According to quantum mechanics, there is a non-zero probability that a single electron from the cup could quantum tunnel through the paper. However, the likelihood of such an event occurring within a reasonable time frame is extremely low.
Quantum tunneling is a phenomenon in which a particle can pass through a barrier that would be classically impossible to penetrate. It arises from the wave-like nature of particles described by quantum mechanics. In the case of the electron in the cup, it is subject to the potential barrier formed by the paper. Although the barrier appears solid at the macroscopic level, at the quantum level, there is always a finite probability that the electron can tunnel through the barrier.
However, the probability of quantum tunneling decreases exponentially with the thickness and height of the barrier. In the case of a typical sheet of paper, the barrier is relatively high compared to the size of the electron, and the probability of tunneling through such a barrier within a short time span is extremely small.
Additionally, the environment and various interactions that the electron experiences can also affect the tunneling probability. Factors such as temperature, electromagnetic fields, and interactions with other particles can influence the likelihood of tunneling.
In practical terms, the probability of a single electron from a cup quantum tunneling under a sheet of paper within a 10-year timeframe is infinitesimally small and can be considered negligible for all practical purposes.