3-Event Probability Calculator
Enter values as percentages (0 to 100). You can input all intersections manually, or check the independence option to auto-calculate overlaps.
How this 3-event probability calculator works
When you deal with three events, the arithmetic can get messy quickly. This calculator gives you a clean way to compute the most useful results: the probability that at least one event happens, none happen, exactly one happens, exactly two happen, and all three happen.
It uses the standard inclusion-exclusion framework. You provide single-event probabilities and overlap terms, and the tool combines them into a full breakdown.
Core formulas for three events
1) Union (at least one event occurs)
The central formula is:
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A∩B) − P(A∩C) − P(B∩C) + P(A∩B∩C)
This avoids double-counting pairwise overlaps and then adds back the triple overlap one time.
2) Complement (none occur)
P(none) = 1 − P(A ∪ B ∪ C)
3) Exactly one and exactly two
- P(exactly one) = P(A)+P(B)+P(C) − 2[P(A∩B)+P(A∩C)+P(B∩C)] + 3P(A∩B∩C)
- P(exactly two) = P(A∩B)+P(A∩C)+P(B∩C) − 3P(A∩B∩C)
- P(all three) = P(A∩B∩C)
Manual mode vs independence mode
Manual mode
Use manual mode when you already know overlap data from experiments, surveys, reliability logs, or historical records. This is common in real-world analytics where events are usually correlated.
Independence mode
Use independence mode if your model assumes events are independent. The calculator will set:
- P(A∩B) = P(A)P(B)
- P(A∩C) = P(A)P(C)
- P(B∩C) = P(B)P(C)
- P(A∩B∩C) = P(A)P(B)P(C)
Independence is mathematically convenient, but it can be unrealistic in many practical systems. If you have measured overlap terms, manual mode is usually more accurate.
Quick example
Suppose:
- P(A) = 45%
- P(B) = 30%
- P(C) = 25%
- P(A∩B) = 12%
- P(A∩C) = 10%
- P(B∩C) = 8%
- P(A∩B∩C) = 4%
Then:
- At least one occurs = 74%
- None occur = 26%
- All three occur = 4%
Use the Load Example button in the calculator to test this instantly.
Common mistakes to avoid
- Mixing decimals and percentages in the same input set.
- Forgetting the triple-overlap term in inclusion-exclusion.
- Assuming independence without evidence.
- Entering inconsistent overlap values that produce impossible probabilities (like negative region sizes).
Where 3-event probability appears in practice
- Finance: three market triggers, risk flags, or correlated defaults.
- Medical screening: symptom/test combinations.
- Quality control: failure modes in systems engineering.
- Marketing analytics: users exposed to multiple campaign channels.
- Cybersecurity: simultaneous alert classes across detection layers.
Final thought
A three-event probability model is often the first step from simple textbook probability into real analytical work. If your input values are reliable, this calculator gives a fast and transparent way to understand overlap, total exposure, and edge cases in one place.