Chapter 13 — Probability

Class 12 Maths Chapter 13 (Probability) covers conditional probability, the multiplication rule, independent events, Bayes' theorem, and the theorem of total probability, building on the axiomatic approach introduced by A.N. Kolmogorov (1903–1987).

NCERT Class 12 Mathematics Chapter 13 introduces the conditional probability of an event E given event F as P(E|F) = P(E∩F)/P(F), provided P(F) ≠ 0. It derives the multiplication rule P(E∩F) = P(E)·P(F|E) and defines independent events as those satisfying P(E∩F) = P(E)·P(F). The chapter then proves the theorem of total probability and Bayes' theorem, which computes the posterior probability P(Ei|A) using prior probabilities and likelihoods. Worked examples include urn problems, HIV testing, defective bolts, and alternating dice games.

1. 01Conditional probability is defined as P(E|F) = P(E∩F)/P(F) when P(F) ≠ 0, reducing the effective sample space to outcomes favourable to F.
2. 02The multiplication rule states P(E∩F) = P(E)·P(F|E) = P(F)·P(E|F), and extends to three or more events.
3. 03Two events E and F are independent if P(E∩F) = P(E)·P(F); independent events with nonzero probabilities cannot be mutually exclusive.
4. 04The theorem of total probability states that for a partition {E1, E2, …, En} of sample space S, P(A) = Σ P(Ej)·P(A|Ej).
5. 05Bayes' theorem gives P(Ei|A) = P(Ei)·P(A|Ei) / Σ P(Ej)·P(A|Ej), allowing computation of reverse (posterior) probabilities from prior probabilities and likelihoods.
6. 06Key historical contributors include Pascal and Fermat (1654 correspondence founding probability theory), Jacob Bernoulli (Binomial distribution), and A.N. Kolmogorov (axiomatic theory, 1933).