Biological fitness is typically measured by the expected rate of reproduction, but strategies with high fitness can also have high probabilities of extinction. Likewise, gambling strategies with a high expected payoff can also have a high risk of ruin. We take inspiration from the gambler’s ruin problem to examine how extinction is related to population growth. Using moment theory we demonstrate how higher moments can impact the probability of extinction and how the first few moments can be used to find bounds on the extinction probability, focusing on s-convex ordering of random variables. This approach generates “best case” and “worst case” scenarios to provide upper and lower bounds on the probability of extinction.