Probability Axioms & Bayes

The probability axioms are three conditions for measuring uncertainty — non-negativity $P(A) \ge 0$, normalization $P(\Omega) = 1$, and countable additivity $P(\bigcup_{i} A_i) = \sum_i P(A_i)$ for disjoint $A_i$. They yield conditional probability $P(A|B) = \frac{P(A \cap B)}{P(B)}$, the law of total probability $P(B) = \sum_i P(B|A_i)P(A_i)$, and Bayes' theorem $P(A_i|B) = \frac{P(B|A_i)P(A_i)}{\sum_j P(B|A_j)P(A_j)}$, i.e. posterior $\propto$ likelihood $\times$ prior.

the starting point of every probabilistic calculation — classic interview problems (positive test → disease probability, spam classification) and the shared foundation of Bayesian inference and generative modeling via the $p(x,z)$ factorization.

intuition ignores base rates — with a 1% prevalence and 99% sensitivity/specificity, a positive test yields only about 50% posterior probability of disease, not 99%; Bayes' rule forces in the prior and evidence normalization, preventing the systematic overestimation of the base rate fallacy.

State the three probability axioms and prove $P(A^c) = 1 - P(A)$ and $P(A \cup B) = P(A) + P(B) - P(A \cap B)$?

Derive Bayes' theorem and the law of total probability? What is the base rate fallacy?

Prevalence 1%, sensitivity and specificity 99% — what is the posterior given a positive test?

Which terms are the prior, likelihood, evidence and posterior, and how do they jointly shape inference?

Conditional independence vs independence? Why does $P(A|B) = P(A)$ mean $A,B$ are independent?