Probability of 3 Events Calculator

Enter the probabilities of Event A, Event B, and Event C (as percentages between 0 and 100) to calculate key outcomes for three independent events. Your results include the probability that at least one event occurs (P(A∪B∪C)), the probability all three occur (P(A∩B∩C)), the probability exactly one occurs, and the probability none occur.

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Enter the probability of Event A as a percentage (0–100).

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Enter the probability of Event B as a percentage (0–100).

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Enter the probability of Event C as a percentage (0–100).

Results

P(A ∪ B ∪ C) — At Least One Event Occurs

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P(A ∩ B ∩ C) — All Three Events Occur

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Exactly One Event Occurs

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Exactly Two Events Occur

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P(A' ∩ B' ∩ C') — No Events Occur

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Frequently Asked Questions

What are independent events in probability?

Independent events are events where the occurrence of one does not affect the probability of any other. For example, rolling a die and flipping a coin are independent — the result of one has no influence on the other. This calculator assumes all three events A, B, and C are independent.

What is the formula for the probability that at least one of three events occurs?

The formula is 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). For independent events, each joint probability is the product of the individual probabilities, e.g. P(A∩B) = P(A) × P(B).

How do I calculate the probability that all three events occur?

For independent events, the probability that all three occur is simply the product of their individual probabilities: P(A∩B∩C) = P(A) × P(B) × P(C). If the events are not independent, you would need to use conditional probabilities instead.

How is the probability of exactly one event occurring calculated?

The probability that exactly one of three independent events occurs is: P(A)×(1−P(B))×(1−P(C)) + (1−P(A))×P(B)×(1−P(C)) + (1−P(A))×(1−P(B))×P(C). This sums the cases where only A occurs, only B occurs, and only C occurs.

What is the probability of none of the three events occurring?

The probability that none of the three independent events occur is: P(A')×P(B')×P(C') = (1−P(A))×(1−P(B))×(1−P(C)). It equals 1 minus the probability that at least one event occurs.

What are the rules of probability I should know?

Key rules include: probabilities always fall between 0 and 1; the complement rule states P(A') = 1 − P(A); the addition rule gives P(A∪B) = P(A) + P(B) − P(A∩B); and the multiplication rule for independent events gives P(A∩B) = P(A) × P(B). These rules extend naturally to three events.

What is the probability of an impossible event vs. a certain event?

An impossible event has a probability of 0 — it can never occur. A certain event has a probability of 1 — it always occurs. All real-world events have probabilities strictly between 0 and 1, expressed as decimals, fractions, or percentages.

Can this calculator be used for dependent events?

No — this calculator is designed specifically for independent events. For dependent events, the joint probability requires conditional probabilities: P(A∩B) = P(A) × P(B|A), which changes the calculation significantly. If your events influence each other, the results from this tool will not be accurate.