How can I double check my math answers?

Resolve equations using alternative methods (e.g., check quadratic roots by plugging them back into ax² + bx + c), verify decimals using scientific solvers, and review step-by-step worked calculations.

Where are these syllabus concepts applied in real life?

Trigonometry is used in architecture, mechanics and navigation. Quadratic equations model rocket trajectories and profit boundaries. Basic arithmetic and fractions are applied in accounting, budgeting and lab chemistry.

Back to Class X Mathematics

Classes X, XI & XII Mathematics

Chapter 14: Probability

Standard NCERT & CBSE aligned study curriculum. Master concepts, track accuracy, revise weak areas, and challenge yourself with 9 customized practice modes.

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Chapter Overview

Welcome to Class X Mathematics: Probability. This chapter forms a core structural component of the math syllabus, designed to build analytical rigor and key formula models.

Use the detailed subtopic guide below to review standard definitions, key mathematical rules, and study guidelines.

Prerequisite Concepts

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Detailed Subtopics Study Guide

Review detailed conceptual explanations, mathematical equations, and guidelines for each subtopic in this chapter:

1Classical definition of probability review

Concept Explanation

The classical definition of probability defines the likelihood of an event E occurring as the ratio of the number of outcomes favorable to E to the total number of equally likely outcomes in the sample space.

Mathematical Representation

P(E) = \frac{n(E)}{n(S)} \quad \text{where } 0 \le P(E) \le 1

Study Guideline: Ensure all outcomes are equally likely. The probability of any event is always between 0 (impossible event) and 1 (sure event).

2Complementary events

Concept Explanation

The complement of an event E is the event 'not E', denoted as E'. The sum of the probability of an event and its complementary event is always exactly 1.

Mathematical Representation

P(E) + P(E') = 1 \implies P(\text{not } E) = 1 - P(E)

Study Guideline: Use the complement rule to simplify calculations when finding 'at least one' probability: P(at least one) = 1 - P(none).

3Probability of cards, coins, and dice games

Concept Explanation

Standard probability scenarios use playing cards (52 cards: 26 red, 26 black; 4 suits of 13 cards each), coins (1 coin: 2 outcomes; 2 coins: 4 outcomes; 3 coins: 8 outcomes), and dice (1 die: 6 outcomes; 2 dice: 36 outcomes).

Mathematical Representation

n(S)_{\text{coins}} = 2^k, \quad n(S)_{\text{dice}} = 6^d, \quad n(S)_{\text{cards}} = 52

Study Guideline: List the sample space systematically. For two dice, write down the 36 pairs (1,1) to (6,6) to avoid missing favorable combinations.

4Impossible and Sure events

Concept Explanation

An impossible event is an event that can never occur; its probability is 0 (e.g., rolling a 7 on a standard die). A sure (or certain) event is guaranteed to happen; its probability is 1 (e.g., rolling a number less than 7).

Mathematical Representation

P(\emptyset) = 0 \, (\text{Impossible}), \quad P(S) = 1 \, (\text{Sure})

Study Guideline: If your calculated probability is less than 0 or greater than 1, you have made a mathematical error.