Chapter 13 — Probability

Chapter 13 of the Class 12 Maths NCERT textbook, "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).

• Probability under given information — The chapter's starting point is conditional probability — how the chance of an event shifts once another event is known to have occurred — which effectively shrinks the sample space and leads naturally to the multiplication rule for combined events.
• Independence of events — It defines events as independent when one's occurrence tells you nothing about the other's, and draws out a subtle consequence: independent events with non-zero probability can never be mutually exclusive, sharpening students' intuition about how events interact.
• Total probability and Bayes' theorem — By partitioning a sample space, the theorem of total probability assembles an event's chance from its parts, and Bayes' theorem reverses the reasoning to update prior beliefs into posterior ones — applied to urn problems, medical testing and defect detection.

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).