Delay discounting and probability discounting decision making tasks in rodent models have high translational potential. However, it is unclear whether the discounted value of the large reward option is the main contributor to variability in animals’ choices in either task, which may limit translatability to human discounting data. Male and female mice underwent sessions of delay and probability discounting in sequence to assess how choice behavior adapts over experience with each task. To control for “anchoring” (persistent choices based on the initial delay or probability), mice experienced “Worsening” schedules where the large reward was offered under initially favorable delay or probability conditions that became less favorable during testing, followed by “Improving” schedules where the large reward was offered under initially unfavorable conditions that improved over a session. During delay discounting, both male and female mice showed elimination of anchoring effects over training. In probability discounting, both sexes of mice continued to show some anchoring even after months of training. One possibility is that noisy action selection could contribute to these anchoring effects, rather than persistent fluctuations in value discounting. We fit choice behavior in individual animals using models that included both a value-based discounting parameter and a decision noise parameter that captured variability in choices deviating from value maximization. Changes in anchoring behavior over time were tracked by changes in our decision noise parameter, not the value parameter. Thus, changes in discounting behavior in mice can result from changes in exploration of the environment rather than changes in reward valuation.