Previous work has demonstrated that search for a target in noise is consistent with the predictions of the optimal search strategy, both in the spatial distribution of fixation locations and in the number of fixations observers require to find the target. In this study we describe a challenging visual-search task and compare the number of fixations required by human observers to find the target to predictions made by a stochastic search model. This model relies on a target-visibility map based on human performance in a separate detection task. If the model does not detect the target, then it selects the next saccade by randomly sampling from the distribution of saccades that human observers made. We find that a memoryless stochastic model matches human performance in this task. Furthermore, we find that the similarity in the distribution of fixation locations between human observers and the ideal observer does not replicate: Rather than making the signature doughnut-shaped distribution predicted by the ideal search strategy, the fixations made by observers are best described by a central bias. We conclude that, when searching for a target in noise, humans use an essentially random strategy, which achieves near optimal behavior due to biases in the distributions of saccades we have a tendency to make. The findings reconcile the existence of highly efficient human search performance with recent studies demonstrating clear failures of optimality in single and multiple saccade tasks.