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. Also try the Dice Probability Calculator.
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. See also our calculate Combined Probability (Fraction), Event 1 Probability (Fraction) & Event 2 Probability (Fraction) — Probability Fraction.
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).
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.
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.
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.
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.
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.
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. You might also find our calculate Total Probability P(A), P(A) as Percentage & Sum of P(Bᵢ) — Should Equal 1 — Total Probability Theorem useful.