If an individual with expected utility and a reasonable level of wealth rejects a small actuarially favorable gamble, it implies a very high degree of risk aversion. It also predicts (counterfactually) the rejection of more sizable and very attractive bets. If additional background uncertainty affects wealth, this result also applies to non-expected utilities. The authors describe a set of reasonable conditions under which an individual may reject the small bet but accept the large bet, even in the presence of background uncertainty. The two critical assumptions that the authors use are rank-dependent utility and a discrete distribution for background risk. Plausible calibrations can reconcile large/small bet risk attitudes and the empirical evidence on limited stock market participation in the presence of labor income risk. Copyright 2011, Oxford University Press.