2.4 Independent Events

Independent Events

Events $A$ and $B$ in probability space $S$ are independent if the occurrence of one of them does not influence the occurrence of the other.
In other words, $B$ is independent of $A$ if $P (B) = P (B | A)$
$⟹ P (A \cap B) = P (A) P (B | A) = P (A) P (B)$
So we can use the following definition of independence:
Events $A$ and $B$ are independent if $P (A \cap B) = P (A) P (B)$ ; otherwise they are dependent

Example

A fair coin is tossed 3 times. Let $A$ be the event that there is at least one head, and $B$ the event that there is at least one tail. Are events $A$ and $B$ independent?

$| S | = 2^{3} = 8$
$P (A) = \frac{7}{8}$
$P (B) = \frac{7}{8}$
$A \cap B =$ The event that there is at least one head and at least one tail
$P (A \cap B) = \frac{6}{8} = \frac{3}{4}$
$P (A \cap B) \neq P (A) P (B)$ and hence events $A$ and $B$ are dependent