# 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) $ for all sets $ S $ of events.

## 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) $