While individuals are often viewed as rational actors, engaging in action that promises success, sometimes they act despite low odds. We report two studies that investigate hope as a motivational resource during times when the odds of success seem low. We argue that when people are personally invested in the cause, their hope leaps with emerging possibility (low likelihood) of a positive outcome, but linearly aligns with likelihood for more probable outcomes (i.e., hope is a cubic function of likelihood). Crucially, hope then motivates support for collective action, in this case support for climate action, thus illuminating the possible antecedents for collective action against the odds. In Study 1, with a highly invested sample, hope mediated the relationship between cubic likelihood and support for climate change action. Study 2 extended these findings, showing that for individuals strongly invested in the outcome (but not for those less invested), hope arose with possibility but not probability of success, leading to greater support for climate change action. Hope's unique motivational role arises when the odds are low, when success is only possible rather than probable. These then represent the conditions that facilitate support for collective action against the odds.