holds:
$ 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
$ P( \bigcap_{i \in S} A_i ) = \prod_{i \in S} P(A_i) $
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) $
