 Mixture monotonicity

Assume $0\le q<p\le 1$ and write $L_q=qA+(1-q)B$. The independence axiom for lottery preferences gives $A\succ L_q$ because $1-q>0$. Set $\lambda =(p-q)/(1-q)>0$; then $L_p=\lambda A+(1-\lambda )L_q$. A second use of the same axiom gives $L_p\succ L_q$. Thus a higher probability of the strictly preferred outcome strictly improves a two-outcome lottery.