NCERT Mathematics Class 12 - Chapter 13: Probability - Notes

Learning Objectives

• Understand conditional probability and multiplication theorem
• Learn about independent events and Bayes' theorem
• Study random variables, probability distributions, and Bernoulli trials
• Apply binomial distribution

Key Concepts

Conditional Probability

P(A|B) = P(A ∩ B)/P(B), provided P(B) ≠ 0. Read as "probability of A given that B has occurred." Properties: 0 ≤ P(A|B) ≤ 1. P(S|B) = 1. P((A ∪ C)|B) = P(A|B) + P(C|B) - P((A ∩ C)|B).

Multiplication Theorem: P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B). For three events: P(A ∩ B ∩ C) = P(A) × P(B|A) × P(C|A∩B).

Independent Events

Events A and B are independent if P(A ∩ B) = P(A) × P(B). Equivalently: P(A|B) = P(A) or P(B|A) = P(B). If A and B are independent, then A and B', A' and B, A' and B' are also independent. Note: Mutually exclusive ≠ independent. In fact, if P(A) > 0, P(B) > 0, and A, B are mutually exclusive, then they CANNOT be independent.

Bayes' Theorem

If E₁, E₂, ..., Eₙ are mutually exclusive and exhaustive events (partition of S), then for any event A: P(E_i|A) = P(E_i)P(A|E_i) / Σ P(E_j)P(A|E_j). P(E_i) = prior probability. P(E_i|A) = posterior probability. Total Probability Theorem: P(A) = Σ P(E_j)P(A|E_j) (used in denominator of Bayes').

Random Variables and Probability Distribution

A random variable X is a real-valued function on the sample space. Probability distribution: Table showing all values of X and their probabilities. ΣP(X = x_i) = 1. Mean (Expectation): E(X) = μ = Σx_iP(x_i). Variance: Var(X) = E(X²) - [E(X)]² = Σx_i²P(x_i) - μ². Standard deviation: σ = √Var(X).

Bernoulli Trials and Binomial Distribution

Bernoulli trial: Experiment with exactly two outcomes (success with probability p, failure with probability q = 1-p). Trials are independent. Binomial Distribution: X ~ B(n, p). Probability of exactly r successes in n trials: P(X = r) = C(n,r) p^r q^n-r. Mean: E(X) = np. Variance: Var(X) = npq. Standard deviation: σ = √(npq).

Properties: P(X = r) is maximized at r = [(n+1)p] (mode). Sum of all probabilities = (p + q)ⁿ = 1ⁿ = 1. If n is large and p is small, Binomial can be approximated by Poisson distribution (not in NCERT but useful for JEE).

Summary

Conditional probability modifies probability based on given information. Bayes' theorem reverses conditional probabilities using prior and likelihood. Independent events satisfy P(A ∩ B) = P(A)P(B). Random variables have probability distributions with mean and variance. Binomial distribution models the number of successes in n independent Bernoulli trials.

Important Terms

• Conditional probability: P(A|B) = P(A∩B)/P(B)
• Independent events: P(A∩B) = P(A)P(B)
• Bayes' theorem: Updates probability using new evidence
• Random variable: Function assigning numbers to outcomes
• Bernoulli trial: Experiment with two outcomes (success/failure)
• Binomial distribution: P(X=r) = C(n,r)p^rq^n-r
• Expectation: Mean value of random variable; E(X) = np for binomial

Quick Revision

• P(A|B) = P(A∩B)/P(B); P(A∩B) = P(B)P(A|B)
• Independent: P(A∩B) = P(A)P(B); mutually exclusive ≠ independent
• Bayes': P(E_i|A) = P(E_i)P(A|E_i) / ΣP(E_j)P(A|E_j)
• Total probability: P(A) = ΣP(E_j)P(A|E_j)
• E(X) = Σx_iP(x_i); Var(X) = E(X²) - [E(X)]²
• Binomial: P(X=r) = C(n,r)p^rq^n-r; Mean = np; Var = npq
• Σ all P(x_i) = 1 for any probability distribution