Independent Events

A coin has no memory because each flip is a fresh, physical event unconnected to the last one. Five tails in a row don't change the coin's shape or the forces acting on it, so the probability of heads on flip six is still 0.5. That's the intuition behind independence: one event's outcome carries no information about another's.

To check independence formally, compare P(A given B) with P(A). If they match, B happening told you nothing new about A, and the events are independent. If they differ, B changed the odds of A, so the events are dependent. This same comparison works in reverse: you could also check whether P(B given A) equals P(B). When events are independent, there's a shortcut too: P(A and B) = P(A) x P(B), with no conditioning needed.

This idea sits right next to mutually exclusive events, and the two get confused constantly. Independence is about one event giving no information about another. Mutual exclusivity is about two events being unable to happen together at all. Those are nearly opposite ideas, not the same one. Independence also connects forward to the binomial setting, where every trial must be independent of the others for the binomial formula to apply — that's why coin flips and dice rolls work, but drawing cards without replacement does not.