The formula
How it works
Combine the probabilities of two independent events to find the chance that both happen, that at least one happens, or that neither does. Enter each probability as a percentage.
FAQ
What does “independent” mean here?
Two events are independent when one does not affect the other — like two coin flips. This calculator assumes independence; if the events influence each other, the “and” rule needs a conditional probability instead.
Why isn’t “A or B” just A plus B?
Because simply adding double-counts the case where both happen. The formula subtracts that overlap, P(A)·P(B), so each outcome is counted exactly once.
What is the difference between independent and mutually exclusive events?
Independent events don’t affect each other’s odds, while mutually exclusive events can never happen together. Two mutually exclusive events are actually the opposite of independent, since knowing one occurred tells you the other definitely didn’t.
Can a probability be greater than 100%?
No — a valid probability always falls between 0% and 100%. This calculator clamps any input outside that range so the results stay meaningful.
How is the “neither” probability calculated?
Neither event happens when both fail, so it’s the product of each event’s complement: \((1 - P(A))(1 - P(B))\). It’s a useful check, since “neither” plus “at least one” should always add up to 100%.
Does this work for more than two events?
This calculator only handles two events, but the same logic extends further — for independent events, “all happen” is still the product of every individual probability.
Why do the four outcomes add up to more than 100%?
They don’t all belong to one partition — “at least one” already includes the “both” case, so it overlaps with it. The outcomes that add up cleanly to 100% are “both”, “exactly one”, and “neither”.
About the probability calculator
This calculator combines the probabilities of two independent events to work out the chances of different outcomes: both happening, at least one happening, or neither. Probability measures how likely something is on a scale from 0 to 100%, and combining events is where it gets genuinely useful — and where intuition often goes wrong. Laying out the “and”, “or” and “neither” results side by side makes the logic clear.
How to use it
Enter the probability of each event as a percentage. The calculator returns the probability that both A and B occur, that at least one occurs, that neither occurs, and that exactly one does. For example, with two events each at 50%, both happen 25% of the time, at least one happens 75% of the time, and neither happens 25% of the time. The events are treated as independent, so one has no effect on the other.
The formula
For independent events, both happening is the product, . At least one happening uses , where the subtraction removes the double-counted overlap. The chance of neither is , and exactly one is the “or” probability minus the “and” probability. Every probability stays between 0 and 1.
Where it is used
Combining probabilities is fundamental to risk, gaming and decision-making. Insurers use it to price the chance of multiple claims, engineers to work out the reliability of systems with several parts, and gamblers to judge the odds of parlays and combined bets. It also underlies weather forecasts, medical test results and quality control — anywhere the chance of a combined outcome matters.