We (Hahn & Warren, 2009) recently proposed a new account of the systematic errors and biases that appear to be present in people's perception of randomly generated events. In a comment on that article, Sun, Tweney, and Wang (2010) critiqued our treatment of the gambler's fallacy. We had argued that this fallacy was less gross an error than it might at first appear, once the nature of people's actual experience was taken into account. In support of this claim, we had advanced a series of interconnected arguments. One of these involved a betting game that seemed extremely similar to the gambler's fallacy but that nevertheless involved a winning strategy. Sun et al. challenged the idea of this game, arguing that we had failed to take its context into account. Here, we implement the original game, demonstrate that it indeed involves a winning strategy as originally claimed, and identify where Sun et al.'s critique went wrong. Finally, key aspects of our general proposal are clarified in light of Sun et al.'s comments. (Contains 3 footnotes and 5 figures.)