Union vs intersection in probability

Union vs intersection: the short answer

These are the two ways to combine events. The union of $A$ and $B$, written $A \cup B$, is the event that $A$ happens, or $B$ happens, or both. The intersection, written $A \cap B$, is the event that $A$ and $B$ both happen at the same time.

The one-word tests are 'or' for union and 'and' for intersection. They are linked by the general addition rule, which uses the intersection to correct the union, so getting the two straight is the first step in almost every probability problem.

What a union is

The union $A \cup B$ is the event that $A$ or $B$ (or both) occurs. The 'or' here is inclusive: outcomes in $A$ only, in $B$ only, and in both all count as part of the union. Rolling an even number or a number greater than 4 on a die, for example, is a union.

You find its probability with the general addition rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. You add the two probabilities, then subtract the intersection once, because outcomes in both events got counted twice when you added. When the events cannot both happen, the intersection is 0 and the rule collapses to simple addition.

What an intersection is

The intersection $A \cap B$ is the event that both $A$ and $B$ occur, sometimes called the joint probability. Drawing a card that is both a heart and a face card is an intersection: it must satisfy both conditions at once.

How you compute $P(A \cap B)$ depends on whether the events are independent. When they are independent, $P(A \cap B) = P(A) \cdot P(B)$. When they are not, use the general multiplication rule $P(A \cap B) = P(A) \cdot P(B \mid A)$, where $P(B \mid A)$ is the probability of $B$ given that $A$ has occurred. If the two events are mutually exclusive, they cannot both happen, so $P(A \cap B) = 0$.

Union vs intersection side by side

How the addition rule ties them together

The union and the intersection meet in the general addition rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. The subtraction is the whole point. When you add $P(A)$ and $P(B)$, any outcome that lies in both events is counted once in each sum, so it is counted twice total. Subtracting the intersection removes that double count exactly once.

This is why the intersection is not just a separate topic; you need it to get the union right. If the events are mutually exclusive, the intersection is 0 and there is nothing to subtract, which is the special case many students remember as 'just add them.' That shortcut only works when $P(A \cap B) = 0$.

The classic mix-up and how to avoid it

The classic error is swapping 'and' with 'or', which sends you to the wrong operation entirely. A close second is using $P(A \cup B) = P(A) + P(B)$ when the events overlap, which double-counts the shared outcomes and gives a probability that is too big, sometimes above 1.

Three habits keep them apart. First, translate the sentence: 'and' means intersection, 'or' means union. Second, always subtract the intersection in the addition rule unless you have checked the events are mutually exclusive. Third, sanity-check the size: a union is at least as likely as either event alone, while an intersection is no more likely than the smaller of the two, and a final probability above 1 always signals a double-count error.

Where this fits in AP Statistics

Union and intersection are core to Unit 2. The intersection appears in topic 2.5 on mutually exclusive events, and the union with the general addition rule is topic 2.7, alongside independence. Both the addition rule and conditional probability are printed on the Fall 2026 formula sheet, so you are expected to apply them, not memorize them from scratch.

A frequent trap gives events that are not mutually exclusive and rewards students who remember to subtract the intersection. Watching the language of a question, 'and' versus 'or', decides which rule to reach for, and it connects to telling disjoint from independent events, which are easy to confuse but mean different things.