When are two events independent?

The meaning of independence

Two events are independent when knowing that one occurred gives no information about whether the other will happen. In formal terms, the conditional probability P(A|B) equals the unconditional probability P(A). Tossing a fair coin twice provides a simple example: getting heads on the first flip does not change the 1/2 probability of getting heads on the second flip because the coin has no memory.

Independence is not the same as causation. Two events can be statistically dependent simply because they are measured within the same group or outcome space, even if neither event causes the other.

The multiplication rule as a mathematical test

You do not determine independence by intuition or by reading the story. You determine it by calculation. Two events A and B are independent if and only if P(A and B) = P(A) * P(B).

To test whether two events are independent, follow three steps: first, find the actual probability of their intersection, P(A and B); second, compute the product P(A) * P(B); third, compare the two values. If the numbers match exactly, the events are independent. If they differ, the events are dependent.

For example, consider rolling a single fair six-sided die. Let event A be getting an even number {2, 4, 6}, so P(A) = 3/6 = 1/2. Let event B be rolling a number less than 3 {1, 2}, so P(B) = 2/6 = 1/3. The product P(A) * P(B) is (1/2) * (1/3) = 1/6. The only outcome in both events is {2}, which has probability 1/6. Because 1/6 = 1/6, the events are independent.

Two common mistakes to avoid

First, never assume sampling without replacement is independent. If a bag holds 3 red marbles and 2 blue marbles, the probability of drawing red first is 3/5. Drawing without replacement leaves 4 marbles in the bag, changing the second draw's denominator and creating dependence.

Second, do not confuse disjoint events with independent events. Disjoint (mutually exclusive) events cannot happen together, meaning P(A and B) = 0. If two disjoint events each have positive probability, knowing one happened guarantees the other did not. That is complete dependence, never independence.