Probability Calculator

Calculation Type

BasicComplement (A')Union (A ∪ B)Intersection (A ∩ B)Conditional (A|B)

P(A) - Probability of Event A

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Probability Rules Reference

Complement Rule:

P(A') = 1 - P(A)

Addition Rule:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Multiplication Rule (Independent):

P(A ∩ B) = P(A) × P(B)

Conditional Probability:

P(A|B) = P(A ∩ B) / P(B)

Choose the event question

The calculator handles supplied probabilities using five modes: a basic probability, complement, union, independent intersection and conditional probability. It does not fit probabilities from observations or calculate binomial, Poisson or other distribution functions.

Probabilities must be from 0 to 1 and refer to the same experiment and population. For P(A∪B), enter P(A), P(B) and the overlap P(A∩B). A valid overlap lies between max(0,P(A)+P(B)−1) and min(P(A),P(B)); individually valid numbers can still form an incompatible combination.

Worked examples

With P(A)=0.4, P(B)=0.3 and overlap 0.15, the union is 0.4+0.3−0.15=0.55. It means A or B or both, not exactly one of them.

For P(A|B), divide the overlap by P(B). Here 0.15/0.3=0.5. The reversed conditional P(B|A) would be 0.15/0.4=0.375. The denominator is the event after “given.” P(B)=0 makes the elementary ratio undefined and is rejected.

The independent-intersection mode multiplies marginal probabilities. For the same marginals it gives 0.12, which differs from the specified overlap 0.15. That mode assumes independence; it does not test or establish it. For dependent events, use P(A∩B)=P(A)P(B|A).

A complement uses 1−P(A), giving 0.6 for P(A)=0.4.

Before using the result

Clarify outcomes, overlapping categories and population. Supplied model probabilities differ from observed proportions, which carry sampling and selection uncertainty. A probability describes a model or defined empirical distribution, not a guarantee about the next short run.

Continue

Read probability rules and conditional probability, or use the Bayes calculator to reverse a conditional with a base rate.