I struggle to explain discrete and continuous random variable, what it mean to integrate in terms of Ito, covariance as a measure for how much two variables jointly vary, and the like.
It would be great if there was a mental image so that one could easily transfer existing intuition in the deterministic variable to random variable. In differential geometry, for example, there is always this image of surfaces, direction and curves, which extends even to exterior and Clifford algebra.
What I am looking for is a real life intuition for the mapping from sample space to measurable space, which also maps the tools used in probability theory to something intuitive.
So far I explain it in terms of mixtures of fluids and probability densities, Fokker-Planck style.