Assume X, Y, and Z are lotteries. Denote "X is preferred to Y" as X succeeds Y, and indifference between them by X~Y. One version of the probability axioms are then given by the following, the last of which is the independence axiom: 1. Completeness: for all X, Y either X succeeds Y, Y succeeds X or X~Y. 2. Transitivity: X succeeds Y, Y succeeds Z⟹X succeeds Z. 3. Continuity: for all X succeeds Y succeeds Z, exists a unique p such that p X + (1 - p) Z~Y. 4. Independence: if X succeeds Y, then p X + (1 - p) Z succeeds p Y + (1 - p) Z for all Z and p element (0, 1).

Allais paradox | probability axioms | Saint Petersburg paradox