--- title: "Intersection — AP Stats Definition & Probability Guide" description: "Intersection P(A ∩ B) is the probability that events A and B both happen. It's the heart of mutually exclusive events (Topic 4.4) and joint probability in AP Stats." canonical: "https://fiveable.me/ap-stats/key-terms/intersection" type: "key-term" subject: "AP Statistics" unit: "Unit 2" --- # Intersection — AP Stats Definition & Probability Guide ## Definition In AP Statistics, the intersection of events A and B is the event where both occur at the same time, written A ∩ B. Its probability, P(A ∩ B), is called the joint probability, and if A and B are mutually exclusive (disjoint), then P(A ∩ B) = 0 because they can't happen together. ## What It Is The intersection of two [events](/ap-stats/unit-2/estimating-probabilities-using-simulation/study-guide/ABwpnnUf4VCXeVbB9q72 "fv-autolink") A and B is the outcome where **both happen simultaneously**. The symbol is ∩, and the probability P(A ∩ B) is called the **[joint probability](/ap-stats/key-terms/joint-probability "fv-autolink")** (that's straight from the CED, VAR-4.C.1). If you're a Venn diagram person, the intersection is the overlapping middle region where the two circles share outcomes. The intersection is also how [AP Stats](/ap-stats "fv-autolink") defines **mutually exclusive (disjoint) events**. Two events are mutually exclusive if they cannot occur at the same time, which means their intersection is empty and P(A ∩ B) = 0 (VAR-4.C.2). In the Venn diagram picture, the circles don't touch at all. So the intersection isn't just a piece of notation. It's the test you run to decide whether two events overlap. ## Why It Matters Intersection lives in **[Unit 4](/ap-stats/unit-4 "fv-autolink"): Probability, Random Variables, and Probability Distributions**, specifically **[Topic 4.4](/ap-stats/unit-4/setting-up-test-for-population-mean/study-guide/1gAGgo2P3abc5G0sLvMr "fv-autolink"): Mutually Exclusive Events**. The learning objective is 4.4.A, which asks you to explain why two events are or are not mutually exclusive, and the only way to do that is to reason about the intersection. If you can find even one outcome in A ∩ B, the events are NOT mutually exclusive. If the intersection is impossible, they are. Beyond Topic 4.4, the intersection threads through almost everything else in Unit 4. The [addition rule](/ap-stats/key-terms/addition-rule "fv-autolink") subtracts P(A ∩ B) so the overlap isn't double-counted, conditional probability puts P(A ∩ B) in the numerator of its formula, and checking independence means comparing P(A ∩ B) to P(A)·P(B). Get comfortable with ∩ now and the rest of the probability unit gets much easier. ## Connections ### Mutually Exclusive Events (Unit 4) [Mutually exclusive](/ap-stats/unit-2/mutually-exclusive-events/study-guide/iBljImMDLJ8bSWeWXbi6 "fv-autolink") is defined BY the intersection. Two events are disjoint exactly when P(A ∩ B) = 0. The classic exam example is scoring above 80 and below 70 on the same exam. One student can't do both, so the intersection is empty. ### Union (Unit 4) Union (∪, 'A or B') and intersection (∩, 'A and B') are partners in the addition rule, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). You subtract the intersection because outcomes in the overlap got counted twice. When events are disjoint, that subtraction term is just 0. ### [Joint Probability (Unit 4)](/ap-stats/key-terms/joint-probability) Joint probability is literally another name for P(A ∩ B). On two-way tables, the joint probability of a row event and a column event is the count in a single cell divided by the grand total. That cell IS the intersection. ### [Complement (Unit 4)](/ap-stats/key-terms/complement) An event and its [complement](/ap-stats/key-terms/complement "fv-autolink") are the cleanest example of mutually exclusive events. A and A' share no outcomes, so P(A ∩ A') = 0, and together they cover the whole sample space. ## On the AP Exam Intersection shows up mostly in multiple-choice questions about Topic 4.4, where the stem describes two events and asks whether they're mutually exclusive, or asks under what condition they would be. The move is always the same. Hunt for an outcome that lands in both events. For example, with E = 'heads on a coin' and F = 'rolling less than 4 on a die,' the outcome (heads, 2) sits in both, so E and F are not mutually exclusive. With C = 'studies with classical music' and J = 'studies with jazz,' they're only mutually exclusive if no student does both, meaning P(C ∩ J) = 0. No released FRQ has asked you to define 'intersection' by itself, but probability FRQs routinely require it in disguise. You'll compute P(A ∩ B) from a two-way table, plug it into the addition rule, or use it to justify whether two events are independent. Always justify a 'mutually exclusive' claim with the actual condition, P(A ∩ B) = 0, not just intuition. ## Intersection vs Union Intersection (∩) means BOTH events happen ('A and B'); union (∪) means AT LEAST ONE happens ('A or B'). A quick memory hook is that ∩ looks like an 'n' for 'aNd' and ∪ looks like a 'u' for 'Union/or.' The classic trap answer swaps them, so when a problem says 'or,' reach for union and the addition rule; when it says 'and' or 'both,' you want the intersection. ## Key Takeaways - The intersection A ∩ B is the event where both A and B occur, and P(A ∩ B) is called the joint probability (VAR-4.C.1). - Two events are mutually exclusive (disjoint) if and only if they cannot happen at the same time, which means P(A ∩ B) = 0 (VAR-4.C.2). - To decide whether events are mutually exclusive, look for a single outcome that belongs to both events; finding one proves they are NOT disjoint. - In the addition rule, P(A ∪ B) = P(A) + P(B) − P(A ∩ B), you subtract the intersection so the overlap isn't counted twice. - On a two-way table, P(A ∩ B) is the count in one cell divided by the grand total. - Mutually exclusive and independent are not the same thing; disjoint events with nonzero probabilities are actually dependent, because one happening guarantees the other doesn't. ## FAQs ### What is the intersection in AP Stats? The intersection of events A and B, written A ∩ B, is the event where both A and B happen at the same time. Its probability, P(A ∩ B), is called the joint probability, and it's the foundation of Topic 4.4 on mutually exclusive events. ### If two events are mutually exclusive, is their intersection zero? Yes. That's the definition in the CED (VAR-4.C.2). Mutually exclusive (disjoint) events cannot occur together, so P(A ∩ B) = 0. For example, 'scored above 80' and 'scored below 70' on the same exam can't both be true for one student. ### Does mutually exclusive mean the same thing as independent? No, and this is the biggest trap in Unit 4. Mutually exclusive means the events can't happen together, so P(A ∩ B) = 0. Independent means P(A ∩ B) = P(A)·P(B). If both events have nonzero probability, disjoint events are automatically dependent, since knowing one happened tells you the other didn't. ### What's the difference between intersection and union? Intersection (∩) is 'and,' the event where both occur. Union (∪) is 'or,' the event where at least one occurs. They're linked by the addition rule, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). ### How do I find P(A ∩ B) on a problem? It depends on the setup. On a two-way table, divide the single cell where A and B overlap by the grand total. If A and B are independent, multiply P(A)·P(B). If they're mutually exclusive, P(A ∩ B) = 0 with no calculation needed. ## Related Study Guides - [2.5 Mutually Exclusive Events](/ap-stats/unit-2/mutually-exclusive-events/study-guide/iBljImMDLJ8bSWeWXbi6) ## Structured Data ```json {"@context":"https://schema.org","@graph":[{"@type":"LearningResource","@id":"https://fiveable.me/ap-stats/key-terms/intersection#resource","name":"Intersection — AP Stats Definition & Probability Guide","url":"https://fiveable.me/ap-stats/key-terms/intersection","learningResourceType":"Concept explainer","educationalLevel":"AP® / High School","about":{"@id":"https://fiveable.me/ap-stats/key-terms/intersection#term"},"audience":{"@type":"EducationalAudience","educationalRole":"student"},"isPartOf":{"@type":"Collection","name":"AP Statistics Key Terms","url":"https://fiveable.me/ap-stats/key-terms"},"publisher":{"@type":"Organization","name":"Fiveable","url":"https://fiveable.me"}},{"@type":"DefinedTerm","@id":"https://fiveable.me/ap-stats/key-terms/intersection#term","name":"Intersection","description":"In AP Statistics, the intersection of events A and B is the event where both occur at the same time, written A ∩ B. 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