independence of events
Two events are independent when knowing one happened tells you nothing about the other. The classic picture is two separate coin tosses: learning the first landed heads does not change the chance the second is heads — it stays 1/2. Independence is the formal way of saying 'these two things do not influence each other's probability'.
The clean definition is multiplicative: A and B are independent exactly when P(A and B) = P(A) times P(B). Equivalently, whenever P(B) is not zero, P(A given B) = P(A) — conditioning on B leaves A's probability unchanged. The two phrasings agree, and the multiplicative one is preferred because it stays sensible even when a probability is zero. To test independence you do not appeal to intuition about 'connected' events; you check whether the joint probability really equals the product of the individual probabilities.
The single most important warning: independent is NOT the same as mutually exclusive (disjoint). Disjoint events cannot happen together, so if one occurs the other definitely does not — that is the strongest possible dependence, not independence. In fact, two events that each have positive probability cannot be both independent and mutually exclusive, since P(A and B) = 0 for disjoint events but P(A)P(B) > 0 for independent ones. People constantly conflate the two; keep them firmly apart.
Roll one fair die. Let A = 'even' and B = 'at most 4'. P(A) = 3/6, P(B) = 4/6, and P(A and B) = P(get 2 or 4) = 2/6. Check: P(A)P(B) = (1/2)(2/3) = 1/3 = 2/6. They match, so A and B are independent here — even though both describe the same single roll.
Independence is a numerical fact (P(A and B) = P(A)P(B)), not a story about physically separate events.
Disjoint with positive probabilities is the OPPOSITE of independent. And independence depends on the probabilities, not on whether the events 'seem' related — the die example proves two facets of one roll can be independent.