In this thesis, we study noise-induced tipping in a piecewise-smooth, linear, one-dimensional periodically forced system perturbed by weak additive noise. This problem is motivated by a recent model of energy flux in Arctic sea ice. We determine the most probable tipping events using path integral techniques. Specifically, we calculate local minimizes of the Onsager-Machlup functional using solutions to the corresponding Euler-Lagrange equations and a gradient flow applied to the functional. We also prove an extension of Kramer's law to determine bounds for the expected tipping time. Using these methods, we determine the most probable transition path from a frozen state to an unfrozen state and determine the seasons that are most susceptible to tipping.