Conditional probability, the multiplication rule, independent events, the theorem of total probability and Bayes' theorem. Our own step-by-step solutions to Exercises 13.1 to 13.3 and the Miscellaneous Exercise, one page per exercise, set out for step marks.
Each exercise has its own page with every question solved step by step. Pick the one you are on.
What it tests. \(P(E|F) = \dfrac{P(E \cap F)}{P(F)}\) (\(P(F) \ne 0\)): restrict the sample space to F. With equally likely outcomes, \(P(E|F) = \dfrac{n(E \cap F)}{n(F)}\). \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\).
What it tests. \(P(E \cap F) = P(E)P(F|E)\); for draws without replacement multiply the changing fractions. E and F are independent when \(P(E \cap F) = P(E)P(F)\) (then their complements are independent too), so 'at least one' \(= 1 - P(\text{none})\).
What it tests. If \(E_1, \ldots, E_n\) partition the sample space: \(P(A) = \sum P(E_i)P(A|E_i)\) (total probability) and \(P(E_i|A) = \dfrac{P(E_i)P(A|E_i)}{\sum_j P(E_j)P(A|E_j)}\) (Bayes). Name the events, write the priors \(P(E_i)\) and the likelihoods \(P(A|E_i)\), then substitute; a tree diagram keeps the products straight.
What it tests. Mixed practice with conditional probability, independence and Bayes' theorem; list small sample spaces completely.
Probability has 39 original board-style questions (MCQ, assertion–reason, short and long answers, case studies) with step mark schemes, revision notes and a four-step Route to 95. Three sample questions are open to everyone; a free account opens the rest.
Also useful: free MCQs and case studies for Probability · Class 12 formula sheet · official CBSE board and sample papers · our sample papers with marking scheme · the Route to 95 plan
Textbook: NCERT Mathematics Class 12, Parts I and II (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.