Probability Concepts for Class 12 Maths – A Quick Guide

Probability is a fundamental concept in mathematics that helps us measure uncertainty in real-world scenarios. From rolling dice to making predictions in statistics, probability concepts help us understand the various events and experiments.

I hope, you found my previous articles and the handwritten downloadable notes helpful. I would love to hear from you and here also, at the end of this article, you can download a handwritten PDF summarizing the concepts for quick revision.

Table of Contents show

1. Basics of Probability: Types of Events

Random Experiment & Sample Space

A random experiment is an action where the outcome is uncertain.

The set of all possible outcomes is called the Sample Space ( $S$ ).

Each possible outcome is a sample point.

Example: Tossing a coin → Sample space:
$S = {H, T}$

Types of Events:

Sure event ( $S$ ) – Always happens.
Impossible event ( $\emptyset$ ) – Never happens.
Simple event – Contains one outcome (sample point).
Compound event – Contains multiple outcomes (sample points).

Download the handwritten PDF at the end for a summary of event types with examples.

2. Basics of Probability Using Sets & Venn Diagram

Union and Intersection of Events

Using Sets and Venn diagrams, probability relations can be understood visually.

Union (A or B):
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Intersection (A and B):
$P(A \cap B) = P(A) + P(B) - P(A \cup B)$

And there are more types of events and their respective probabilities, as mentioned in the image below.

Probability Basics — Basics of Probability

3. Conditional Probability

When an event’s probability depends on another event, it is given by:

$P(A | B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) \neq 0$

Example: If a card is drawn from a deck, what is the probability that it’s a King given that it’s a face card?

4. Multiplication Theorem

This theorem is useful for probability calculations involving multiple events.

$P(A \cap B) = P(A) \cdot P(B | A)$

5. Independent Events

Two events A and B are independent if one does not affect the probability of the other.

$P(A | B) = P(A)$ ,
$P(B | A) = P(B)$

Thus, for independent events, we have
$P(A \cap B) = P(A) \cdot P(B)$

Example: Tossing two dice – The outcome of one does not influence the other.

A detailed handwritten PDF is available for download at the end.

6. Law of Total Probability

If $E_1, E_2, ..., E_n$ are mutually exclusive events that form a partition of the sample space, then:

$P(A) = P(A | E_1) P(E_1) + P(A | E_2) P(E_2) + ... + P(A | E_n) P(E_n)$

This is useful when dealing with complex probability distributions.

7. Bayes’ Theorem

Used for updating probabilities based on new evidence. The formula is useful to get the probability of ’causes’. For example,

$P(E_i | A) = \frac{P(A | E_i) P(E_i)}{\sum P(A | E_i) P(E_i)}$ for any i = 1, 2, 3, …, n

Common applications include medical testing, spam filtering, machine learning and AI predictions.

8. Probability Distribution

A random variable ( $X$ ) assigns numerical values to outcomes.

For discrete random variables, the probability distribution satisfies:
$P(X = x_i) \geq 0, \quad \sum P(X = x_i) = 1$

9. Mean, Variance & Standard Deviation

Mean (Expected Value):
$E(X) = \sum x_i P(x_i)$

Variance:
$Var(X) = E(X^2) - [E(X)]^2$

Standard Deviation:
$\sigma = \sqrt{Var(X)}$

Formulas and examples are included in the downloadable PDF for quick reference as also shown in the image below.

10. Miscellaneous

Follwing are some of the key mathematical tools frequently used in probability:

Permutations & Combinations:
${}^nP_r = \frac{n!}{(n - r)!}$ ,
${}^nC_r = \frac{n!}{r!(n - r)!}$

Binomial Theorem:
$(a + b)^n = \sum_{r=0}^{n} {}^nC_r a^{n-r} b^r$

Infinite Geometric Series:
$S = \frac{a}{1 - r}, \quad |r| < 1$

Final Thoughts & Downloadable PDF

Probability concepts are widely used in mathematics, data science, and real-world predictions. A strong grasp of Conditional Probability, Bayes’ Theorem, Probability Distributions, Mean and Variance is essential for solving advanced problems.

For a quick revision, download the handwritten PDF below, summarizing the key probability concepts, formulas, and examples.

PROBABILITY CONCEPTS Download

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Probability Concepts for Class 12 Maths – A Quick Guide

1. Basics of Probability: Types of Events