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Adiabatic heating of solar wind electrons at the Earth's bow shock and itsforeshock region produces transversely anisotropic hot electrons that, in turn,generate intense high-frequency whistler-mode waves. These waves are oftendetected by spacecraft as narrow-band, electromagnetic emissions in thefrequency range of [0.1,0.5] of the local electron gyrofrequency. Resonantinteractions between these waves and electrons may cause electron accelerationand pitch-angle scattering, which can be important for creating the electronpopulation that seeds shock drift acceleration. The high intensity andcoherence of the observed whistler-mode waves prohibit the use of quasi-lineartheory to describe their interaction with electrons. In this paper, we aim todevelop a new theoretical approach to describe this interaction, thatincorporates nonlinear resonant interactions, gradients of the backgrounddensity and magnetic field, and the fine structure of the waveforms thatusually consist of short, intense wave-packet trains. This is the first of twoaccompanying papers. It outlines a probabilistic approach to describe thewave-particle interaction. We demonstrate how the wave-packet size affectselectron nonlinear resonance at the bow shock and foreshock regions, and how toevaluate electron distribution dynamics in such a system that is frequented byshort, intense whistler-mode wave-packets. In the second paper, thisprobabilistic approach is merged with a mapping technique, which allows us tomodel systems containing short and long wave-packets.

Electron resonant interaction with whistler-mode waves around the Earth's bow shock I: the probabilistic approach