Change the parameters of the distributions that keep appearing in the material and watch the shape, mean, and spread respond.
Poisson for counts, Gamma for magnitudes.
The Poisson has the striking property that its mean equals its variance (both λ), which is why it is the natural first model for counts of independent events in a fixed interval, arrivals in a queue, words in a document, mutations along a genome. When real counts are more variable than this, you reach for the negative binomial.
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Build probability from the ground up: possible worlds, the sample space, the two basic axioms, and the addition and multiplication rules, each derived rather than asserted, with worked numeric examples.