If an agent's preferences over subjectively uncertain acts are consistent with his or her having a subjective probability distribution over the states of nature, then those preferences can induce consistent preferences over "objectively" risky lotteries. Such "probabilistically sophisticated" behavior thus allows us to treat decision making under situations of uncertainty in an analogous manner to those under risk. This paper first characterizes exactly what probabilistic sophistication entails for an agent's beliefs about the likelihood of states of nature. Secondly, it presents characterizations of probabilistically sophisticated individuals whose induced lottery preferences obey neither the Independence Axiom (unlike Savage (1954, 1972)) nor a monotonicity property that shares some of the nature of Independence (unlike Machina and Schmeidler (1992)).