Independence and Mutual Exclusivity

Two events \( A \) and \( B \) are called independent if the occurrence of one does not affect the probability of the other, which means \( P(A \mid B) = P(A) \).

This is equivalent to saying \( P(A \cap B) = P(A) \cdot P(B) \), a key identity used frequently in both theory and application.

For example, flipping a coin and rolling a die are independent experiments because they don't influence one another's outcome.

In contrast, two events are mutually exclusive if they cannot occur at the same time, meaning \( A \cap B = \emptyset \), and thus \( P(A \cap B) = 0 \).

A roll of $2$ and a roll of $3$ on a single die are mutually exclusive since both outcomes cannot occur on a single trial.

Importantly, independence and mutual exclusivity are not the same; mutually exclusive events are always dependent because knowing one occurred changes the probability of the other to zero.