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.

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In Multiple Events

$ \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) $