In probability, a union refers to the event that at least one of two (or more) events occurs. Written as A ∪ B, it covers every outcome found in event A, event B, or both. When you see the word “or” in a probability question, you’re almost always dealing with a union.
How Union Differs From Intersection
The easiest way to grasp union is to contrast it with its counterpart, intersection. A union asks “What is the probability that A or B happens?” An intersection asks “What is the probability that A and B happen?” The union is always at least as large as the intersection because it includes every outcome where either event shows up, not just the outcomes where both do.
Picture two overlapping circles in a Venn diagram. The intersection is only the sliver where the circles overlap. The union is everything inside both circles combined, including that overlap. Anything outside the circles, still in the sample space, is not part of the union.
The Addition Rule
To calculate the probability of a union, you use the addition rule:
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
You subtract the intersection because, when you add P(A) and P(B) separately, you count the overlapping outcomes twice. Removing P(A ∩ B) corrects that double count. This one formula handles every two-event union problem you’ll encounter.
When Events Can’t Overlap
Some events are mutually exclusive, meaning they can never happen at the same time. Rolling a 2 and rolling a 5 on a single die, for example. When events are mutually exclusive, their intersection is zero, so the formula simplifies to:
P(A ∪ B) = P(A) + P(B)
You can safely add the probabilities straight across because there’s no overlap to worry about.
Two Worked Examples
Concrete numbers make this click faster than abstract formulas.
Drawing a Card
Suppose you draw one card from a standard 52-card deck and want the probability of getting a spade or a face card. There are 13 spades and 12 face cards (jacks, queens, kings across all four suits). But 3 cards are both spades and face cards (the jack, queen, and king of spades), so the events overlap.
P(spade ∪ face card) = 13/52 + 12/52 − 3/52 = 22/52 ≈ 0.423
Without subtracting those 3 overlapping cards, you’d overcount and get 25/52, which is wrong.
Rolling Two Dice
Roll two dice and add the faces. What’s the probability of getting a sum of 8 or rolling doubles? There are 36 equally likely outcomes. Five of them produce a sum of 8: (2,6), (3,5), (4,4), (5,3), (6,2). Six produce doubles: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6). The pair (4,4) appears in both lists, so these events are not mutually exclusive.
P(sum of 8 ∪ doubles) = 5/36 + 6/36 − 1/36 = 10/36 ≈ 0.278
Extending to Three or More Events
When you need the union of three events, the same logic applies, but you have to account for every possible overlap. The formula, called the inclusion-exclusion principle, looks like this:
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(A ∩ C) − P(B ∩ C) + P(A ∩ B ∩ C)
First you add all three individual probabilities. Then you subtract every pair’s overlap, because each pair got double-counted. Finally, you add back the three-way overlap, because subtracting the pairs removed it one too many times. The pattern continues for four or more events, alternating between subtracting and adding progressively larger overlaps.
Why Union Matters Beyond Textbooks
Union isn’t just a classroom exercise. Any time you assess the chance that “at least one thing goes wrong” or “at least one test comes back positive,” you’re calculating a union. If a factory has two machines and you want the probability that at least one breaks down during a shift, that’s a union of the two breakdown events. If a patient takes two diagnostic tests and you want the probability that at least one detects a condition, that’s also a union.
The concept is so fundamental that it appears in the axioms that define probability itself. One of the Kolmogorov axioms, the foundational rules of probability theory, states that for mutually exclusive events, the probability of their union equals the sum of their individual probabilities. Every other union formula builds from that starting point.
Common Mistakes to Avoid
- Adding probabilities without subtracting the overlap. Unless the events are mutually exclusive, simply adding P(A) + P(B) will overcount the shared outcomes and give you a number that’s too high.
- Confusing union with intersection. “Or” maps to union (∪). “And” maps to intersection (∩). Mixing these up flips the entire calculation.
- Assuming events are mutually exclusive when they aren’t. Before using the simplified formula, verify that no single outcome belongs to both events. The dice example above shows how easy it is to miss an overlap like (4,4).