Probability quantifies uncertainty under a model. Statistics uses data to estimate, test, or update that model. The two subjects share language — random variables, distributions, expectation — but ask different questions: “what should happen if the model is true?” versus “what do the data suggest about the world?”
Events, conditionals, and independence
An event is a set of outcomes we care about. Conditional probability $P(A\mid B)$ reweights the sample space to outcomes inside $B$. Independence means learning $B$ does not change the probability of $A$ — a modeling assumption, not a vibe. Confusion between $P(A\mid B)$ and $P(B\mid A)$ is one of the most common reasoning errors in applied work; Bayes’ theorem is the disciplined bridge between them. See Bayes’ theorem without the fog.
Random variables and expectation
A random variable assigns a number to each outcome. Its distribution describes how those numbers are spread. Expectation is a probability-weighted average — a center of mass for the distribution, not necessarily a “typical” single draw when the distribution is skewed.
Statistics: estimates with humility
Sample means, variances, and proportions estimate population quantities. Confidence intervals and hypothesis tests formalize uncertainty — but only under assumptions. Always ask what was sampled, what could confound, and whether the model is even plausible.