Why can two mutually exclusive events with nonzero probability never be independent?

## Two ideas that sound alike and are not Interviewers ask this because the everyday words pull in the wrong direction. In ordinary speech, 'independent' and 'unrelated' and 'separate' all blur together, and disjoint events feel maximally 'separate'. Probability uses the words precisely, and the two definitions are about different things. **Mutual exclusivity (also called disjointness) is a statement about outcomes.** Two events `A` and `B` are mutually exclusive when they cannot occur on the same trial — as sets of outcomes they do not intersect, so `P(A and B) = 0`. Rolling a single die once, 'the roll is 3' and 'the roll is even' are mutually exclusive: no single outcome is in both. **Independence is a statement about information.** `A` and `B` are independent when the multiplication rule holds in its unconditional form: ``` P(A and B) = P(A) * P(B) ``` Equivalently, whenever `P(B) > 0`, independence means `P(A|B) = P(A)`, where the conditional probability is defined as ``` P(A|B) = P(A and B) / P(B) ``` The conditional form is the one to say out loud in an interview, because it names what independence actually claims: knowing that `B` happened does not move your probability for `A` at all. ## The one-line proof Suppose `A` and `B` are mutually exclusive and both have positive probability. Then `P(A and B) = 0` by exclusivity, but `P(A) * P(B) > 0` because a product of two positive numbers is positive. The two quantities differ, so the independence condition fails. Equivalently, `P(A|B) = 0 / P(B) = 0`, while `P(A) > 0` — the conditional probability collapses to zero, the largest possible move away from the unconditional value. So far from being unrelated, mutually exclusive events are perfectly informative about each other in one direction: observing `B` proves `A` did not occur. ## The degenerate case If `P(A) = 0` (or `P(B) = 0`), then `P(A and B) = 0 = P(A)P(B)`, so the events are technically both mutually exclusive and independent. This is not a deep fact — a probability-zero event carries no information because it is (almost) never observed. Mentioning the edge case shows care; leading with it instead of the main argument does not. ## Getting the arithmetic right The two definitions also lead to different formulas, and mixing them up is the most common downstream error. - **Addition** — `P(A or B) = P(A) + P(B) - P(A and B)` always. For mutually exclusive events the last term is zero, so `P(A or B) = P(A) + P(B)`. Exclusivity is what licenses simply adding. - **Multiplication** — `P(A and B) = P(A) * P(B|A)` always. For independent events `P(B|A) = P(B)`, so the joint is the product of marginals. Independence is what licenses simply multiplying. Said compactly: exclusivity simplifies 'or', independence simplifies 'and'. A candidate who can state that sentence has the distinction. ## A concrete contrast Draw two cards from a shuffled 52-card deck. - *Without replacement*, 'the first card is an ace' and 'the second card is an ace' are **not independent**: given the first is an ace, only 3 aces remain among 51 cards, so the conditional probability drops from 4/52 to 3/51. They are not mutually exclusive either — both can be aces. - *With replacement* (shuffle the first card back in), the same two events are **independent**: the conditional probability is 4/52 either way, and the joint is `(4/52) * (4/52)`. - On a **single** draw, 'the card is an ace' and 'the card is a king' are mutually exclusive — and therefore dependent, because learning it is a king drives the probability of ace to zero. ## Related traps worth naming Independence is preserved under complements: if `A` and `B` are independent, so are `A` and 'not `B`', since `P(A and not B) = P(A) - P(A and B) = P(A) - P(A)P(B) = P(A)(1 - P(B))`. Exclusivity has no such property — the complement of a disjoint event usually overlaps heavily. Also note that independence is a property of a probability assignment, not of physical unrelatedness. Two events can be independent by numerical coincidence in one model and dependent in another, so the honest way to answer 'are these independent?' is always to check the product rule, never to reason from whether the events feel connected.