Würfelsätze

Würfelsätze is a German term that refers to a group of propositions or theorems about dice experiments, primarily used in probability theory and statistics. The phrase is common in didactic or problem-collection contexts and does not denote a single canonical theorem. Instead, Würfelsätze describes basic, frequently encountered results that illustrate fundamental concepts such as distributions, expected values, variances, and independence in the context of dice.

Typical contents of Würfelsätze include results for simple dice experiments. For two fair six‑sided dice, the sum S takes values from 2 to 12 with a triangular distribution: P(S = 2) = 1/36, P(S = 3) = 2/36, P(S = 4) = 3/36, up to P(S = 7) = 6/36, and then symmetric decreases to P(S = 12) = 1/36. The expected value is E[S] = 7, and the variance Var(S) = 35/6. Other common results cover independent throws, e.g., the probability of at least one six in two throws is 11/36, and properties of sums under repeated throws.

Generalizations to n dice extend these ideas. The sum X of n fair dice has mean E[X] = 3.5n and variance Var(X) = (35/12)n. The distribution is the n-fold convolution of the uniform {1,...,6} distribution and can be analyzed via counting, generating functions, or normal approximation for large n. Würfelsätze thus serve as accessible foundations for exploring probability concepts in education.