Probability – Class 10th Notes (Mathematics)

 

Introduction to Probability

Probability is a branch of mathematics that deals with the likelihood or chance of different outcomes. It is a measure of how likely an event is to occur. The concept of probability helps us in predicting the chance of something happening or not happening.

In everyday life, we come across situations like:

• Will it rain tomorrow?
• What are the chances of winning a lottery?
• What is the likelihood of getting heads when flipping a coin?

All these scenarios can be understood using probability.

Definition of Probability

The probability of an event is a number between 0 and 1 where:

• 0 indicates that the event cannot happen.
• 1 indicates that the event is certain to happen.

For example:

• The probability of getting a Head when tossing a fair coin is 0.5.
• The probability of drawing a red ball from a bag with 5 red balls and 5 blue balls is 0.5.

Formula of Probability

The formula for calculating probability is:

$$\text{Probability (P)} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}$$

Where:

• Number of favorable outcomes refers to the outcomes that satisfy the condition of the event.
• Total number of possible outcomes refers to all the possible outcomes of the experiment.

Important Terms in Probability

Before solving problems on probability, it’s important to understand some basic terms:

1. Experiment: An operation which can produce some well-defined outcomes. Example: Tossing a coin.

2. Outcome: The result of a single trial of an experiment. Example: When tossing a coin, the outcome can either be Head or Tail.

3. Event: A collection of one or more outcomes of an experiment. Example: Getting a Head in a coin toss.

4. Sample Space (S): The set of all possible outcomes of an experiment. Example: When rolling a die, the sample space is {1, 2, 3, 4, 5, 6}.

Types of Events

1. Certain Event: An event that is sure to happen. The probability of a certain event is 1.

   • Example: In a bag of only red balls, the event of drawing a red ball is a certain event.

2. Impossible Event: An event that cannot happen. The probability of an impossible event is 0.

   • Example: Drawing a blue ball from a bag containing only red balls is an impossible event.

3. Equally Likely Events: Events are equally likely if each event has the same chance of occurring.

   • Example: When tossing a fair coin, getting a head and getting a tail are equally likely events.

Examples and Practice Problems

Example 1: Tossing a Coin

A fair coin is tossed. What is the probability of getting a head?

• Solution:
  Sample space S = {Head, Tail}
  Number of favorable outcomes (getting a head) = 1
  Total number of possible outcomes = 2
  Therefore, the probability of getting a head: $$P(\text{Head}) = \frac{1}{2}$$

Example 2: Rolling a Die

A die is rolled. What is the probability of getting a number greater than 4?

• Solution:
  Sample space S = {1, 2, 3, 4, 5, 6}
  Favorable outcomes = {5, 6} (numbers greater than 4)
  Total number of possible outcomes = 6
  Therefore, the probability of getting a number greater than 4: $$P(\text{number greater than 4}) = \frac{2}{6} = \frac{1}{3}$$

Example 3: Drawing a Card from a Deck

A card is drawn from a well-shuffled deck of 52 cards. What is the probability of drawing a King?

• Solution:
  Total number of cards = 52
  Number of Kings in a deck = 4
  Therefore, the probability of drawing a King: $$P(\text{King}) = \frac{4}{52} = \frac{1}{13}$$

Probability in Daily Life

Probability is not just limited to mathematics but also has real-life applications:

1. Weather Forecasting: Meteorologists use probability to predict the chance of rain, snowfall, or other weather conditions.
2. Insurance: Insurance companies calculate the probability of events like accidents or illness to determine the premium.
3. Games and Sports: Probability is used to determine the chances of winning a game or a team winning a match.

Class 10 Probability Syllabus Overview

The Class 10 syllabus for Probability primarily covers the following:

• Basic probability concepts.
• Probability of simple events like tossing a coin, rolling a die, or drawing cards.
• Solving problems using the probability formula.

Common Questions in Exams:

1. What is the probability of getting an even number when a die is rolled?
2. A bag contains 3 red, 5 green, and 2 blue balls. A ball is drawn at random. What is the probability that the drawn ball is green?
3. If a coin is tossed twice, what is the probability of getting exactly one head?

Tips for Scoring Well in Probability

1. Understand the Formula: Memorize the basic probability formula and understand how to use it in different situations.
2. Practice Different Problems: Practice as many problems as you can, covering all types of questions like coin toss, dice roll, and card drawing.
3. Know the Sample Space: Clearly define the sample space for each experiment, as this will help in calculating the total number of outcomes.
4. Revise Thoroughly: Go through your class notes and NCERT solutions for a clear understanding of the concepts.

Probability is an interesting and practical topic in Class 10 Mathematics. By mastering the basics of probability, students can solve complex problems in exams and understand real-world applications. Regular practice and thorough understanding of the concepts will help students score well in this topic.