This paper investigates how the general public behaves when confronted with low probability events and ambiguity in an insurance context. It reports the results of a questionnaire completed by a large representative sample of the French population that aims at separating attitudes toward risk, imprecision and conflict and at determining if there is a demand for ambiguous and extreme event risk. The data show is a strong distinction between two aspects of the problem: the decision of purchasing insurance and the willingness to pay. In the decision to insure, more than 25% of the respondents refuse to buy insurance and people are more willing to insure in a risky situation than in an ambiguous one. This certain taste for risk can be explained by the respondents' observable characteristics. In addition, it highlights a lack of confidence in the insurance markets. When it comes to willingness to pay, people exhibit ambiguity seeking behaviors. They are willing to pay more under risk than under ambiguity (embracing here imprecision and conflict), revealing that people consider ambiguous situations as inferior. Furthermore, respondents behave differently under imprecision and conflict. They exhibit a preference for consensual information and dislike conflicts. However, the willingness to pay is poorly correlated with observable characteristics.