We explore hurricane and ocean reanalysis data to understand how rapid intensification (RI) of tropical cyclones is impacted by the upper ocean density structure, with an emphasis on barrier layer (BL) thickness and thermocline depth in the eastern Caribbean Sea and adjacent western tropical North Atlantic. This analysis leverages transition path theory (TPT), supported by basic statistical methods. In TPT, Markov chains are constructed by discretizing data series related to weather system intensity, changes in intensity, translational speed, and BL thickness and thermocline depth. These series are viewed as trajectories in abstract state spaces, following a memoryless stochastic process. RI imminence is rigorously framed using a newly derived TPT statistic, which gives the time distribution to first reach a target -- the RI state -- from a source -- for instance, the state determined by a certain BL range and system intensity -- conditional on connecting paths exhibiting minimal detours. Increased RI frequency is observed in the eastern Caribbean and nearby Atlantic, influenced by river runoff, primarily in tropical storms and category 2 hurricanes. RI frequently correlates with a well-developed BL; however, increased translational speed is necessary for RI. TPT shows a stronger connection between RI and thermocline depth than BL presence, with RI likelihood rising for hurricanes with a thin BL, especially category 1. Across all strength categories, a deep thermocline consistently elevates RI probability, a factor missed by basic statistical analysis. Furthermore, translational speed is crucial, with faster, stronger hurricanes more susceptible to RI, while slower systems are less so.