Independence

If $A,B \subset S$ and we have $P(A) > 0, P(B) > 0$ , then in general it is not the case that $P(A | B) = P(A)$ . We then wish to ask when is this the case? With some thought we can see that if $B$ does not effect $A$ then this equality will hold. If this is the case we say $A$ is independent of $B$ .

We can also see that if $A$ is independent of $B$ then we have

$\begin{align*} P(A | B) = \frac{P(A \cap B)}{P(B)} = P(A) \end{align*}$

we see that this implies that $P(A \cap B) = P(A)P(B)$ , and in fact we have the following definition:

Definition: Two events $A$ and $B$ are called independent if

$P(A \cap B) = P(A)P(B)$

and called dependent otherwise.

Exercise: Prove that if $A$ is independent of itself then $P(A) = 0$ or $P(A) = 1$ .

Theorem: If $A$ and $B$ are independent, then $A$ and $B^c$ are independent as well.

Proof:

Definition: Events $A,B,C$ are called independent if:

$\begin{align*} P(A \cap B) &= \\ P(A \cap C) &= \\ P(B \cap C) &= \\ P(A \cap B \cap C) &= \end{align*}$