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Mutually inclusive events are events that can happen at the same time: at least one outcome belongs to both. In probability terms, A and B are mutually inclusive when P(A ∩ B) > 0. They are the opposite case to mutually exclusive events, which cannot occur together and have P(A ∩ B) = 0. The practical consequence is the one most people miss: when events are mutually inclusive, you must subtract the overlap before adding probabilities, or you count the shared outcomes twice.
What does mutually inclusive mean? (Definition)
Two events are mutually inclusive if they can occur at the same time, meaning they share at least one outcome. “Mutually inclusive” is the common name for this overlapping case; statistics textbooks usually define the contrasting term, mutually exclusive, and treat everything else as events that overlap. OpenStax Introductory Statistics 2e defines mutually exclusive events as events that “cannot occur at the same time”, so that A and B “do not share any outcomes and P(A AND B) = 0” (section 3.2, checked 2026-10-29). Mutually inclusive is the complement of that statement: they do share outcomes, and P(A AND B) is greater than zero.
The same book defines an outcome as being in the event “A AND B” if it is “in both A and B at the same time”, and in “A OR B” if it is in A, in B, “or is in both A and B” (section 3.1, checked 2026-10-29). The “in both” outcomes are the overlap. Mutually inclusive simply means that overlap is not empty.
Mutually inclusive vs mutually exclusive (exclusive vs inclusive)
| Feature | Mutually exclusive | Mutually inclusive |
|---|---|---|
| Can A and B happen together? | No | Yes |
| Shared outcomes | None | At least one |
| P(A ∩ B) | 0 | Greater than 0 |
| P(A ∪ B) | P(A) + P(B) | P(A) + P(B) − P(A ∩ B) |
| Venn diagram | Two separate circles | Two overlapping circles |
The first, second and third rows follow directly from the OpenStax definitions above. The fourth row is the general addition rule from section 3.3, which OpenStax writes as “P(A OR B) = P(A) + P(B) − P(A AND B)” and notes reduces to a simple sum when the events are mutually exclusive because P(A AND B) = 0. For the exclusive side of this contrast, see our guide to disjoint events and what disjoint means.
Formula: P(A or B) with the overlap subtracted
For any two events A and B on the same sample space:
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- P(A ∪ B): probability that A happens, B happens, or both (“A or B”). The union.
- P(A) and P(B): the probability of each event on its own.
- P(A ∩ B): probability that both happen together (“A and B”). The intersection, or overlap.
The subtraction is there because P(A) and P(B) each already include the overlapping outcomes. Adding them counts the overlap twice; subtracting P(A ∩ B) once leaves it counted exactly once. If you want the “and” side of the calculation in more depth, see the joint probability formula.
Worked example with a die
Roll one fair six-sided die. The sample space is {1, 2, 3, 4, 5, 6}, with six equally likely outcomes. Let A = “the roll is even” and B = “the roll is greater than 3”.
- A = {2, 4, 6}, so P(A) = 3/6 = 0.5.
- B = {4, 5, 6}, so P(B) = 3/6 = 0.5.
- The overlap is A ∩ B = {4, 6}, so P(A ∩ B) = 2/6 ≈ 0.333. Because the overlap is not empty, A and B are mutually inclusive.
- P(A ∪ B) = 3/6 + 3/6 − 2/6 = 4/6 ≈ 0.667.
- Check by listing the union directly: A ∪ B = {2, 4, 5, 6}, which is 4 of 6 outcomes. The two methods agree.
If you had skipped the subtraction you would have got 3/6 + 3/6 = 1.0, which would claim you are certain to roll an even number or a number above 3. That is wrong: a roll of 1 or 3 is neither. The extra 2/6 is the double-counted 4 and 6.
A research-design example (illustrative)
The same arithmetic applies to survey and study data. This is an illustrative example with invented counts, not real survey results. Suppose 200 respondents are asked which analysis tools they use, and the question is “select all that apply”. 120 use R, 90 use Python and 40 use both.
- P(R) = 120/200 = 0.60 and P(Python) = 90/200 = 0.45.
- P(R ∩ Python) = 40/200 = 0.20, so the two categories are mutually inclusive.
- P(R ∪ Python) = 0.60 + 0.45 − 0.20 = 0.85, which is 170 respondents (120 + 90 − 40). The other 30 use neither.
This is why check-all-that-apply response options are mutually inclusive categories: percentages for the options can add to more than 100%, and the number of distinct people is the union, not the sum. For how that response format is built, see survey question types.
Interpretation: how to read the result
| What you find | What it means |
|---|---|
| P(A ∩ B) = 0 | The events are mutually exclusive: no shared outcomes. Add the probabilities directly. |
| P(A ∩ B) > 0 | The events are mutually inclusive: they can occur together. Subtract the overlap when finding “A or B”. |
| P(A ∩ B) = P(A) × P(B) | The events are independent (OpenStax section 3.2). This is a separate property from overlap. |
| P(A ∩ B) ≠ P(A) × P(B) | The events are not independent, whether or not they overlap. |
Is mutually inclusive the same as independent?
No. OpenStax states that “independent and mutually exclusive do not mean the same thing” (section 3.2), and the same holds for inclusive events: overlapping does not tell you whether the events are independent. Independence is a test on numbers, P(A ∩ B) = P(A) × P(B), not a feature of the Venn diagram.
Using the die again, A = “even” has P(A) = 0.5 and B = “greater than 3” has P(B) = 0.5. Then P(A) × P(B) = 0.25, but P(A ∩ B) ≈ 0.333. The events are mutually inclusive and not independent.
Now let C = “the roll is 4 or less” = {1, 2, 3, 4}, so P(C) = 4/6 ≈ 0.667. The overlap with A is {2, 4}, so P(A ∩ C) = 2/6 ≈ 0.333, and P(A) × P(C) = 0.5 × 0.667 ≈ 0.333. These events are mutually inclusive and independent. For the comparison with mutually exclusive events, see independent vs mutually exclusive events.
Assumptions and common mistakes
- Forgetting the subtraction. Adding P(A) and P(B) for overlapping events double-counts the overlap and can push the result above 1, which is impossible for a probability.
- Treating “not exclusive” as “independent”. Overlap and independence are separate questions, as the die example shows.
- Mixing sample spaces. The formula assumes A and B are defined on the same sample space with the same outcomes counted.
- Reading survey percentages as a partition. With check-all-that-apply questions the categories overlap, so the percentages need not sum to 100%.
- Confusing “at least one” with “exactly one”. P(A ∪ B) includes the overlap. If you want exactly one of the two, subtract the overlap twice, or see the probability of at least one event for related reasoning.
How to report it (APA example)
The APA Style numbers and statistics handout (7th edition, updated 2024-09-11) asks for italics on letters used as statistical symbols, a space before and after a mathematical operator, the term rather than the symbol in running text, and no zero before the decimal point for statistics that cannot exceed 1, such as a proportion. A sentence reporting the survey example above could read:
Of 200 respondents, 120 used R, 90 used Python and 40 used both; the probability of using at least one of the two tools was P = .85.
In a table, report the counts for each category and for the overlap, so readers can see that the categories are not a partition.
How to calculate it in Excel, R or Python
Each snippet computes the union probability from the counts in the survey example (n = 200, 120 in A, 90 in B, 40 in both). The result is 0.85.
Excel, with the counts in cells B1 (n = 200), B2 (120), B3 (90) and B4 (40):
=(B2 + B3 - B4) / B1R:
n <- 200; a <- 120; b <- 90; ab <- 40
p_union <- (a + b - ab) / n
p_union # 0.85Python, which can also count the overlap directly from the die outcomes:
n, a, b, ab = 200, 120, 90, 40
print((a + b - ab) / n) # 0.85
omega = {1, 2, 3, 4, 5, 6}
even, high = {2, 4, 6}, {4, 5, 6}
print(len(even & high) / len(omega)) # 0.333..., the overlap
print(len(even | high) / len(omega)) # 0.666..., the unionFAQ
What is the meaning of mutually inclusive?
Two events are mutually inclusive when they can occur at the same time, so they share at least one outcome and P(A ∩ B) is greater than zero.
Define mutually inclusive: what is the definition of mutually inclusive events?
Events A and B are mutually inclusive if there is at least one outcome that belongs to both. Their union is found with P(A) + P(B) − P(A ∩ B).
What is the difference between exclusive and inclusive events?
Mutually exclusive events cannot happen together (no overlap, P(A ∩ B) = 0). Mutually inclusive events can (overlap exists, P(A ∩ B) > 0). The exclusive case lets you add probabilities directly; the inclusive case needs the overlap subtracted.
Can two events be both mutually inclusive and independent?
Yes. In the die example, “even” and “4 or less” overlap in {2, 4} and satisfy P(A ∩ C) = P(A) × P(C). Overlap and independence are separate properties.
Can mutually inclusive events have a probability that adds to more than 1?
Their individual probabilities can add to more than 1, as with 0.60 + 0.45 in the survey example. The probability of the union cannot, because the overlap is subtracted.
Sources
- OpenStax, Introductory Statistics 2e, section 3.1 Terminology, openstax.org/books/introductory-statistics-2e/pages/3-1-terminology (checked 2026-10-29).
- OpenStax, Introductory Statistics 2e, section 3.2 Independent and Mutually Exclusive Events, openstax.org/books/introductory-statistics-2e/pages/3-2-independent-and-mutually-exclusive-events (checked 2026-10-29).
- 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).
- American Psychological Association, Numbers and Statistics Guide (APA Style 7th edition handout, updated 2024-09-11), apastyle.apa.org/instructional-aids/numbers-statistics-guide.pdf (checked 2026-10-29).
