Wahrscheinlichkeitsrechner

This calculator works out how two named events, A and B, relate to one another. It covers the four relationships that come up in practice: independence, mutual exclusivity, conditional probability and the union of two events.

Every probability model built on two events can be expressed with the same four numbers — both, A only, B only, neither — and those four always add to 1. Once you have them, any question about the pair can be answered, including the odds in favour of an outcome.

When the events are independent, the overlap is simply the product P(A) × P(B). Everything else follows by subtraction: A only is P(A) minus the overlap, B only is P(B) minus the overlap, and the remainder is neither. The union is then the overlap plus the two exclusive regions, which equals P(A) + P(B) − P(A ∩ B).

Conditional probability inverts that overlap: P(A | B) is the part of B that overlaps with A, expressed as a fraction of all of B. Rearranging the same identity gives P(A ∩ B) = P(A | B) × P(B), which is how the conditional question reconstructs the overlap from a known conditional probability.

Bayes’ theorem runs the identity in the other direction. Given a likelihood P(B | A) and a prior P(A), the posterior is the product divided by the total probability of the evidence P(B). This is the calculation behind every diagnostic test result you have ever been given.

The headline probability depends on the question you asked, so always check the label next to it — P(A ∩ B) for independence, P(A | B) for the conditional and Bayes questions, P(A ∪ B) for the union question.

The relationship verdict is the useful part. "Independent" means the overlap matches the product; "Not independent" means the events are linked; "Mutually exclusive" means the overlap is zero and the union is simply the sum of the two probabilities.

Odds in favour are the probability divided by one minus it. A probability of 0.25 is odds of 1 to 3, which sounds very different until you notice it is the same number written differently.

• The base rate. Bayes’ theorem is dominated by how common the condition is, which is why a highly accurate test still produces mostly false positives in a rare disease.
• Whether you are asking about one draw or a long run. P(A ∩ B) for a single pair of events is not the same as the proportion of times a coincidence occurs over many repetitions.
• Whether the probabilities you supply are themselves estimates. Small errors in a base rate or a sensitivity figure propagate straight through to the posterior.
• The direction of conditioning. Reversing P(A | B) into P(B | A) changes the answer unless the events really are independent.

• Reading a diagnostic test result and wanting the probability of the condition rather than the probability of the result.
• Working out whether two events in a dataset are independent before modelling them separately.
• Checking that a set of event probabilities describes a possible situation at all.
• Any coursework or exam question involving independence, mutual exclusivity, conditional probability or Bayes’ rule.

• This is arithmetic on the inputs you supply. It cannot tell you whether those inputs are right, and no precision in the arithmetic compensates for a wrong base rate.
• The calculator assumes classical probability with two events only. Repeated trials, competing risks and non-independent observations all need more structure than two probabilities and one overlap.
• Bayes’ theorem is exact for a single hypothesis. Comparing several competing hypotheses requires a further step — normalising across them — which this calculator does not attempt.