Probability of 3 Events Calculator

A three-event probability problem has eight possible occurrence patterns: all three events, each of the three possible pairs without the remaining event, each event alone, and no events. This calculator summarizes four especially useful combinations for independent events. It reports the probability that all three occur, at least one occurs, none occurs, and exactly one occurs. Inputs use the standard zero-to-one probability scale, while results are shown as percentages. When events are independent, their joint probability is found by multiplication. Therefore, the chance of all three is P(A) times P(B) times P(C). The chance of no events uses the complement of every input: one minus P(A), multiplied by one minus P(B), multiplied by one minus P(C). At least one event is most efficiently calculated with the complement rule. Subtract the probability of no events from one rather than adding seven separate cases. Exactly one event requires three mutually exclusive cases. In the first case A occurs while B and C do not. In the second, B occurs while A and C do not. In the third, C occurs while A and B do not. Multiply within each case and then add the three results. Because these cases cannot happen at the same time, their probabilities may be added without subtracting overlaps. These formulas require mutual independence, which is stronger than merely observing that event pairs appear unrelated. Independence means every relevant combination factors into its component probabilities. Repeated spins of a properly operating wheel may satisfy that assumption, but operational risks, medical symptoms, and financial events often share causes. If one occurrence changes another event's chance, conditional probabilities or a joint probability table are needed. Use probabilities derived from a consistent population, period, and definition. A calculated percentage expresses long-run chance under the model; it does not predict which event will occur on one particular trial. Check whether the three inputs and independence assumption are defensible before using a result in a decision. For dependent events, construct the eight joint outcomes directly or apply conditional multiplication in the correct order.