NCERT Class 9 Maths Ganita Manjari Chapter 7 Solutions: The Mathematics of Maybe: Introduction to Probability

Probability is a fascinating branch of mathematics that helps us understand uncertainty and chance in real life. In NCERT Class 9 Maths Ganita Manjari Chapter 7 – “The Mathematics of Maybe”, students learn how to measure and interpret the likelihood of events.

Whether it's predicting rain, winning a match, or being selected in a lucky draw—probability helps us make sense of such uncertain situations.

7.1 What is Probability?

Probability is a type of measurement, just like length, area, or volume. However, instead of measuring physical quantities, it measures the likelihood (chance) of an event occurring.

It helps us answer questions like:

• Will it rain today?
• Will our school win the hockey match?
• Will I be selected in a lucky draw?

These are called random events.

We know the possible outcomes, but we cannot predict the exact result in advance. This uncertainty is what probability deals with.

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7.1.1 What is Randomness?

Randomness refers to situations where outcomes cannot be predicted with certainty.

Examples:

• Tossing a coin
• Rolling a dice
• Picking a random name

Even though all possible outcomes are known, the exact result is uncertain.

7.1.2 The Probability Scale

Probability is measured on a scale from 0 to 1.

0≤P(E)≤10 \leq P(E) \leq 10≤P(E)≤1

Interpretation:

• 0 → Impossible Event
• 1 → Certain Event
• Between 0 and 1 → Likely Event

Example:
If probability = 0.75 → There is a 75% chance the event will occur.

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Probability Formula

P(E)=Number of favourable outcomesTotal number of outcomesP(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}P(E)=Total number of outcomesNumber of favourable outcomes?

7.2 Measuring Probability Objectively

Probability can be measured using two main approaches:

7.2.1 Experimental Probability

This is based on actual experiments or observations.

P(E)=Number of times event occursTotal number of trialsP(E) = \frac{\text{Number of times event occurs}}{\text{Total number of trials}}P(E)=Total number of trialsNumber of times event occurs?

Example:
If a coin is tossed 10 times and head appears 6 times:
Probability of head = 6/10 = 0.6

7.2.2 Theoretical Probability

This is based on logical reasoning and all possible outcomes, without performing experiments.

Example:

• Probability of head in a fair coin = 1/2

7.2.3 Analysing Statistical Data Using Probability

Probability helps in analyzing real-world data, such as:

• Weather reports
• Sports statistics
• Market trends

7.3 Elements of Probability: Sample Space and Events

Sample Space

The set of all possible outcomes is called the sample space.

Example:
Coin toss → {Head, Tail}

7.3.2 Events

An event is a subset of the sample space.

Example:
Getting a head → Event

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7.4 Tree Diagrams

Tree diagrams help us visualize all possible outcomes in a step-by-step way.

Example:

Two coin tosses:

• HH
• HT
• TH
• TT

Total outcomes = 4

Tree diagrams are very useful in solving complex probability problems.

Solved Examples

Example 1: Coin Toss

Probability of getting Head:

• Total outcomes = 2
• Favorable outcomes = 1

Answer = 1/2

Example 2: Dice Roll

Probability of even number:

• Even numbers = {2, 4, 6}
• Total outcomes = 6

Answer = 3/6 = 1/2

Example 3: Card Selection

Probability of red card:

• Red cards = 26
• Total cards = 52

 Answer = 1/2

NCERT Exercise Solutions (Sample)

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Q1. Classify events:

Q2. Probability of Tail

Answer = 1/2

Q3. Number less than 3 on dice

Answer = 2/6 = 1/3

Real-Life Applications

Probability is used in:

• Weather forecasting
• Data analysis
• Games & sports
• Stock market
• AI & Machine Learning

Important Tips

Understand randomness clearly
Always write sample space
Practice numerical questions
Revise formulas regularly

Exam Strategy

• Focus on NCERT questions
• Practice examples daily
• Use tree diagrams for clarity
• Avoid calculation mistakes

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