Probability of 3 Events Calculator

Calculate union, intersection, and compound probabilities for three independent events.

Last updated: March 2026

Event Probabilities

P(A)

P(B)

P(C)

P(A ∩ B ∩ C) - All three occur

0.060000

6.0000%

P(A ∪ B ∪ C) - Union0.790000

P(at least one)0.790000

P(exactly one)0.440000

P(exactly two)0.290000

P(none occur)0.210000

Pairwise Intersections

P(A∩B)

0.1200

P(A∩C)

0.1500

P(B∩C)

0.2000

What is Three-Event Probability?

Three-event probability extends basic probability theory to calculate the likelihood of various combinations when three independent events are involved. This calculator helps determine not just whether individual events occur, but complex scenarios like "all three happen," "exactly two happen," or "at least one happens."

For independent events A, B, and C, the probability calculations follow specific formulas. The intersection (all three occurring) is P(A) × P(B) × P(C). The union (at least one occurring) uses the inclusion-exclusion principle: P(A∪B∪C) = P(A) + P(B) + P(C) − P(A∩B) − P(A∩C) − P(B∩C) + P(A∩B∩C). This accounts for overlaps to avoid double-counting.

These calculations are essential in risk assessment, quality control, reliability engineering, and game theory. For example, calculating the probability that at least one of three backup systems fails, or the chance of winning when you need exactly two out of three events to succeed.

How to Calculate Multi-Event Probabilities

The Formulas (Independent Events)

All three: P(A∩B∩C) = P(A) × P(B) × P(C)

Union: P(A∪B∪C) = P(A) + P(B) + P(C) − P(A∩B) − P(A∩C) − P(B∩C) + P(A∩B∩C)

None: P(A'∩B'∩C') = (1−P(A)) × (1−P(B)) × (1−P(C))

At least one: P(at least one) = 1 − P(none)

Exactly one: P(A∩B'∩C') + P(A'∩B∩C') + P(A'∩B'∩C)

Understanding Independence

Events are independent when the occurrence of one doesn't affect the probability of the others.

✓ Coin flips: getting heads on flip 1 doesn't change the probability for flip 2

✓ Dice rolls: rolling a 6 on die A doesn't affect die B

✗ Drawing cards without replacement: NOT independent (first card affects second)

✗ Weather days in sequence: NOT independent (yesterday's weather affects today)

Example: Three Coin Flips

Flipping three fair coins:

Given:

P(A = heads) = 0.5

P(B = heads) = 0.5

P(C = heads) = 0.5

All heads:

P(A∩B∩C) = 0.5 × 0.5 × 0.5 = 0.125 (12.5%)

At least one heads:

P(none) = (1−0.5) × (1−0.5) × (1−0.5) = 0.125

P(at least one) = 1 − 0.125 = 0.875 (87.5%)

Exactly two heads:

HHT: 0.5 × 0.5 × 0.5 = 0.125

HTH: 0.5 × 0.5 × 0.5 = 0.125

THH: 0.5 × 0.5 × 0.5 = 0.125

Total: 0.375 (37.5%)

Frequently Asked Questions

What does 'independent events' mean?

Independent events are those where the outcome of one doesn't affect the probability of the others. For example, coin flips are independent, but drawing cards from a deck without replacement is not—the first card affects what's left for the second draw.

What's the difference between union and intersection?

Intersection (∩) means 'AND'—all events occur together. Union (∪) means 'OR'—at least one event occurs. P(A∩B∩C) is all three happening; P(A∪B∪C) is one or more happening.

Why subtract and add back in the union formula?

The inclusion-exclusion principle prevents double-counting. When you add P(A) + P(B) + P(C), overlaps are counted multiple times, so you subtract pairwise intersections, then add back the triple intersection which was over-subtracted.

Can I use this for dependent events?

No, this calculator assumes independence. For dependent events, you need conditional probabilities: P(A∩B) = P(A) × P(B|A), where P(B|A) is the probability of B given that A occurred. The formulas become more complex.

What does 'exactly two' mean?

'Exactly two' means precisely two events occur and one doesn't. For coins, it's HHT, HTH, or THH—not HHH. It's calculated by adding the probabilities of all combinations where exactly two succeed.

How do I verify my results?

All possible outcomes must sum to 1. Check: P(none) + P(exactly one) + P(exactly two) + P(all three) = 1. For three fair coins: 0.125 + 0.375 + 0.375 + 0.125 = 1.0 ✓

What's the complement rule?

P(at least one) = 1 − P(none). It's often easier to calculate the probability that nothing happens and subtract from 1 than to enumerate all ways at least one thing can happen.

Can probabilities exceed 1?

No! Probabilities range from 0 (impossible) to 1 (certain). If your calculation gives a value over 1, there's an error—possibly the events aren't actually independent or there's a calculation mistake.