Gegenwahrscheinlichkeiten

Gegenwahrscheinlichkeit is the probability of the complement of an event in probability theory. It refers to the likelihood that the opposite or an alternative outcome to a given event occurs. The Gegenwahrscheinlichkeit of an event A is written as P(A^c) (or P(Ā)) and by definition equals 1 minus the probability of A: P(A^c) = 1 − P(A), provided the probabilities are defined.

A common intuition is that the Gegenwahrscheinlichkeit represents all outcomes in the sample space that are not in A. For example, in a fair six-sided die, if A is the event “rolling an even number,” then P(A) = 3/6 = 1/2, and the Gegenwahrscheinlichkeit P(A^c) = 1 − 1/2 = 1/2, corresponding to rolling an odd number. In a fair coin toss, A = “heads” has P(A) = 0.5, so P(A^c) = 0.5 as well.

The concept extends to any event, including biased situations: if P(A) = 0.7, then P(A^c) = 0.3. The Gegenwahrscheinlichkeit also behaves predictably under conditioning: P(A^c | B) = 1 − P(A | B).

Gegenwahrscheinlichkeiten are a basic tool for simplifying calculations, especially when dealing with complements of events or when partitioning the sample space. They also relate to other probability rules, such as P(A ∪ B) = P(A) + P(B) − P(A ∩ B), via complementary reasoning.