CBSE Class 12 Mathematics: Probability (Class 12) — Important Questions with Answers 2026

Short Answer Questions (2-3 Marks)

• Q: State and prove Bayes' Theorem.
  Ans: Bayes' Theorem: If E₁, E₂, ..., Eₙ are mutually exclusive and exhaustive events with P(Eᵢ) > 0, and A is any event with P(A) > 0, then: P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ) / ΣP(Eⱼ)P(A|Eⱼ). Proof: P(Eᵢ∩A) = P(Eᵢ)P(A|Eᵢ) = P(A)P(Eᵢ|A). So P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ)/P(A). Since E₁...Eₙ are exhaustive: P(A) = ΣP(Eⱼ∩A) = ΣP(Eⱼ)P(A|Eⱼ). Substituting gives Bayes' formula.
• Q: A bag contains 4 red and 6 blue balls. Two balls are drawn at random. What is P(both red)?
  Ans: Total balls = 10. P(both red) = C(4,2)/C(10,2) = 6/45 = 2/15. Alternatively: P(1st red) × P(2nd red|1st red) = 4/10 × 3/9 = 12/90 = 2/15.
• Q: Three machines A, B, C produce 25%, 35%, 40% of items. Defective rates are 5%, 4%, 2%. An item is defective. Find P(from machine C).
  Ans: P(A) = 0.25, P(B) = 0.35, P(C) = 0.40. P(D|A) = 0.05, P(D|B) = 0.04, P(D|C) = 0.02. P(D) = P(A)P(D|A) + P(B)P(D|B) + P(C)P(D|C) = 0.0125 + 0.014 + 0.008 = 0.0345. By Bayes': P(C|D) = P(C)P(D|C)/P(D) = 0.008/0.0345 = 0.232 ≈ 23.2%.

Exam Tips for This Chapter

• Revise all definitions and laws from Probability (Class 12) — commonly asked as 1-2 mark questions
• Practice diagrams related to Probability (Class 12) — neat labelled diagrams carry 2-3 marks
• For numericals, always show formula → substitution → answer with correct units
• Previous year analysis shows Probability (Class 12) carries significant marks in the board exam