CBSE Class 12 Maths: Probability — Complete Notes 2026

Conditional Probability

• P(A|B) = P(A∩B)/P(B); probability of A given B has occurred
• Multiplication theorem: P(A∩B) = P(A) × P(B|A) = P(B) × P(A|B)
• Independent events: P(A|B) = P(A); P(A∩B) = P(A)×P(B); occurrence of B does not affect probability of A

Bayes' Theorem

• P(Bₖ|A) = P(Bₖ) × P(A|Bₖ) / Σᵢ P(Bᵢ) × P(A|Bᵢ); finds posterior probability given evidence
• Partition of sample space: mutually exclusive and exhaustive events B₁, B₂, …, Bₙ
• Classic problem: two bags with balls; draw one ball; given colour, find which bag it came from

Random Variables

• Random variable X: function assigning numerical value to each outcome in sample space
• Discrete RV: finite or countably infinite values; Continuous RV: uncountably infinite (measurement data)
• Probability distribution: table of X values and their probabilities; Σ P(X = xᵢ) = 1

Mean and Variance

• Mean (Expected Value) E(X) = Σ xᵢ P(xᵢ); weighted average of values
• E(X²) = Σ xᵢ² P(xᵢ); Variance Var(X) = E(X²) − [E(X)]²
• Standard deviation = √Var(X); measure of spread

Binomial Distribution

• Conditions: n independent trials; each trial has 2 outcomes (success p, failure q=1−p); X = no. of successes
• P(X = r) = ⁿCᵣ × pʳ × qⁿ⁻ʳ; r = 0, 1, 2, …, n
• Mean = np; Variance = npq; Standard deviation = √(npq)

CBSE Problem Types

• Bag and ball problems: use Bayes theorem; find probability of picking from bag given ball colour
• Binomial distribution: P(at least 3 successes in 6 trials with p = 0.4): P(X ≥ 3) = 1 − P(X ≤ 2)
• Construct probability distribution table: find P for each value; verify sum = 1; find mean and variance

CBSE Board Focus

• Probability: 8–10 marks; Bayes theorem and Binomial distribution are two separate marks-heavy topics
• Bayes theorem: identify events, write prior probabilities, find P(A|Bᵢ), apply formula systematically
• Binomial: state distribution, find mean and variance, calculate specific probability