Definition
Two events $A$ and $B$ are independent if the conditional probability of one event given the other is just the probability of the event, i.e. \(P(B|A) = P(B) \text{ and } P(A|B) = P(A).\)
Intuition
If two events are independent, then the occurrence of one has no impact on the likelihood of the other. A good example of independent events is two dice, one after another. You can probably convince yourself that the outcome of the first roll has no effect on the outcome of the second.
Properties
Multiplication Rule
The probability of the intersection of two independent events is just the product of their individual probabilities:
\[P(A \cap B) = P(A) P(B).\]We can derive this quickly with conditional probability. Suppose that events $A$ and $B$ are independent. Then
\[P(A|B) = \frac{P(A \cap B)}{P(B)} = P(A) \Rightarrow P(A \cap B) = P(A)P(B).\]The multiplication rule is both a necessary and sufficient condition for independence, and is often a much easier way to prove the independence of two events than the definition.