Probability of 3 Events Calculator
Free probability of 3 events calculator. P(A∩B∩C), P(A∪B∪C), at least one, exactly one/two. Inclusio
Why This Statistical Analysis Matters
Why: Statistical calculator for analysis.
How: Enter inputs and compute results.
P(A∩B∩C), P(A∪B∪C) — Inclusion-Exclusion for Three Events
Independent and dependent events. At least one, exactly one, exactly two. Full step-by-step breakdown.
Real-World Scenarios — Click to Load
Inputs (0–100% or 0–1)
Probability Breakdown
Outcome Distribution
Probability Comparison
Calculation Breakdown
⚠️For educational and informational purposes only. Verify with a qualified professional.
Key Takeaways
- • Inclusion-exclusion: P(A∪B∪C) = P(A)+P(B)+P(C) − P(A∩B) − P(A∩C) − P(B∩C) + P(A∩B∩C)
- • Independence: P(A∩B∩C) = P(A)×P(B)×P(C) only when events are independent
- • P(at least one) = 1 − P(none) — complement rule is often easier
- • P(exactly one) sums three terms: A-only, B-only, C-only
- • P(exactly two) sums three terms: AB, AC, BC (each without the third)
Did You Know?
How It Works
1. Independent Events
When A, B, C don't influence each other: P(A∩B∩C) = P(A)×P(B)×P(C). Dice, coins, draws with replacement.
2. Inclusion-Exclusion Principle
To avoid double-counting overlaps: add singles, subtract pairs, add triple. P(A∪B∪C) = P(A)+P(B)+P(C) − P(A∩B) − P(A∩C) − P(B∩C) + P(A∩B∩C).
3. At Least One
P(at least one) = 1 − P(none). For independent: P(none) = (1−P(A))(1−P(B))(1−P(C)).
4. Exactly One / Exactly Two
Exactly one: A-only + B-only + C-only. Exactly two: AB + AC + BC (each excluding the third event).
5. Dependent Events
Use P(B|A) and P(C|A∩B) to compute P(A∩B) = P(A)×P(B|A) and P(A∩B∩C) = P(A)×P(B|A)×P(C|A∩B).
Expert Tips
Check independence first
If events influence each other, use conditional probabilities.
At least one vs. union
For independent events, P(at least one) = 1−P(none) is often easier than inclusion-exclusion.
Distribution sums to 1
P(none) + P(exactly one) + P(exactly two) + P(all three) = 1.
Use presets for intuition
Load dice, free throws, or rain examples to see how formulas behave.
Independent vs Dependent
| Feature | Independent | Dependent |
|---|---|---|
| P(A∩B∩C) | P(A)×P(B)×P(C) | P(A)×P(B|A)×P(C|A∩B) |
| P(none) | (1−P(A))(1−P(B))(1−P(C)) | Requires full model |
| P(at least one) | 1 − P(none) | 1 − P(none) |
| Inclusion-exclusion | Full formula | Needs pair intersections |
| Inputs needed | P(A), P(B), P(C) | P(B|A), P(C|A∩B) |
Why Use This Calculator?
| Feature | This Calculator | Manual | Wolfram |
|---|---|---|---|
| P(A∩B∩C), P(A∪B∪C) | ✅ | ⚠️ Error-prone | ✅ |
| Independent & dependent | ✅ | ✅ | ⚠️ |
| Step-by-step breakdown | ✅ | ❌ | ❌ |
| Charts & visualization | ✅ | ❌ | ⚠️ |
| Copy & share | ✅ | ❌ | ❌ |
| AI analysis | ✅ | ❌ | ❌ |
Frequently Asked Questions
What is the inclusion-exclusion principle?
For three events: P(A∪B∪C) = P(A)+P(B)+P(C) − P(A∩B) − P(A∩C) − P(B∩C) + P(A∩B∩C). It corrects for overcounting overlaps.
When can I use P(A)×P(B)×P(C) for all three?
Only when A, B, C are independent. Otherwise use conditional probabilities.
How do I compute P(at least one)?
P(at least one) = 1 − P(none). For independent: P(none) = (1−P(A))(1−P(B))(1−P(C)).
What is P(exactly one)?
Probability that exactly one of A, B, C occurs: A-only + B-only + C-only.
What is P(exactly two)?
Probability that exactly two occur: P(A∩B∩not C) + P(A∩C∩not B) + P(B∩C∩not A).
Do the probabilities sum to 1?
Yes. P(none) + P(exactly one) + P(exactly two) + P(all three) = 1.
How do I handle dependent events?
Use P(B|A) and P(C|A∩B). Then P(A∩B) = P(A)×P(B|A), P(A∩B∩C) = P(A)×P(B|A)×P(C|A∩B).
Can I use percentages?
Yes. Enter 50 for 50%, or 0.5 — the calculator accepts both 0–1 and 0–100.
3 Events by the Numbers
Official Data Sources
Disclaimer: This calculator provides probability computations for educational and professional reference. For dependent events, ensure P(B|A) and P(C|A∩B) are correctly specified.