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Probability (Class 12)

NCERT Class 12 Mathematics Chapter 13

Probability for Class 12 Maths and JEE: conditional probability, the multiplication theorem, independent events, Bayes’ theorem and random variables.

5 topics Updated 8 Nov 2026

For: Class 12 and JEE, unless a topic says otherwise

In the syllabus: JEE Class 12 syllabus

With your textbook

NCERT Mathematics Part II (Class 12), Chapter 13 “Probability”, reprint 2026-27.

Open the official NCERT chapter (PDF, 2.9 MB, ncert.nic.in)

Checked on 19 Oct 2026.

Learn in this order

  1. Conditional probability: definition, properties and examples

    Conditional probability for Class 12 Maths and JEE: P(E|F) = P(E ∩ F)/P(F), the reduced sample space, properties, coin and dice examples, NCERT questions.

    NCERT Section 13.2 9 min read 6 solved examples 14 practice questions

  2. Multiplication theorem on probability

    Multiplication theorem of probability for Class 12 Maths and JEE: P(E ∩ F) = P(E) P(F|E), three events, probability trees, draws without replacement.

    NCERT Section 13.3 7 min read 4 solved examples 9 practice questions

  3. Independent events in probability

    Independent events for Class 12 Maths and JEE: P(E ∩ F) = P(E) P(F), independent versus mutually exclusive, three events, complements, NCERT questions.

    NCERT Section 13.4 10 min read 6 solved examples 12 practice questions

  4. Bayes’ theorem: partition, total probability and examples

    Bayes’ theorem for Class 12 Maths and JEE: partitions, the theorem of total probability, one path over all paths on a tree, NCERT examples and practice.

    NCERT Section 13.5 13 min read 6 solved examples 13 practice questions

  5. Probability distribution of a random variable

    Building the table, probabilities that add up to 1, mean and variance, with dice and card examples.

    JEE only 10 min read 4 solved examples 9 practice questions

Key formulas in this chapter

Conditional probability

  • P(E|F) = P(E ∩ F) / P(F), P(F) ≠ 0
  • P(S|F) = P(F|F) = 1
  • P(E′|F) = 1 − P(E|F)

Multiplication theorem

  • P(E ∩ F) = P(E) P(F|E) = P(F) P(E|F)
  • P(E ∩ F ∩ G) = P(E) P(F|E) P(G|E ∩ F)

Independent events

  • independent: P(E ∩ F) = P(E) P(F)
  • P(A ∪ B) = 1 − P(A′) P(B′) for independent A, B
  • three events: all pairs and the triple multiply

Bayes’ theorem

  • P(A) = P(E) P(A|E) + … + P(E) P(A|E)
  • P(E|A) = P(E) P(A|E) / Σ P(E) P(A|E)

Probability distribution

  • p = P(X = x), p ≥ 0, p + … + p = 1
  • μ = E(X) = Σ x p
  • Var(X) = Σ (x − μ) p

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