Probability
1.0Master Theoretical Outcomes, Sample Spaces, and Deck Determinations in Minutes
Unlock the mathematical logic governing chance and predictability. Learn how to construct exhaustive sample spaces, master the fractional balance dividing favorable events from total possible outcomes, and analyze classical coin, dice, and playing card distributions to ace your Class 10 board exams.
2.0Learning Outcomes
After completing this chapter, you will be able to:
- Define probability and related terms.
- Understand experiments, outcomes, sample space, and events.
- Calculate the probability of simple events.
- Determine the probability of complementary events.
- Apply probability concepts to solve real-life situations.
- Interpret probability values correctly.
- Solve NCERT, competency-based, and CBSE Board examination questions confidently.
3.0Introduction to the Probability
Have you ever wondered how likely it is to get a head when tossing a coin, roll a six on a die, or pick a red ball from a bag? The branch of mathematics that helps us measure the likelihood of such events is called Probability.
Probability is widely used in weather forecasting, sports analysis, insurance, finance, medicine, artificial intelligence, and scientific research. In this chapter, you'll learn how to calculate the probability of simple events using the classical approach, understand terms such as experiment, outcome, sample space, and event, and solve real-life problems involving chance and uncertainty. This chapter is one of the easiest and highest-scoring units in the CBSE Class 10 Mathematics syllabus.
Probability is one of the most fascinating branches of mathematics because it helps us measure uncertainty and predict the likelihood of future events. Whether you're tossing a coin, rolling a die, checking the weather forecast, or analyzing sports statistics, probability plays an important role in everyday decision-making. It allows us to make logical predictions based on available information rather than relying on guesswork.
For CBSE Class 10 students, Probability is an essential topic that forms the foundation for higher studies in mathematics, statistics, economics, artificial intelligence, data science, finance, and machine learning. A strong understanding of probability not only helps in scoring well in board examinations but also develops analytical and logical thinking skills.
4.0What is Probability?
Probability is the branch of mathematics that deals with the chance or likelihood of an event occurring. It provides a numerical measure of uncertainty, helping us predict how likely an event is to happen.
The value of probability always lies between 0 and 1, where:
- 0 indicates an impossible event.
- 1 indicates a certain event.
- A value between 0 and 1 represents the likelihood of an event occurring.
For example:
- The probability of getting Head when tossing a fair coin is 1/2.
- The probability of rolling a 7 on a standard six-sided die is 0 because it is impossible.
- The probability of getting a number less than 7 on a die is 1 because it is certain.
Probability helps us answer questions such as:
- What is the chance of rain tomorrow?
- What is the probability of drawing an Ace from a deck of cards?
- What are the chances of winning a lottery?
- What is the likelihood of getting an even number on a die?
These questions demonstrate how probability is applied in both everyday life and scientific decision-making.
5.0Why is Probability Important?
Probability is used across various fields because it enables informed decision-making under uncertain conditions. Some common applications include:
- Predicting weather conditions.
- Insurance risk assessment.
- Medical diagnosis and disease prediction.
- Stock market analysis.
- Artificial Intelligence and Machine Learning.
- Quality control in manufacturing.
- Sports analytics and performance prediction.
- Election surveys and opinion polls.
- Scientific research and experiments.
Understanding probability helps students develop logical reasoning and data interpretation skills, which are valuable across multiple disciplines.
6.0Important Terminology
Before solving probability problems, it is essential to understand some fundamental terms.
Experiment: An experiment is an activity or process that produces one or more outcomes.
Examples:
Tossing a coin
Rolling a die
Drawing a card from a deck
Selecting a student from a class
Trial: A trial is one performance of an experiment.
Examples:
Tossing a coin once
Rolling a die once
If a coin is tossed ten times, it consists of 10 trials.
7.0Outcome
An outcome is a possible result obtained from an experiment.
Examples:
Coin Toss:
Rolling a Die:
Sample Space: The sample space is the complete set of all possible outcomes of an experiment.
It is usually represented by S.
Examples
Coin Toss
S = {H, T}
Number of outcomes = 2
Rolling a Die
S = {1, 2, 3, 4, 5, 6}
Number of outcomes = 6
Two Coin Tosses
S = {HH, HT, TH, TT}
Number of outcomes = 4
Event: An event is a collection of one or more outcomes from the sample space.
Examples:
When rolling a die,
Event A = Getting an even number
A = {2, 4, 6}
Event B = Getting a prime number
B = {2, 3, 5}
An event may consist of:
One outcome (Elementary Event)
Multiple outcomes (Compound Event)
8.0Equally Likely Outcomes
Outcomes are said to be equally likely if each outcome has the same chance of occurring.
Examples include:
- Tossing a fair coin
- Rolling a fair die
- Drawing one card from a well-shuffled deck
Since no outcome is favoured over another, all outcomes have equal probability.
9.0Types of Probability
Probability can be classified into two major categories depending on how it is calculated.
1. Classical (Theoretical) Probability: Classical Probability is based on logical reasoning and assumes that all outcomes are equally likely.
It is calculated using the formula:
Probability = Number of Favourable Outcomes ÷ Total Number of Possible Outcomes
For example, when rolling a fair die, the probability of getting an even number is:
Favourable outcomes = {2, 4, 6}
Total outcomes = 6
Probability = 3/6 = 1/2
Classical probability is widely used in solving textbook problems involving coins, dice, and playing cards.
2. Experimental (Empirical) Probability: Experimental Probability is determined by conducting an experiment and observing the results.
It is calculated using the formula:
Experimental Probability = Number of Times an Event Occurs ÷ Total Number of Trials
For example, if a coin is tossed 100 times and heads appear 47 times,
Experimental Probability of Head = 47/100
Unlike theoretical probability, experimental probability depends on actual observations and may vary from one experiment to another.
As the number of trials increases, experimental probability approaches theoretical probability.
Probability Formula
The classical probability of an event E is calculated as:
P(E) = Number of Favourable Outcomes / Total Number of Possible Outcomes
Where:
P(E) represents the probability of event E.
The numerator represents the outcomes that satisfy the event.
The denominator represents all possible outcomes in the sample space.
This formula forms the basis of almost every probability problem in CBSE Class 10 Mathematics and is widely used in statistics, science, economics, and data analysis.
10.0Basic Properties of Probability
- Every probability value follows these important rules:
- Probability always lies between 0 and 1.
- P(Impossible Event) = 0
- P(Sure Event) = 1
- The sum of probabilities of all outcomes in a sample space is 1.
- Probability of an event and its complement always adds up to 1.
- The probability of an event can never be negative or greater than 1.
- These properties help verify whether a calculated probability is mathematically valid and serve as the foundation for solving more advanced probability questions.n outcome is a possible result obtained from an experiment.
Examples:
Coin Toss:
Head (H)
Tail (T)
Rolling a Die:
1, 2, 3, 4, 5, or 6
11.0Probability Formula
The probability equation defines the likelihood of the happening of an event. It is the ratio of favorable outcomes to the total favorable outcomes. The probability formula can be expressed as,
i.e., P(A) = n(A)/n(S)
where,
- P(A) is the probability of an event 'B'.
- n(A) is the number of favorable outcomes of an event 'B'.
- n(S) is the total number of events occurring in a sample space.
12.0Solved Examples
Question: A die is rolled once. Find the probability of getting a multiple of 3.
Solution: Sample Space
{1, 2, 3, 4, 5, 6}
Multiples of 3
{3, 6}
Favourable Outcomes = 2
Probability
= 2/6
= 1/3
Answer: 1/3
Question: One card is drawn randomly from a well-shuffled deck of 52 cards. Find the probability of drawing a Queen.
Solution: Total Cards = 52
Queens = 4
Probability
= 4/52
= 1/13
Answer: 1/13
13.0Important topics in Class 10 Maths: Probability
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15.0Supporting Study Materials
This study material, containing comprehensive CBSE Notes and NCERT Solutions for Chapter 14 of Class 10 Maths, is fully updated according to the latest NCERT guidelines. Complete with side-by-side card distribution charts, step-by-step sample space grids, and clear complementary probability formulas, this guide ensures absolute clarity for your board examinations.
30-Second Quick Review: Probability
- Probability measures the chance of an event occurring.
- Formula for Probability: Probability = (Number of Favourable Outcomes) ÷ (Total Number of Outcomes)
- Probability of any event lies between 0 and 1.
- Probability of an impossible event = 0.
- Probability of a certain event = 1.
- Sum of probabilities of an event and its complement is 1.
- Sample space is the set of all possible outcomes.
- Event is a subset of the sample space.
- Probability cannot be negative or greater than 1.
16.0Previous Year Questions (PYQs) on Probability
Question: A bag contains 5 red, 3 blue, and 2 green balls.
One ball is selected at random. Find the probability of selecting:
(i) A red ball
(ii) A green ball
Answer
Total balls = 10
(i) Probability = 5/10 = 1/2
(ii) Probability = 2/10 = 1/5
Question: A die is thrown once. Find the probability of getting:
(i) An even number
(ii) A prime number
Answer Sample Space: {1, 2, 3, 4, 5, 6}
(i) Even numbers = {2, 4, 6}
Probability = 3/6 = 1/2
(ii) Prime numbers = {2, 3, 5}
Probability = 3/6 = 1/2
17.0Recommended Next Topics