Most studies assessing animal decision-making under risk rely on probabilities that are typically larger than 10%. To study Decision-Making in uncertain conditions, we explore a novel experimental and modelling approach that aims at measuring the extent to which rats are sensitive - and how they respond - to outcomes that are both rare (probabilities smaller than 1%) and extreme in their consequences (deviations larger than 10 times the standard error). In a four-armed bandit task, stochastic gains (sugar pellets) and losses (time-out punishments) are such that extremely large - but rare - outcomes materialize or not depending on the chosen options. All rats feature both limited diversification, mixing two options out of four, and sensitivity to rare and extreme outcomes despite their infrequent occurrence, by combining options with avoidance of extreme losses (Black Swans) and exposure to extreme gains (Jackpots). Notably, this sensitivity turns out to be one-sided for the main phenotype in our sample: it features a quasi-complete avoidance of Black Swans, so as to escape extreme losses almost completely, which contrasts with an exposure to Jackpots that is partial only. The flip side of observed choices is that they entail smaller gains and larger losses in the frequent domain compared to alternatives. We have introduced sensitivity to Black Swans and Jackpots in a new class of augmented Reinforcement Learning models and we have estimated their parameters using observed choices and outcomes for each rat. Adding such specific sensitivity results in a good fit of the selected model - and simulated behaviors that are close - to behavioral observations, whereas a standard Q-Learning model without sensitivity is rejected for almost all rats. This model reproducing the main phenotype suggests that frequent outcomes are treated separately from rare and extreme ones through different weights in Decision-Making.