Definition
Two events A and B are independent if the occurrence of one does not change the probability of the other. The formal condition is P(A and B) = P(A) × P(B), which is equivalent to P(A|B) = P(A) when P(B) > 0.
Independence is the property that makes probability calculations tractable: when events are independent, joint probabilities factor into products, and complex models reduce to manageable pieces. It is also the property most easily — and dangerously — assumed when it does not actually hold.
Why it matters
How it works
To check if events are independent, compare P(A and B) to the product P(A) × P(B). If they match, the events are independent and you can treat them as separate. If they do not match, the events are dependent, and you must compute joint probabilities directly or via conditioning.
For example, successive flips of a fair coin are independent — past flips give no information about future ones, which is why the gambler's fallacy is wrong. But two cards drawn without replacement from a deck are dependent — knowing the first card changes the composition of the deck for the second draw.