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 
.
