Use this probability calculator to combine two events (A and B). It computes union, intersection, exactly one event, neither event, and conditional probabilities.
What is a combined events calculator?
A combined events calculator helps you evaluate probabilities when two events are considered together. In statistics and probability, this usually means finding one or more of the following:
- P(A or B): at least one of the two events happens (union).
- P(A and B): both events happen together (intersection).
- P(exactly one): only A or only B occurs, but not both.
- P(neither): neither event occurs.
These calculations appear everywhere: risk analysis, quality control, exams, sports analytics, and daily decision-making.
Core formulas used in this calculator
1) General addition rule
The most important equation for combined events is:
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
We subtract the overlap once so we do not double-count outcomes that belong to both A and B.
2) Independent events
If A and B are independent, then:
P(A ∩ B) = P(A) × P(B)
This is useful when one event does not affect the probability of the other.
3) Mutually exclusive events
If A and B cannot happen at the same time, then:
P(A ∩ B) = 0
So the union becomes P(A ∪ B) = P(A) + P(B).
How to use the calculator correctly
- Select input format (decimal or percent).
- Enter P(A) and P(B).
- Choose the event relationship:
- Independent if the events do not influence each other.
- Mutually exclusive if both cannot happen together.
- Custom intersection if you already know P(A and B).
- Click Calculate to see full results.
Worked examples
Example A: Independent events
Suppose P(A)=0.4 and P(B)=0.5, independent. Then:
- P(A and B)=0.4×0.5=0.20
- P(A or B)=0.4+0.5−0.20=0.70
- P(neither)=1−0.70=0.30
Example B: Mutually exclusive events
Suppose P(A)=0.25 and P(B)=0.35, mutually exclusive:
- P(A and B)=0
- P(A or B)=0.25+0.35=0.60
- P(neither)=0.40
Example C: Custom intersection
If P(A)=0.7, P(B)=0.6, and P(A and B)=0.5:
- P(A or B)=0.7+0.6−0.5=0.8
- P(exactly one)=0.8−0.5=0.3
- P(neither)=0.2
Common mistakes to avoid
- Adding probabilities without subtracting overlap. This can produce impossible values above 1.
- Confusing independent and mutually exclusive events. They are different concepts.
- Entering mixed formats. Keep inputs all decimals or all percentages based on your selected mode.
- Using impossible intersections. For valid probabilities, intersection must lie in a feasible range.
Why this matters in real life
Combined event probability is practical. You can estimate overlapping risks in project management, evaluate customer behavior in marketing funnels, measure pass/fail overlap in education data, and model system failures in engineering. A reliable calculator saves time and reduces arithmetic errors, especially when you need multiple derived values at once.
Quick FAQ
Can P(A or B) ever be greater than 1?
No. If your result is above 1, at least one input or assumption is invalid.
Can mutually exclusive events be independent?
Usually no (except trivial cases involving zero-probability events). If two non-zero events are mutually exclusive, knowing one happened guarantees the other did not.
What if I only know P(A), P(B), and P(A|B)?
You can compute intersection first using P(A and B)=P(A|B)×P(B), then use the addition rule for union.