additionsregeln

Additionsregeln, or addition rules in German, describe how to compute the size or probability of a union when more than one outcome or set can occur. They provide a general method for combining counts or probabilities and are closely related to the inclusion–exclusion principle.

In probability theory, the basic addition rule states that the probability of the union of two events A and B is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are disjoint, the intersection is empty and the rule reduces to P(A ∪ B) = P(A) + P(B). If A and B are independent, one can also use P(A ∪ B) = P(A) + P(B) − P(A)P(B). The rule generalizes to more events via inclusion–exclusion: P(∪i A_i) = sum P(A_i) − sum P(A_i ∩ A_j) + sum P(A_i ∩ A_j ∩ A_k) − ...

In set theory, the analogous rule relates to the sizes of unions: |A ∪ B| = |A| + |B| − |A ∩ B|, with the same generalization to larger families of sets through inclusion–exclusion. These formulas allow counting or probability calculations without double-counting overlapping outcomes.

Example: If P(A) = 0.4, P(B) = 0.3, and P(A ∩ B) = 0.2, then P(A ∪ B) = 0.4 + 0.3 − 0.2 = 0.5. If A and B are disjoint, P(A ∪ B) = 0.4 + 0.3 = 0.7.

Applications of additionsregeln appear in statistics, combinatorics, survey sampling, and probability puzzles, where accurate union calculations are essential and overlap between events must be accounted for.

---