  The fitness of a biological strategy is typically measured by its expected
reproductive rate, the first moment of its offspring distribution. However,
strategies with high expected rates can also have high probabilities of
extinction. A similar situation is found in gambling and investment, where
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. We discuss how moments can be
used to find bounds on the extinction probability, focusing on s-convex
ordering of random variables, a method developed in actuarial science. This
approach generates "best case" and "worst case" scenarios to provide upper and
lower bounds on the probability of extinction. Our results demonstrate that
even the most fit strategies can have high probabilities of extinction.
