  We prove that a random choice rule satisfies Luce's Choice Axiom if and only
if its support is a choice correspondence that satisfies the Weak Axiom of
Revealed Preference, thus it consists of alternatives that are optimal
according to some preference, and random choice then occurs according to a tie
breaking among such alternatives that satisfies Renyi's Conditioning Axiom. Our
result shows that the Choice Axiom is, in a precise formal sense, a
probabilistic version of the Weak Axiom. It thus supports Luce's view of his
own axiom as a "canon of probabilistic rationality."
