NCERT Solutions for Class 12 Mathematics Chapter 13: Probability – Free PDF Download

Chapter 13 of Class 12 Maths is Probability, the final chapter of the syllabus. It covers conditional probability, multiplication theorem, independent events, Bayes theorem, and probability distributions. These Class 12 Mathematics Chapter 13 solutions are also useful as quick revision notes before exams.

Conditional Probability

The conditional probability of event A given event B has occurred is P(A|B)=P(A∩B)/P(B), provided P(B)>0.

Multiplication Theorem

P(A∩B)=P(A)·P(B|A)=P(B)·P(A|B). Extended to 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) and P(B|A)=P(B).

Total Probability Theorem and Bayes Theorem

If E₁,E₂,…,E_n are mutually exclusive and exhaustive events (a partition of the sample space), the law of total probability gives P(A)=ΣP(E_i)P(A|E_i). Bayes theorem reverses conditioning: P(E_i|A)=[P(E_i)P(A|E_i)]/[ΣP(E_j)P(A|E_j)].

Random Variables and Probability Distributions

A random variable X assigns a numerical value to each outcome. Its probability distribution lists all values with their probabilities (summing to 1). Mean E(X)=Σx_ip_i; Variance Var(X)=E(X²)−[E(X)]².

Bernoulli Trials and Binomial Distribution

Bernoulli trials are repeated independent trials with only two outcomes (success/failure). The binomial distribution gives P(X=r)=ⁿC_rp^rq^n−r, where p=probability of success, q=1−p.

Frequently Asked Questions

What is the formula for conditional probability?
P(A|B)=P(A∩B)/P(B), for P(B)>0.

When are two events independent?
When P(A∩B)=P(A)·P(B).

What is Bayes theorem used for?
To find the probability of a cause given an observed effect, reversing the direction of conditioning.

Written by Satish

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