Probability Class 12 Notes: Conditional Probability, Bayes’ Theorem and Distributions

Quick answer: Probability is 8 marks in Class 12 boards and the Bayes’ theorem question is almost fixed. This post gives the rules and the exact way to write the Bayes solution.

Rules

• Conditional: P(A|B) = P(A ∩ B)/P(B)
• Multiplication: P(A ∩ B) = P(A) P(B|A)
• Independent: P(A ∩ B) = P(A) P(B)
• Total probability: P(A) = Σ P(Eᵢ) P(A|Eᵢ)
• Bayes: P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ) / Σ P(Eⱼ)P(A|Eⱼ)

Bayes standard example

Bag I has 3 red, 4 black; Bag II has 5 red, 6 black. A bag is chosen at random and a red ball is drawn. Probability it came from Bag II?

E₁ = Bag I, E₂ = Bag II, P(E₁) = P(E₂) = 1/2. P(R|E₁) = 3/7, P(R|E₂) = 5/11.

P(E₂|R) = (1/2 × 5/11)/(1/2 × 3/7 + 1/2 × 5/11) = (5/11)/(3/7 + 5/11) = (5/11)/(68/77) = 35/68.

Independence check

A die: A = even, B = ≤ 4. P(A) = 1/2, P(B) = 2/3, P(A ∩ B) = 2/6 = 1/3 = P(A)P(B) → independent.

Random variables

• Mean μ = Σ xᵢ pᵢ; variance σ² = Σ xᵢ²pᵢ − μ²
• Example: X = number of heads in 2 tosses: P(0) = 1/4, P(1) = 1/2, P(2) = 1/4 → mean 1, variance 1/2

Bernoulli trials / binomial

• P(X = r) = nCr pʳ qⁿ⁻ʳ; mean np; variance npq
• Example: probability of exactly 2 sixes in 4 throws = 4C2 (1/6)²(5/6)² = 25/216

Writing tip

Define events in words first, write all given probabilities, then apply the formula — step marks are given for each.

Common mistakes

• Confusing P(A|B) with P(B|A)
• Forgetting the denominator (total probability) in Bayes
• Using nCr without pʳqⁿ⁻ʳ

Related: Permutations and combinations.

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Frequently asked questions

Are these Probability Class 12 Notes notes enough for the exam?

They cover every formula, definition and question type that appears in school and board papers for this chapter. Use them with the NCERT examples, exercise questions and the last five years' papers to be fully prepared.

How should I revise Probability Class 12 Notes?

Revise the notes after 1 day, 7 days and 30 days. Solve five questions each time instead of only reading, and keep an error list of the steps you get wrong.

What if I get stuck on a probability class 12 question?

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