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Calculate three-event probabilities
On the defaults — 50%, 60%, and 70% — all three happen 21% of the time and at least one happens 94% of the time. Enter your own probabilities below.
FREE TOOL · STATISTICS
Probability of Three Events
Three independent events, each with its own chance of happening. Enter the three probabilities and get every combination that matters: all three happen, at least one happens, exactly one happens, or none happen — with every formula shown.
How three-event probability works
When events are independent — one outcome does not change the odds of another, like three separate coin flips — combination probabilities are simple multiplication. The four questions people ask are all answered by the same four products, and the calculator shows every one.
Output · formula · meaning
| Output | Formula | What it means |
|---|---|---|
| All three happen | P(A) × P(B) × P(C) | 0.50 × 0.60 × 0.70 = 21% — everything lines up at once. |
| None happen | (1−P(A)) × (1−P(B)) × (1−P(C)) | 0.50 × 0.40 × 0.30 = 6% — all three miss together. |
| At least one happens | 1 − P(none) | 1 − 6% = 94% — the complement shortcut: flip the "none" answer. |
| Exactly one happens | A-only + B-only + C-only | 6% + 9% + 14% = 29% — add the three "only this one hits" cases. |
Worked example: 50%, 60%, and 70%
All three: 0.50 × 0.60 × 0.70 = 0.21 = 21%. None: 0.50 × 0.40 × 0.30 = 0.06 = 6%. At least one: 1 − 0.06 = 94% — high, because three decent chances rarely all fail at once. Exactly one: A-only (0.50 × 0.40 × 0.30 = 6%) plus B-only (0.50 × 0.60 × 0.30 = 9%) plus C-only (0.50 × 0.40 × 0.70 = 14%) = 29%. Notice all-three (21%) is much lower than any single event — multiplication shrinks probabilities.
Who uses this
- Students: probability homework and exams — "all three", "at least one", and "exactly one" are the classic three questions.
- Game designers: estimating how often three independent mechanics (crit, drop, proc) all trigger together.
- Risk planners: combining independent failure odds — if three backups each fail 1% of the time, all three fail together 0.0001% of the time.
- Curious minds: three dice all landing six: (1/6)³ ≈ 0.46% — try it with 16.67% three times.
Accuracy & limitations
Every result is exact arithmetic from your inputs — the only assumption is independence. If the events influence each other (drawing cards without replacement, weather across consecutive days), multiplication understates or overstates the truth and you need conditional probabilities instead. Independence means one outcome tells you nothing about the others; when in doubt, it is the thing to verify, not assume.
For random draws and games, try the dice roller and the random number generator — probability in action.
Cite this calculatorBright Side Kit. “Probability of Three Events.” brightoolbox.com, 2026.
Bright Side Kit. "Probability of Three Events." https://brightoolbox.com/probability-three-events/ Accessed October 22, 2026.Frequently asked questions
Why do you multiply the probabilities for “all three”?
Independence means the events don’t interfere. If A happens 50% of the time, then within that half of the time B happens 60% of the time — that’s 0.50 × 0.60 = 30% of all trials where both happen. Add C the same way: 30% × 70% = 21%. Each multiplication narrows the slice of outcomes where everything lines up.
What is the trick for “at least one”?
Flip it: calculate “none” and subtract from 1. Counting "A happens, or B happens, or C happens" directly is a tangle of overlaps. But "none happen" is one clean product: (1−P(A)) × (1−P(B)) × (1−P(C)). At-least-one is everything else, so 1 − P(none). For 50/60/70: 1 − 6% = 94%.
How do I calculate “exactly one”?
Add the three “only this one” cases. A happens while B and C miss: P(A)×(1−P(B))×(1−P(C)). Same for B-only and C-only. For 50/60/70: 6% + 9% + 14% = 29%. The three cases can’t overlap (only one event happens in each), so they add directly.
What if the events are NOT independent?
Then multiplication is wrong. If one event changes another’s odds — drawing cards without replacement, rain on consecutive days — you need conditional probabilities: P(A and B) = P(A) × P(B|A). This calculator assumes independence, so verify that assumption before trusting the answers.
What is the difference between independent and mutually exclusive?
They are opposites in a sense. Independent events can both happen (coin flips). Mutually exclusive events cannot both happen (one die roll landing 1 AND 6) — and if knowing one happened makes the other impossible, they are definitely not independent. Don’t mix them up on exams.
Can I enter probabilities as decimals instead of percents?
Enter percents here. Type 50 for a half chance, not 0.5 — the calculator divides by 100 for you. Inputs must be between 0 and 100; a probability above 100% (or below 0%) is rejected as an error.