Written and maintained by CASRAI Editorial Board
Last updated
Disjoint means having nothing in common. Two sets are disjoint when their intersection is empty, and two events are disjoint when they cannot both happen on the same trial. Statistics books usually call disjoint events mutually exclusive; the two words name the same idea.
Definition in one paragraph: what does disjoint mean?
Sets A and B are disjoint when A ∩ B = ∅, meaning no element belongs to both. A collection of more than two sets is disjoint when every pair of different sets in it has an empty intersection (Aᵢ ∩ Aⱼ = ∅ for i ≠ j); MathWorld adds that disjoint sets are also said to be mutually exclusive. In probability, the sets are events (sets of outcomes of one random process), so disjoint events are events that share no outcome. If you define “disjoint” by the empty intersection, the probability statement follows: disjoint events satisfy P(A ∩ B) = 0, which is how OpenStax states the rule for mutually exclusive events.
What are disjoint events, and how is “disjoint” different from “mutually exclusive”?
It is not different. “Disjoint” is the word from set theory and “mutually exclusive” is the word from probability teaching; ProofWiki’s definition of disjoint events says as much. Rolling one die and getting a 2 or a 5 are disjoint events, because a single roll cannot show both.
Formula, with every symbol explained
Two rules cover almost every use of the idea. For events A and B defined on the same sample space:
- Disjoint test: A ∩ B = ∅, and as a consequence P(A ∩ B) = 0.
- Addition rule (general): P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
- Addition rule when A and B are disjoint: P(A ∪ B) = P(A) + P(B), because the subtracted term is 0.
Symbols: P(A) is the probability of event A; ∪ (“or”, union) is the set of outcomes in A, in B, or in both; ∩ (“and”, intersection) is the set of outcomes in both; ∅ is the empty set; the sample space is the set of all possible outcomes. The general addition rule is the one OpenStax Introductory Statistics 2e gives in section 3.3, where it also states that P(A AND B) = 0 for mutually exclusive events.
Worked example with real numbers
Roll one fair six-sided die. The sample space is {1, 2, 3, 4, 5, 6}, and each face has probability 1/6.
Disjoint pair
Let A = {1, 2} and B = {5, 6}. Nothing is in both, so A ∩ B = ∅ and P(A ∩ B) = 0. Then P(A) = 2/6 and P(B) = 2/6, so P(A ∪ B) = 2/6 + 2/6 = 4/6 = 2/3. Counting directly agrees: A ∪ B = {1, 2, 5, 6} has 4 of the 6 faces.
A pair that is not disjoint
Let A = {1, 2, 3} and B = {3, 4}. Both contain the face 3, so A ∩ B = {3} and P(A ∩ B) = 1/6. Adding naively gives 3/6 + 2/6 = 5/6, which is wrong because the face 3 is counted twice. The general rule gives P(A ∪ B) = 3/6 + 2/6 − 1/6 = 4/6 = 2/3, and counting A ∪ B = {1, 2, 3, 4} confirms 4 faces out of 6.
Three or more events: pairwise disjoint
Let A = {1, 2}, B = {3, 4} and C = {5, 6}. Check each pair: A ∩ B, A ∩ C and B ∩ C are all empty, so the three events are pairwise disjoint. Their probabilities add: 2/6 + 2/6 + 2/6 = 6/6 = 1. Here the three events also cover the whole sample space, but disjoint events do not have to: A = {1, 2} and B = {5, 6} above are disjoint and cover only 4 of 6 faces.
The OpenStax example (Example 3.14) uses the same logic with real-world numbers: a traveller can afford only one of two trips, with P(A) = 0.6 for New Zealand and P(B) = 0.35 for Alaska, so P(A AND B) = 0 and P(A OR B) = 0.6 + 0.35 = 0.95.
Interpretation: reading the result
| What you find | What it means | Which rule to use |
|---|---|---|
| A ∩ B = ∅ (so P(A ∩ B) = 0) | A and B are disjoint (mutually exclusive) | P(A ∪ B) = P(A) + P(B) |
| A ∩ B has at least one outcome | A and B are not disjoint; they overlap | P(A ∪ B) = P(A) + P(B) − P(A ∩ B) |
| Disjoint, with P(A) > 0 and P(B) > 0 | The events are dependent: P(A ∩ B) = 0 but P(A)P(B) > 0 | Do not use the multiplication rule for independent events |
| Independent, with P(A) > 0 and P(B) > 0 | The events cannot be disjoint | P(A ∩ B) = P(A)P(B) > 0 |
Is disjoint the same as dependent? Disjoint versus independent
For events that can actually happen, disjoint events are dependent, not independent. Independence means P(A ∩ B) = P(A)P(B). If A and B are disjoint with P(A) = 1/3 and P(B) = 1/3, then P(A ∩ B) = 0 but P(A)P(B) = 1/9, so the product rule fails. Intuitively, once you know A happened, you know B did not, so A carries information about B. MathWorld notes the same point: disjointness is a different property from independence. The full test for each is laid out in independent vs mutually exclusive events. The one exception is an event with probability 0, which is trivially independent of everything.
Is disjoint set union? What is the opposite of disjoint?
No. A union is an operation that builds a new set (A ∪ B). Disjoint is a property that two sets either have or lack. Any two sets have a union, whether or not they are disjoint; what disjointness changes is how you count or weigh that union. For two disjoint events nothing is double counted, so the plain sum P(A) + P(B) is the probability of the union.
The opposite of disjoint is overlapping (also written non-disjoint): the sets share at least one element, so A ∩ B ≠ ∅. For overlapping events use the general addition rule above. Do not confuse the opposite of disjoint with a complement: the complement of A is everything outside A, and A together with its complement is always disjoint.
Assumptions and common mistakes
- Same sample space. The words only make sense for events defined on one random process, such as one roll of one die. A coin flip and a die roll are separate processes whose events are not “disjoint” in this sense. For how events combine across steps, see the joint probability formula.
- Pairs, not just the whole group. For three or more events, every pair must be disjoint, not merely “no outcome is in all of them”.
- Adding probabilities of overlapping events. It double counts the overlap (5/6 instead of 4/6 in the example above).
- Treating disjoint as independent. Disjoint events with positive probabilities are dependent.
- Assuming disjoint events cover everything. They only do when they are also exhaustive.
When one event is conditioned on another, disjointness has a sharp effect: if A and B are disjoint and P(B) > 0, then P(A | B) = 0. See conditional probability and Bayes’ theorem and the Bayes’ theorem probability tree for the general case. For a case where several outcome categories are exclusive by design, see the multinomial distribution.
How to report it
Disjointness is usually stated in a methods or analysis note rather than reported as a test result. A sentence in a methods note could read:
The two outcome categories were mutually exclusive (disjoint), so P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B) = 0.35 + 0.60 = 0.95.
State the sample space the events were defined on.
How to calculate it in Excel, R or Python
Each snippet checks that A = {1, 2} and B = {5, 6} on a six-sided die are disjoint and returns P(A ∪ B) = 4/6.
Excel
Put the faces 1 to 6 in cells A2:A7. Overlap count (0 means disjoint): =SUMPRODUCT((A2:A7<=2)*(A2:A7>=5)). Probability of A or B: =SUMPRODUCT(--(((A2:A7<=2)+(A2:A7>=5))>0))/ROWS(A2:A7), which returns 0.6667.
R
A <- c(1, 2); B <- c(5, 6)
length(intersect(A, B)) == 0 # TRUE: disjoint
length(union(A, B)) / 6 # 0.6666667Python
A, B = {1, 2}, {5, 6}
A.isdisjoint(B) # True
len(A | B) / 6 # 0.6666666666666666FAQ
What does disjoint or mutually exclusive mean?
Both mean the two sets or events share no element or outcome, so they cannot both occur on one trial and P(A ∩ B) = 0.
Is disjoint set union?
No. Union is an operation that combines sets; disjoint describes two sets whose intersection is empty.
Is disjoint the same as dependent?
Not by definition, but disjoint events with positive probabilities are always dependent, because P(A ∩ B) = 0 is not equal to P(A)P(B).
What is the opposite of disjoint?
Overlapping (non-disjoint): the sets share at least one element. Use P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
What is the meaning of disjoint in statistics?
Events that cannot happen together, such as rolling a 1 and a 6 on a single roll of one die.
Sources
- OpenStax, Introductory Statistics 2e, section 3.3 “Two Basic Rules of Probability”: openstax.org/books/introductory-statistics-2e/pages/3-3-two-basic-rules-of-probability (checked 2026-10-29).
- Wolfram MathWorld, “Disjoint Sets”: mathworld.wolfram.com/DisjointSets.html (checked 2026-10-29).
- ProofWiki, “Definition: Disjoint Events”: proofwiki.org/wiki/Definition:Disjoint_Events (checked 2026-10-29).
