Probability of 3 Events Calculator

Enter the probability of three independent events to find P(all), P(at least one), P(none), and P(A union B union C).

P(3) Probability of 3 Events Calculator
P(A) — Probability of Event A (%)
%
P(B) — Probability of Event B (%)
%
P(C) — Probability of Event C (%)
%

What is the Probability of 3 Events?

When three separate events A, B, and C can each independently occur or not occur, you often need to answer four distinct questions: What is the probability that all three happen? What is the probability that at least one happens? What is the probability that none happen? And what is the probability that A, B, or C (or some combination) happen, expressed as the union?

For independent events — meaning the occurrence of one event has no effect on the probability of the others — the mathematics is elegant and systematic. The probability that all three events occur simultaneously is simply the product of their individual probabilities: P(A) × P(B) × P(C). This is the multiplication rule for independent events extended from two events to three.

The probability that at least one event occurs is best computed using the complement rule: instead of laboriously adding all the ways at least one can happen (exactly one, exactly two, or all three), you compute P(none) and subtract from 1. P(none) = (1−P(A)) × (1−P(B)) × (1−P(C)), then P(at least one) = 1 − P(none). This is one of the most powerful shortcuts in probability theory.

The union P(A ∪ B ∪ C) uses the inclusion-exclusion principle, one of the cornerstones of combinatorics and probability. You start by adding all three individual probabilities, subtract each pairwise intersection (which you have double-counted), then add back the triple intersection (which you have subtracted once too many). The formula is: P(A) + P(B) + P(C) − P(A∩B) − P(A∩C) − P(B∩C) + P(A∩B∩C).

Real-world applications of three-event probability are widespread. Risk analysts compute the probability that three independent systems all fail (reliability engineering). Medical researchers ask: given a 70% chance of test A detecting disease, 60% for test B, and 80% for test C, what is the probability at least one test detects it? Sports analysts estimate the probability that three different athletes all perform above their expected level in the same game. Financial modelers assess the joint probability of three market conditions occurring simultaneously.

Formulas

All three events (joint probability):

P(A ∩ B ∩ C) = P(A) × P(B) × P(C)
P(A), P(B), P(C) = individual event probabilities (0 to 1, or % ÷ 100)
Valid only when A, B, and C are mutually independent

None of the three events:

P(none) = (1 − P(A)) × (1 − P(B)) × (1 − P(C))

At least one event occurs:

P(at least one) = 1 − P(none)

Union (A or B or C or any combination):

P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A)P(B) − P(A)P(C) − P(B)P(C) + P(A)P(B)P(C)

Note: For independent events, P(A∩B) = P(A)×P(B), P(A∩C) = P(A)×P(C), P(B∩C) = P(B)×P(C), and P(A∩B∩C) = P(A)×P(B)×P(C).

How to Use This Calculator

  1. Enter P(A), P(B), and P(C) as percentages (e.g., 60 for 60%, not 0.60). All three events must be independent of each other.
  2. Values must be between 0 and 100 inclusive. A value of 0 means the event never occurs; 100 means it always occurs.
  3. Click Calculate to see all four probability measures simultaneously: P(all), P(at least one), P(none), and P(union).
  4. Interpret the results: P(union) equals P(at least one) for independent events. P(all three) will always be the smallest value; P(at least one) will always be the largest.
  5. Verify: P(at least one) + P(none) must equal exactly 100%. If your results do not sum to 100%, recheck your inputs.

Example Calculations

Example 1 — General Three Events

P(A) = 60%, P(B) = 40%, P(C) = 30%

1
P(all three) = 0.60 × 0.40 × 0.30 = 0.072 = 7.20%
2
P(none) = (1−0.60)(1−0.40)(1−0.30) = 0.40 × 0.60 × 0.70 = 0.168 = 16.80%
3
P(at least one) = 1 − 0.168 = 0.832 = 83.20%
4
P(union) = 60+40+30 − 24 − 18 − 12 + 7.2 = 83.20%
P(all) = 7.20%  ·  P(at least one) = 83.20%  ·  P(none) = 16.80%
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Example 2 — Equal Probabilities

P(A) = 50%, P(B) = 50%, P(C) = 50%

1
P(all three) = 0.53 = 0.125 = 12.50%
2
P(none) = 0.53 = 0.125 = 12.50%
3
P(at least one) = 1 − 0.125 = 0.875 = 87.50%
P(all) = 12.50%  ·  P(at least one) = 87.50%  ·  P(none) = 12.50%
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Example 3 — High Probability Events

P(A) = 80%, P(B) = 70%, P(C) = 60%

1
P(all three) = 0.80 × 0.70 × 0.60 = 0.336 = 33.60%
2
P(none) = 0.20 × 0.30 × 0.40 = 0.024 = 2.40%
3
P(at least one) = 1 − 0.024 = 0.976 = 97.60%
P(all) = 33.60%  ·  P(at least one) = 97.60%  ·  P(none) = 2.40%
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Example 4 — Rare Events (Risk Assessment)

P(A) = 10%, P(B) = 20%, P(C) = 30%

1
P(all three) = 0.10 × 0.20 × 0.30 = 0.006 = 0.60%
2
P(none) = 0.90 × 0.80 × 0.70 = 0.504 = 50.40%
3
P(at least one) = 1 − 0.504 = 0.496 = 49.60%
4
Note: three rare events (10%, 20%, 30%) together still have a ~49.6% chance that at least one occurs.
P(all) = 0.60%  ·  P(at least one) = 49.60%  ·  P(none) = 50.40%
Try this example →

Frequently Asked Questions

How do you find the probability that all 3 independent events occur?+
For three independent events, P(A and B and C) = P(A) × P(B) × P(C). Convert each probability to a decimal before multiplying. Example: P(A)=60%, P(B)=40%, P(C)=30% gives P(all) = 0.60 × 0.40 × 0.30 = 0.072 = 7.2%. The result is always smaller than each individual probability.
What is the formula for P(A ∪ B ∪ C)?+
P(A∪B∪C) = P(A) + P(B) + P(C) − P(A∩B) − P(A∩C) − P(B∩C) + P(A∩B∩C). For independent events: each pairwise term = product of two probabilities, and the triple term = product of all three. This is the inclusion-exclusion principle applied to three sets.
How do you calculate the probability that at least one of 3 events occurs?+
Use the complement: P(at least one) = 1 − P(none occur) = 1 − (1−P(A))(1−P(B))(1−P(C)). This is much simpler than adding P(exactly one) + P(exactly two) + P(all three). For P(A)=0.5, P(B)=0.4, P(C)=0.3: P(none) = 0.5×0.6×0.7 = 0.21, so P(at least one) = 1 − 0.21 = 0.79 = 79%.
What is the probability that none of 3 events occur?+
P(none) = (1−P(A)) × (1−P(B)) × (1−P(C)). Each factor is the probability that the individual event fails to occur. For example, if all three have 50% probability, P(none) = 0.5×0.5×0.5 = 0.125 = 12.5%. This is the probability that A, B, and C all fail simultaneously.
Are these events independent? What changes if they are not?+
This calculator assumes the three events are mutually independent. If events are dependent (correlated), the multiplication rule no longer applies directly. You need conditional probabilities: P(A∩B∩C) = P(A) × P(B|A) × P(C|A∩B). Real-world events often have some correlation, so independence is an assumption that should be validated before using this formula.
P(A)=0.5, P(B)=0.4, P(C)=0.3 — what is P(all three)?+
P(all three) = 0.5 × 0.4 × 0.3 = 0.060 = 6.0%. P(none) = 0.5 × 0.6 × 0.7 = 0.21 = 21%. P(at least one) = 1 − 0.21 = 79%. P(union) = 0.5+0.4+0.3 − 0.2 − 0.15 − 0.12 + 0.06 = 0.79 = 79%.
What is the inclusion-exclusion principle for 3 events?+
The inclusion-exclusion principle for three events states: P(A∪B∪C) = P(A)+P(B)+P(C) − P(A∩B) − P(A∩C) − P(B∩C) + P(A∩B∩C). You first add all three, then subtract pairwise overlaps (which were double-counted), then add back the triple overlap (which was subtracted once too many). The alternating add-subtract-add pattern continues for four or more events.
How is P(at least one) related to P(none)?+
P(at least one) and P(none) are complementary events: they must sum to exactly 1 (or 100%). This means P(at least one) = 1 − P(none). If P(none) = 20%, then P(at least one) = 80%. This complement relationship is the key insight that makes "at least one" problems tractable, since P(none) is a simple product of three terms.
What if two events are mutually exclusive?+
If A and B are mutually exclusive, P(A∩B) = 0 and therefore P(A∩B∩C) = 0 as well. The union formula simplifies to P(A∪B∪C) = P(A) + P(B) + P(C) − P(A∩C) − P(B∩C). Mutually exclusive events with non-zero probability are dependent, so this calculator (which assumes independence) should not be used when mutual exclusivity is present.
How do I verify my 3-event probability calculation?+
Run these sanity checks: (1) P(at least one) + P(none) must equal exactly 100%. (2) P(all three) must be less than or equal to the smallest individual probability. (3) P(union) must equal P(at least one) for independent events. (4) All results must be between 0% and 100%. If any check fails, recheck your input values and ensure all events are truly independent.
Tags: Probability of 3 Events P(A and B and C) Union of Three Events At Least One Probability Inclusion Exclusion Independent Events Probability Calculator Statistics

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📌 Quick Tips

💡For independent events, P(A and B and C) = P(A) × P(B) × P(C). The joint probability is always smaller than any individual probability.
💡P(at least one) = 1 - P(none). It is always easier to compute P(none) first, then subtract from 1.
💡P(A∪B∪C) uses the inclusion-exclusion principle: add all three, subtract all pairwise products, add back the triple product.