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Decision making often requires balancing exploitation of familiar options with exploration of uncertain alternatives. One strategy for exploration is random exploration, in which choices become more variable and less tied to the option that currently appears best. However, it remains unclear whether this variability reflects genuine stochasticity or instead a deterministic structure that traditional cognitive models fail to capture. We examined this question using the Horizon Task in an aggregated dataset of 1,392 participants completing over 1,270,695 trials. In this task, exploration becomes more valuable when participants have more future choices, and choice variability increases under longer horizons. We tested whether more expressive neural network models could recover deterministic structure from this additional variability. Although neural networks provided better overall fits to behavior than the traditional model, long-horizon choices remained consistently less predictable than short-horizon choices. These findings show that limited model expressiveness cannot fully account for the increased variability associated with random exploration, suggesting random exploration truly is substantially random.