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.
The intersection of two events A and B is the outcome where both happen simultaneously. The symbol is ∩, and the probability P(A ∩ B) is called the joint probability (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 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.
Intersection lives in Unit 4: Probability, Random Variables, and Probability Distributions, specifically Topic 4.4: 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 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.
Keep studying AP Statistics Unit 4
Mutually Exclusive Events (Unit 4)
Mutually exclusive 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)
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)
An event and its complement 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.
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 (∩) 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.
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.
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.
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.
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.
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).
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.