Two events are independent when knowing that one occurred does not change the probability of the other.

For events with positive probability,

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

expresses this idea directly.

Using the multiplication rule, independence is equivalently characterized by

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

Example

Roll a fair die and flip a fair coin. Let $A$ be the event “the die shows an even number” and $B$ the event “the coin shows heads”.

The die result does not affect the coin result, so

$$P(A)=\frac12,\qquad P(B)=\frac12,$$

and

$$P(A\cap B)=\frac14=P(A)P(B).$$

Independence is not mutual exclusivity

Mutually exclusive events cannot occur together, so

$$P(A\cap B)=0.$$

If both events have positive probability, this is incompatible with independence because $P(A)P(B)>0$.

Thus “cannot happen together” and “do not affect each other's probabilities” describe very different relationships.

More than two events

For repeated experiments, pairwise independence alone may not be enough. A collection of events is mutually independent when every relevant intersection has probability equal to the product of the individual probabilities.

Independence is therefore a property of a probability model, not merely an informal statement that events seem unrelated.