Assume , , and are lotteries. Denote " is preferred to " as , and indifference between them by . One version of the probability axioms are then given by the following, the last of which is the independence axiom:

1. Completeness: either or .

2. Transitivity: .

3. Continuity: a unique such that .

4. Independence: if , then for all and .

This entry contributed by Ed Pegg, Jr. (author's link)

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Pegg, Ed Jr. "Independence Axiom." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/IndependenceAxiom.html