Recent developments in convolutional neural networks (CNNs) have introduced new ways to model the complex processes of human vision. To date, the comparison of human vision and CNNs has focused on internal representations (i.e., receptive fields), with behavioral comparisons left largely unexplored. Here, we probe the influence of cognitive strategy on the similarity between CNN output and human behavior. We gave study participants a superstitious perception task (i.e., we asked them to detect an assigned target in white noise) while asking them to engage in either an active or passive attentional strategy. Previous research has shown that an active attentional strategy tends to engage central executive functions, whereas a passive strategy allows perceptual processes to unfold with limited central control. The results showed that the pattern of human responses in the superstitious perception task depended significantly on task strategy. Specifically, detecting targets superstitiously (i.e., false alarms) was correlated with evidence of a target's presence in the passive condition, but not in the active condition.Human data were compared to the performance of a CNN performing the same task, with the decision criterion of the CNN set to match the false alarm rates observed in the two strategy conditions of the human participants. CNN responses resembled those of human participants in the passive condition more closely than those in the active condition. This observation suggests that the CNN does a better job of mimicking human behavior when central executive functions are not engaged than when they are engaged. This, in turn, has important implications for what human participants are doing in the superstitious perception task. Namely, it implies that superstitious perception may have two important ingredients that are somewhat dissociable. First, there is the ability to detect weak signals in noise that correspond to the target image. This appears to be what participants are doing under passive strategy conditions; they allow externally generated signals to dominate their perceptual experience. Second, there is the ability to ignore the noise in favor of basing responses solely on internally generated signals. This seems to correspond more closely to what participants are doing under active strategy conditions, when attention is controlled by representations in memory. This research emphasizes the importance of modeling the full range of human responsiveness in even a simple noisy detection task.