• Overview
• Details
• Exercises

Conditional probability updates beliefs given evidence. Bayes’ theorem inverts conditionals, linking prior, likelihood, and posterior: a foundation for inference and decision-making.

• Conditional probability: \(\mathbb{P}(A\mid B)=\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}\), for \(\mathbb{P}(B)>0\).
• Product rule: \(\mathbb{P}(A\cap B)=\mathbb{P}(A\mid B)\,\mathbb{P}(B)=\mathbb{P}(B\mid A)\,\mathbb{P}(A)\).
• Law of total probability: for partition \(\{B_i\}\), \(\mathbb{P}(A)=\sum_i \mathbb{P}(A\mid B_i)\,\mathbb{P}(B_i)\).
• Bayes’ theorem: \(\mathbb{P}(A\mid B)=\frac{\mathbb{P}(B\mid A)\,\mathbb{P}(A)}{\sum_i \mathbb{P}(B\mid A_i)\,\mathbb{P}(A_i)}\).
• Odds form: \(\text{posterior odds} = \text{prior odds} \times \text{likelihood ratio}\).
• Common pitfalls: base-rate neglect, false positive paradox.

1. Medical testing: given sensitivity/specificity and prevalence, compute the posterior probability of disease after a positive test.
2. Derive Bayes’ theorem from the product rule and total probability.
3. Partition example: compute \(\mathbb{P}(A\mid B)\) with three disjoint causes \(A_1,A_2,A_3\).