Free Online Probability Calculator
Calculate single and combined event probabilities.
Enable manual input by clearing the independence checkbox. The intersection cannot exceed either probability.
Probabilities must be between 0 and 1. The calculator reports the union, intersection, complement and conditional probabilities, with a note when a conditional result is undefined.
Working
- Enter P(A) and P(B) to see the formulas.
About the Probability Calculator
Probability quantifies uncertainty on a scale from zero to one. This calculator combines two events, A and B, and reports the chance of either one, both, neither, and one given the other. It supports the independent case, where the events do not influence each other, and a manual intersection case for events that overlap in a way you already know. Results appear as percentages with the matching decimal, making them easy to read in either format.
Union, intersection and the addition rule
The union P(A ∪ B) is the chance that at least one event happens. Counting P(A) plus P(B) double-counts the overlap, so the calculator subtracts the intersection: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B cannot happen together, the intersection is zero and the union is simply the sum. If one event is entirely contained in the other, the union equals the larger probability. The result is always between zero and one when the inputs are valid.
Independence and conditional probability
Independent events satisfy P(A ∩ B) = P(A) × P(B), which is the default mode. Two coin flips are independent; drawing without replacement from a deck is not. Conditional probability asks how the chance of one event changes once the other is known: P(A | B) = P(A ∩ B) / P(B). If B has probability zero, the conditional is undefined rather than zero, and the calculator displays a dash instead of inventing a number. The reverse conditional P(B | A) is computed the same way.
Complements and neither event
The complement P(A′) is one minus P(A), the chance that A does not occur. The chance that neither event occurs is the complement of the union: 1 − P(A ∪ B). This value is useful in risk questions such as “what is the chance that neither alarm triggers?” or “what is the chance that at least one backup fails?”. All these expressions follow from the same two-event structure, so the calculator can derive them without extra inputs.
Manual intersection and validation
When you clear the independence checkbox, the intersection field becomes editable. This is useful when you know the overlap from data, such as the share of customers who bought both products. The intersection must be between zero and the smaller of P(A) and P(B); otherwise the inputs are inconsistent and the calculator stops with an error. Probabilities outside 0 to 1, empty fields, or text values are also rejected. The working section shows the substitution with your numbers.
Frequently asked questions
What does independence mean in practice?
Two events are independent when knowing one occurred does not change the probability of the other. The product rule then applies. Real-world events are often only approximately independent, so treat the assumption as a modeling choice rather than a fact.
Can P(A ∪ B) exceed 1?
Not with valid inputs. The union formula can only exceed 1 if the intersection is smaller than P(A) + P(B) − 1, which is impossible when the intersection is consistent with both events. The calculator validates this and warns if the numbers cannot describe a real probability space.
Why is a conditional probability shown as a dash?
Conditioning on an event with probability zero is undefined because the formula divides by zero. For example, P(A | B) when P(B) = 0 has no meaningful value under standard probability rules.