D.3 Probability

Probability is counting: P(A) = (favourable outcomes) / (total outcomes in the sample space). The complement rule P(A') = 1 − P(A) shortcuts hard counts. Expected number of occurrences = n × P(A) — the long-run average over n trials. For combined events, the addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B) avoids double-counting; Venn, tree, and sample-space diagrams organise problems. Mutually exclusive events satisfy P(A ∩ B) = 0; independent events satisfy P(A ∩ B) = P(A) P(B). Conditional probability P(A | B) = P(A ∩ B) / P(B) restricts the sample space to B — at the heart of the base-rate fallacy in medical testing.