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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About This Chapter

This comprehensive study guide for Probability is designed for Class X students following the CBSE and NCERT Mathematics curriculum. It covers 4 key subtopics including Classical definition of probability review, Complementary events, Probability of cards, coins, and dice games, and 1 more essential concepts. Whether you are preparing for school examinations, CBSE board exams, or competitive tests, this resource provides everything you need to build a strong conceptual foundation and achieve mastery.

The chapter includes 1 key formulas and equations, 1 fully worked step-by-step example problems, interactive practice exercises across 9 difficulty categories, timed mock quizzes, and downloadable worksheets. Each topic is explained with detailed concept definitions, mathematical representations, and expert study guidelines to help you understand not just the "how" but the "why" behind every formula and method.

Mathematics is a subject that rewards consistent practice and conceptual clarity over rote memorization. As you work through this chapter on Probability, focus on understanding the underlying principles first, then gradually increase problem difficulty. Use the practice sections to identify and strengthen weak areas, and refer to the common mistakes section to avoid the pitfalls that most students encounter.

What You'll Learn in This Chapter

By the end of studying Probability for Class X, you will have developed proficiency in the following learning outcomes as outlined by the NCERT syllabus:

Calculate probabilities of single and compound events.

Apply complementary event formulas (P(E) + P(not E) = 1).

Describe impossible (P=0) and sure (P=1) event limits.

Prerequisites for This Chapter

Before studying Probability, make sure you are comfortable with the following prerequisite concepts. A strong foundation in these areas will help you understand new topics faster and solve problems more confidently:

Data HandlingSmart Charts Bar

If any of these prerequisites feel unfamiliar, consider reviewing them first using the Related Chapters section at the bottom of this page. Building a solid base ensures you can tackle Probability with full confidence.

Real-World Applications of Probability

Students often wonder “Where will I use Probabilityin real life?” The answer is: everywhere. The mathematical concepts you learn in this chapter have practical applications across science, engineering, technology, medicine, finance, and everyday problem-solving. Here are some notable examples:

Weather Forecasting Models

Meteorologists use probability distributions to predict the likelihood of rain, storms, and temperature fluctuations days in advance.

Insurance & Risk Assessment

Actuaries calculate premiums by evaluating the probability of events like accidents, illness, or property damage occurring within a population.

Game Theory & Strategy

From card games to stock trading, probability guides optimal decision-making under uncertainty and helps evaluate expected outcomes.

Medical Diagnostics

Doctors use conditional probability (Bayes' Theorem) to assess the likelihood of a disease given positive or negative test results.

Understanding the real-world relevance of Probability not only makes learning more engaging but also helps you appreciate how mathematical thinking is a superpower that opens doors in virtually every career path — from engineering and medicine to finance and technology.

Study Tips for Probability

Follow these expert study strategies to maximize your understanding and exam performance in this chapter. These tips are specifically tailored for the type of content covered in Probability:

📋

Organize Data in Tables First

Always create a frequency distribution table before computing mean, median, or mode. Organized data makes calculations straightforward and reduces counting errors significantly.

🎲

List the Sample Space Completely

For probability problems, write out the complete sample space before calculating probabilities. Missing even one outcome changes the denominator and invalidates your answer.

📊

Choose the Right Central Tendency

Mean is best for symmetric data, median for skewed distributions (income data), and mode for categorical data (favorite colors). Choosing the wrong measure gives misleading results.

Pro Tip: Consistency beats intensity. Studying Probability for 30 minutes daily is far more effective than cramming for 5 hours before the exam. Use the practice sections below to build muscle memory through regular problem-solving.

Detailed Topic Breakdown

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.

Scientific Functions Calculator

Solve trigonometric, exponential, logarithmic, and root functions for standard algebraic values.

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Value (X)45

Function

Calculated Results

Function Output

0.71

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Solve trigonometric, exponential, logarithmic, and root functions for standard algebraic values.

1. Input Parameters

Parameter	Value	Unit
Value (X)	45	—
Function	sin	—

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Metric	Calculated Output
Function Output	0.707

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- operand1 = 45

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- calcResult = 0.7071

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Class X CBSE Board Practice Questions

Review and solve 67 real Board paper questions from the 20 mock sample sets matching Probability.

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Sample Paper 01 | Question 14Section A (1 Mark)

A bag contains 5 pink, 8 blue and 7 yellow balls. One ball is drawn at random from the bag. What is the probability of getting neither a blue nor a pink ball ?

Sample Paper 01 | Question 29Section C (3 Marks)

Two coins are tossed simultaneously. What is the probability of getting (i) At least one head? (ii) At most one tail? (iii) A head and a tail?

Sample Paper 01 | Question 37Section E (4 Marks)

A tiling or tessellation of a flat surface is the covering of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. Historically, tessellations were used in ancient Rome and in Islamic art. You may find tessellation patterns on floors, walls, paintings etc. Shown below is a tiled floor in the archaeological Museum of Seville, made using squares, triangles and hexagons. = 10 sq. units

Sample Paper 02 | Question 14Section A (1 Mark)

The probability of getting a bad egg in a lot of 400 is 0.035. The number of bad eggs in the lot is

Sample Paper 02 | Question 37Section E (4 Marks)

Case Study - 2 In order to conduct sports day activities in your school, lines have been drawn with chalk powder at a distance of 1 m each in a rectangular shaped ground ABCD. 100 flower pots have been placed at the distance of 1 m from each other along AD, as shown in the following figure. Niharika runs ( 1 4 )th distance AD on the 2nd line and posts a green Flag. Preet runs ( 1 5 ) th distance AD on the eighth line and posts are red flags. Taking A as the origin AB along x -axis and AD along y -axis,

Sample Paper 02 | Question 38Section E (4 Marks)

Case Study - 3 Saving money is a good habit and it should be inculcated in children from the beginning. A father brought a piggy bank for his son Aditya. He puts one five -rupee coin of his savings in the piggy bank on the first day. He increases his savings by one five-rupee coin daily. (i) If the piggy bank can hold 190 coins of five rupees in all, find the number of days he can contribute to put the five-rupee coins into it (ii) Find the total money he saved. OR If 6 times the 6th term of an A.P., is equal to 9 times the 9th term, find its 15th term.

Sample Paper 03 | Question 14Section A (1 Mark)

A card is selected from a deck of 52 cards. The probability of being a red face card is

Sample Paper 04 | Question 14Section A (1 Mark)

A card is selected at random from a well shuffled deck of 52 cards. The probability of its being a face card is

Sample Paper 04 | Question 26Section C (3 Marks)

A train covered a certain distance at a uniform speed. If the train would have been 6 km/h faster, it would have taken 4 hours less than the scheduled time. And, if the train were slower by 6 km/hr; it would have taken 6 hours more than the scheduled time. Find the length of the journey.

Sample Paper 04 | Question 28Section C (3 Marks)

Two coins are tossed simultaneously. What is the probability of getting (i) At least one head? (ii) At most one tail? (iii) A head and a tail?

Sample Paper 04 | Question 36Section E (4 Marks)

Saving money is a good habit and it should be inculcated in children from the beginning. A father brought a piggy bank for his son Aditya. He puts one five-rupee coin of his savings in the piggy bank on the first day. He increases his savings by one five-rupee coin daily. (i) If the piggy bank can hold 190 coins of five rupees in all, find the number of days he can contribute to put the five-rupee coins into it (ii) Find the total money he saved. OR If 6 times the 6th term of an A.P., is equal to 9 times the 9th term, find its 15th term.

Sample Paper 05 | Question 27Section C (3 Marks)

All the black face cards are removed from a pack of 52 playing cards. The reaming cards are well shuffled and then a card is drawn at random. Find the probability of getting (i) face card (ii) red card (iii) black card.

Sample Paper 05 | Question 29Section C (3 Marks)

The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.

Sample Paper 06 | Question 10Section A (1 Mark)

Two dice are thrown at the same time. The probability of getting not doublet is

Sample Paper 07 | Question 15Section A (1 Mark)

If P(E) = 0.07, then what is the probability of ‘not E’?

Sample Paper 07 | Question 17Section A (1 Mark)

A box contains cards numbered 6 to 50. A card is drawn at random from the box. The probability that the drawn card has a number which is a perfect square is :

Sample Paper 07 | Question 22Section B (2 Marks)

Two dice are thrown at the same time. Find the probability of getting (i) same number on both dice (ii) different numbers on both dice.

Sample Paper 08 | Question 4Section A (1 Mark)

A bag has 5 white marbles, 8 red marbles and 4 purple marbles. If we take a marble randomly, then what is the probability of not getting purple marble?

Sample Paper 08 | Question 16Section A (1 Mark)

If P(A) denotes the probability of an event A, then

Sample Paper 08 | Question 17Section A (1 Mark)

A card is selected at random from a well shuffled deck of 52 playing cards. The probability of its being a face card is

Sample Paper 08 | Question 21Section B (2 Marks)

A card is drawn at random from a well shuffled pack of 52 cards. Find the probability of getting (i) a red king (ii) a queen or a jack

Sample Paper 08 | Question 38Section E (4 Marks)

Lahari has to buy a scooty. She can buy scooty either making cashdown payment of Rs. 25,000 or by making 15 monthly instalments as below. Ist month = Rs. 3425, IInd month = Rs. 3225, Illrd month = Rs. 3025, IVth month = Rs. 2825 and so on (i) Find the amount of 6th instalment. [1] (ii) Find the total amount paid in 15 instalments. [2] OR (ii) If Lahari pays Rs. 82625 then find the number of instalment. [2] (iii) Lahari paid 10th and 11th instalment together find the amount paid that month.

Sample Paper 09 | Question 10Section A (1 Mark)

All queens, jacks and aces are removed from a pack of 52 playing cards. The remaining cards are well-shuffled and one card is picked up at random from it. The probability of that card to be a king is :

Sample Paper 09 | Question 14Section A (1 Mark)

Two dice are rolled simultaneously. What is the probability that 6 will come up at least once?

Sample Paper 09 | Question 26Section C (3 Marks)

In a teachers' workshop, the number of teachers teaching French, Hindi and English are 48, 80 and 144 respectively. Find the minimum number of rooms required if in each room the same number of teachers are seated and all of them are of the same subject.

Sample Paper 10 | Question 2Section A (1 Mark)

If the probability of a player winning a game is 0.79, then the probability of his losing the same game is:

Sample Paper 10 | Question 4Section A (1 Mark)

From the data l, 4, 7, 9, 16, 21, 25, if all the even numbers are removed, then the probability of getting at random a prime number from the remaining is :

Sample Paper 10 | Question 10Section A (1 Mark)

The probability of getting a chocolate flavoured ice cream at random, in a lot of 600 ice creams is 0.055. The number of chocolate flavoured ice creams in the lot is :

Sample Paper 10 | Question 25Section B (2 Marks)

In a pack of 52 playing cards one card is lost. From the remaining cards, a card is drawn at random. Find the probability that the drawn card is queen of heart, if the lost card is a black card.

Sample Paper 10 | Question 27Section C (3 Marks)

Sample Paper 10 | Question 31Section C (3 Marks)

Due to heavy floods in a state, thousands were rendered homeless. 50 schools collectively decided to provide place and the canvas for 1500 tents and share the whole expenditure equally. The lower part of each tent is cylindrical with base radius 2.8 m and height 3.5 m and the upper part is conical with the same base radius, but of height 2.1 m. If the canvas used to make the tents costs ₹120 per m2, find the amount shared by each school to set up the tents.

Sample Paper 10 | Question 37Section E (4 Marks)

BINGO is game of chance. The host has 75 balls numbered 1 through 75. Each player has a BINGO card with some numbers written on it. The participant cancels the number on the card when called out a number written on the ball selected at random. Whosoever cancels all the numbers on his/her card, says BINGO and wins the game. The table given below, shows the data of one such game where 48 balls were used before Tara said 'BINGO'. Numbers announced Number of times 0-15 8 15-30 9 30-45 10 45-60 12 60-75 9 Based on the above information,

Sample Paper 11 | Question 12Section A (1 Mark)

A card is drawn at random from a well -shuffled pack of 52 cards. The probability that the card drawn is not an ace is:

Sample Paper 11 | Question 17Section A (1 Mark)

Two dice are thrown together. The probability of getting the difference of numbers on their upper faces equals to 3 is:

Sample Paper 11 | Question 20Section A (1 Mark)

Assertion (A): The probability that a leap year has 53 Sunday is 2/7. Reason (R): The probability that a non-leap year has 53 Sunday is 5/7.

Sample Paper 11 | Question 23Section B (2 Marks)

If a fair coin is tossed twice, find the probability of getting 'atmost one head'.

Sample Paper 11 | Question 36Section E (4 Marks)

Jagdhish has a field which is in the shape of a right angled triangle AQC. He wants to leave a space in the form of a square PQRS inside the field from growing wheat and the remaining for growing vegetables (as shown in the figure). In the field, there is a pole marked as O. Based on the above information,

Sample Paper 12 | Question 4Section A (1 Mark)

Two dice are thrown at the same time and the product of numbers appearing on them is noted. The probability that the product is a prime number is

Sample Paper 12 | Question 21Section B (2 Marks)

The probability of selecting a blue marble at random from a jar that contains only blue, black and green marbles is 1/5. The probability of selecting a black marble at random from the same jar is 1/4. If the jar contains 11 green marbles, find the total number of marbles in the jar.

Sample Paper 12 | Question 36Section E (4 Marks)

Aditya plantations have two rectangular fields of the same width but different lengths. They are required to plant 168 trees in the smaller field and 462 trees in the larger field. In both fields, the trees will be planted in the same number of rows but in different number of columns. (i) What is the maximum number of rows in which the trees can be planted in each of the fields? (2) (ii) If the trees are planted in the number of rows obtained in part (i), how many columns will each field have? (iii) If total cost of planted trees in one column is Rs. 500, then find the cost to plant the trees in smaller field. OR If the total cost of planted trees in one column is Rs. 500, the find the cost to plant the trees in larger field.

Sample Paper 12 | Question 37Section E (4 Marks)

Shivani took a pack of 52 cards. She kept aside all the black face cards and shuffled the remaining cards well. Based on the above information

Sample Paper 13 | Question 27Section C (3 Marks)

Sample Paper 14 | Question 18Section A (1 Mark)

A ticket is drawn at random from a bag containing tickets numbered from 1 to 40. The probability that the selected ticket has a number which is a multiple of 5 is

Sample Paper 14 | Question 26Section C (3 Marks)

Cards numbered 1 to 30 are put in a bag. A card is drawn at random from this bag. Find the probability that the number on the drawn card is (i) not divisible by 3. (ii) a prime number greater than 7. (iii) not a perfect square number.

Sample Paper 15 | Question 8Section A (1 Mark)

A card is drawn from a well-shuffled deck of 52 playing cards. Find the probability of getting a red face card.

Sample Paper 15 | Question 9Section A (1 Mark)

Two dice are thrown together. The probability of getting a doublet is:

Sample Paper 15 | Question 30Section C (3 Marks)

Two dice are thrown together. Find the probability of getting a sum of (i) 8, (ii) 7, (iii) less than or equal to 12.

Sample Paper 16 | Question 11Section A (1 Mark)

A card is drawn at random from a pack of 52 cards. The probability that the card drawn is not a face card is:

Sample Paper 16 | Question 15Section A (1 Mark)

A die is thrown once. What is the probability of getting a prime number?

Sample Paper 16 | Question 37Section E (4 Marks)

Case Study 2: Six cards are removed from a deck of 52 cards (kings and queens of red colour). From the remaining cards, one card is drawn. (i) Find total number of remaining cards. (ii) Find probability of getting a face card. (iii) Find probability of getting a black card.

Sample Paper 17 | Question 6Section A (1 Mark)

Two dice are thrown simultaneously. The probability that the product of the numbers appearing on the dice is 7 is

Sample Paper 17 | Question 7Section A (1 Mark)

The probability of guessing the correct

Sample Paper 17 | Question 12Section A (1 Mark)

If a digit is chosen at random from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9; then the probability that this digit is an odd prime number is:

Sample Paper 17 | Question 19Section A (1 Mark)

Assertion(A): In a cricket match, a batsman hits a boundary 9 times out of 45 balls he plays. The probability that in a given ball, he does not hit the boundary is 4/5. Reason(R): P(E) + P(not E) = 1

Sample Paper 17 | Question 22Section B (2 Marks)

One card is drawn at random from a well shuffled deck of 52 cards. Find the probability that the card drawn (i) is queen of hearts; (ii) is not a jack.

Sample Paper 18 | Question 17Section A (1 Mark)

A number x is chosen at random from the numbers -3, -2, -1, 0, 1, 2, 3 the probability that |x| < 2 is

Sample Paper 18 | Question 20Section A (1 Mark)

Assertion (A): The probability of getting number 8 on rolling a die is zero (0). Reason (R): The probability of an impossible event is zero (0).

Sample Paper 18 | Question 30Section C (3 Marks)

One card is drawn at random from a well -shuffled deck of 52 playing cards. Find the probability that the card drawn is : (i) a red king. (ii) not a black card. (iii) an ace of hearts.

Sample Paper 18 | Question 37Section E (4 Marks)

On the roadway, Points A and B, which stand in for Chandigarh and Kurukshetra, respectively, are located nearly 90 kilometres apart. At the same time, a car departs from Kurukshetra and one from Chandigarh. These cars will collide in 9 hours if they are travelling in the same direction, and in 9/7 hours if they are travelling in the other direction. Let X and Y be two cars that are travelling at x and y kilometres per hour from places A and B, respectively. On the basis of the above information,

Sample Paper 18 | Question 38Section E (4 Marks)

Vocational training complements traditional education by providing practical skills and hands -on experience. While education equips individuals with a broad knowledge base, vocational training focuses on job -specific skills, enhancing employability thus making the student self - reliant. Keeping this in view, a teacher made the following table giving the frequency distribution of students/adults undergoing vocational training from the training institute. Age (in years) 15-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 Number of participants 62 132 96 37 13 11 10 4 From the above

Sample Paper 19 | Question 18Section A (1 Mark)

Two dice are thrown together. The probability that they show different numbers is :

Sample Paper 19 | Question 31Section C (3 Marks)

From a pack of 52 playing cards, jacks, queens, kings and aces of red colour are removed. From the remaining a card is drawn at random. Find the probability that the card drawn is (i) a black queen (ii) a red card (iii) a face card.

Sample Paper 19 | Question 33Section D (5 Marks)

(a) In a flight of 2800 km, an aircraft was slowed down due to bad weather. Its average speed is reduced by 100km/h and by doing so, the time of flight is increased by 30 minutes. Find the original duration of the light.

Sample Paper 19 | Question 35Section D (5 Marks)

200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on (see below figure). In how may rows are the 200 logs placed and how many logs are in the top row?

Sample Paper 20 | Question 11Section A (1 Mark)

Two dice are thrown simultaneously. What is the probability of getting doublet?

Sample Paper 20 | Question 25Section B (2 Marks)

A rope by which a cow is tethered is increased from 16mto 23m. How much additional ground does it have now to graze?

Sample Paper 20 | Question 29Section C (3 Marks)

One card is drawn at random from a well-shuffled deck of 52 playing cards. Find the probability that the card drawn is (i) either a red card or a king, (ii) neither a red card nor a queen.

Print-Ready Math Worksheet

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• Title: Class X Math: Probability Printable Sheet
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• Sections: Formulas checklist, 5 assessment questions, worked keys

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Class X Mathematics: Probability

Name: ______________________Date: __________________Score: ______ / 5

Section I: Key Formulas

Complementary Probability

P(E) + P(\bar{E}) = 1

Section II: Answer the Questions

1. Two coins are tossed simultaneously. What is the probability of getting (i) At least one head? (ii) At most one tail? (iii) A head and a tail?