GANITA MANJARI | GRADE 9 | PART I

The Mathematics of Maybe

Introduction to Probability

Complete Notes and Chapter Summary

Chapter 7 | Textbook Pages 155-173

Contents at a Glance

• Introduction

• 7.1 What is Probability?

• 7.1.1 What is Randomness?

• 7.1.2 The Probability Scale

• 7.2 Measuring Probability Objectively

• 7.2.1 Experimental Probability

• 7.2.2 Theoretical Probability

• 7.2.3 Analysing Statistical Data Using Probability

• Gambler's Fallacy

• Fair and Unbiased

• 7.3 Elements of Probability

• 7.3.1 Sample Space

• 7.3.2 Events

• 7.4 Tree Diagrams

• Chapter Summary

Introduction

Probability is a type of measurement used to describe the likelihood of an event. Unlike length, area or volume, it measures how certain or uncertain we are that something will happen.

Events such as rainfall, the result of a hockey match or the selection of a student in a lucky draw involve chance. We may know all the possible outcomes, but we cannot predict the exact outcome with complete certainty.

Probability helps us study such uncertain situations and make reasonable predictions using observations, experiments, statistical data and mathematical reasoning.

7.1 What is Probability?

Probability measures how likely an event is to occur. It deals with chance, uncertainty and the degree of confidence that a particular event will happen.

1. It may rain or may not rain today.

2. A team may win, lose or draw a match.

3. Any one student may be selected in a lucky draw.

These are random events because their exact outcomes cannot be known beforehand.

Subjective Probability

Sometimes people estimate probability using personal judgement or their interpretation of available evidence. One person may consider rain unlikely because the Sun is shining, while another may expect rain because the weather is very hot.

Such an estimate is called subjective probability. Since it can differ from person to person, objective methods are needed for more reliable probability estimates.

7.1.1 What is Randomness?

Randomness refers to a situation in which all possible outcomes may be known, but the exact outcome of a particular trial cannot be predicted. Repeating the same activity may produce different results.

Examples

1. Tossing a coin: The possible outcomes are Heads and Tails, but the result of one toss cannot be known in advance.

2. Rolling a die: The possible outcomes are 1, 2, 3, 4, 5 and 6, but the number appearing on a particular roll is unpredictable.

3. Lucky draw: Each student may have an equal chance of selection, but the selected student cannot be identified beforehand.

Random Experiment

A random experiment is a repeatable activity whose outcome cannot be predicted with certainty. Each performance of the experiment is called an observation or trial.

Probability is the area of mathematics that studies randomness and measures how likely a particular outcome is.

For a fair coin: P(Heads) = 1/2 and P(Tails) = 1/2

Rain is considered random because it depends on complex factors such as temperature, humidity, wind patterns and atmospheric pressure. Weather data can estimate the probability of rain, but not always predict it with complete certainty.

7.1.2 The Probability Scale

Probability is measured on a scale from 0 to 1. The probability of an event E is denoted by P(E).

0 <= P(E) <= 1

Percentage probability = P(E) x 100

For example, 0.75 means 75%, while 0.5 means 50%.

Examples on the Scale

• Getting a number greater than 6 on a standard die is impossible.

• Rolling a 3 on a fair die is less likely because only one face shows 3.

• Getting Heads on a fair coin is an even chance.

• Drawing a card numbered from 2 to 10 from a standard deck is more likely than not.

• Choosing a red sweet from a bag containing only red sweets is certain.

7.2 Measuring Probability Objectively

There are two main methods of estimating probability objectively.

1. Using evidence from experience: Data is collected by repeating an experiment or by analysing statistical data from previous observations. This produces experimental probability.

2. Using theoretical methods: When all possible outcomes are equally likely, probability is found by comparing favourable outcomes with all possible outcomes. This produces theoretical probability.

7.2.1 Experimental Probability: Performing Observations or Experiments

In a random experiment, the result obtained is called an outcome. The collection of all possible outcomes is called the sample space.

• Coin toss: {H, T}

• Die roll: {1, 2, 3, 4, 5, 6}

Experimental Probability = Number of times the event occurred / Total number of trials

Experimental probability is also called the relative frequency of the event.

Example

A die is rolled 50 times and the number 4 appears 8 times.

P(rolling a 4) = 8/50 = 0.16 = 16%

Experimental probability is useful when working with observed outcomes in statistics and data analysis.

Textbook Fact: The game of Snakes and Ladders originated in ancient India from a traditional dice game called Jñān-Chaupaḍ. It taught moral lessons, with ladders representing virtues and snakes representing vices.

7.2.2 Theoretical Probability

Theoretical probability is calculated by assuming that all possible outcomes are equally likely. It describes what we expect in an ideal and perfectly fair situation, so no experiment or collected data is required.

P(E) = Number of favourable outcomes / Total number of possible outcomes

Example 1: Rolling a Die

For a fair six-sided die, there is one favourable outcome for getting 4 and six possible outcomes.

P(rolling a 4) = 1/6 ≈ 0.167 = 16.7%

Example 2: Selecting a Letter

The word PROBABILITY has 11 letters, including 2 letters B.

P(B) = 2/11 ≈ 0.182 = 18.2%

7.2.3 Analysing Statistical Data Using Probability

Statistical data is used to estimate probability in marketing, sales forecasting, insurance, scientific research and social science research.

Example: Favourite Fruits

Suppose 50 students are surveyed: 20 prefer mango, 15 apple, 10 banana and 5 grapes.

P(Mango) = 20/50 = 0.4 = 40%

If the school has 1,500 students, the estimated number who prefer mango is:

40% of 1,500 = (40/100) x 1,500 = 600

Population, Sample and Sampling

• Population: The complete group being studied, such as all 1,500 students.

• Sample: A smaller group selected from the population, such as the 50 surveyed students.

• Sampling: The process of collecting information from a sample to estimate features of the whole population.

A larger and more representative sample generally produces a more reliable estimate. It should include members from different classes or grades so that the results are not biased.

Experimental and Theoretical Probability

Law of Large Numbers

As the number of trials increases, experimental probability generally gets closer to theoretical probability. This is called the Law of Large Numbers. Probability predicts long-term patterns; it does not give the exact result of the next trial.

Gambler's Fallacy

Gambler's Fallacy is the mistaken belief that previous results of independent random events affect the next result.

If a fair coin shows Heads six times in succession, Tails is not certain on the next toss. The coin has no memory.

P(Tails on the next toss) = 1/2

Similarly, even after rolling three consecutive sixes on a fair die:

P(rolling another 6) = 1/6

Each toss or roll is an independent event, so previous outcomes do not change the next probability.

Fair and Unbiased

A coin is called fair or unbiased when it is symmetrical and neither side is more likely to appear than the other.

P(H) = P(T) = 1/2

A random toss means that the coin is allowed to fall freely without interference or deliberate control. Similarly, a fair die is assumed to have six faces that are equally likely to appear.

7.3 Elements of Probability: Sample Spaces and Events

The basic elements used to study probability are random experiment, outcome, sample space, event and favourable outcome. The sample space lists everything that can happen, while an event identifies the outcome or outcomes being studied.

7.3.1 Sample Space

The sample space, denoted by S, is the set of all possible outcomes of a random experiment. Every individual outcome is an element of the sample space. The number of elements is called the sample size and is written as n(S).

A valid sample space must:

• Include every possible outcome.

• List each outcome only once.

• Provide the level of detail required by the question.

Examples

Level of Detail: The sample space depends on the question. If only the presence of rain matters, use {Rain, No Rain}. If rainfall intensity matters, include No Rain, Drizzle, Light Rain and Heavy Rain.

7.3.2 Events

An event is a single outcome or a group of outcomes selected from the sample space. Therefore, an event is a subset of the sample space. Outcomes belonging to the required event are called favourable outcomes.

Examples

Example 1: At least one Head when two coins are tossed:

S = {HH, HT, TH, TT}; E = {HH, HT, TH}

Example 2: Getting a number greater than 4 on a die:

S = {1, 2, 3, 4, 5, 6}; E = {5, 6}

Example 3: Selecting a yellow fruit from {Apple, Banana, Orange}:

E = {Banana}

7.4 Tree Diagrams

A tree diagram is a systematic method of listing all possible outcomes of a multi-step experiment, such as tossing a coin twice or rolling a die three times.

Features

• It begins from a single starting point.

• Each branch represents one possible outcome.

• Branches split again for the next trial.

• Each complete path represents one final outcome.

• It helps list the complete sample space and calculate probabilities.

Example: Tossing a Fair Coin Twice

S = {HH, HT, TH, TT}

There are four equally likely outcomes. Therefore:

P(HH) = 1/4 = 0.25 = 25%

The event of getting one Head and one Tail is {HT, TH}.

P(one Head and one Tail) = 2/4 = 1/2

Chapter Summary

• Probability measures the likelihood of an event.

• A random experiment has known possible outcomes, but its exact result cannot be predicted.

• Probability lies between 0 and 1: 0 <= P(E) <= 1.

• Probability 0 represents an impossible event and probability 1 represents a certain event.

• Experimental probability is based on actual observations and relative frequency.

• Theoretical probability assumes all possible outcomes are equally likely.

• Experimental probability generally approaches theoretical probability as the number of trials increases.

• A sample is a smaller part of a population used for statistical estimation.

• Gambler's Fallacy wrongly assumes that previous independent results affect the next result.

• A fair or unbiased coin gives Heads and Tails equal probabilities.

• A sample space contains every possible outcome exactly once.

• An event is one outcome or a group of outcomes from the sample space.

• Tree diagrams systematically list the outcomes of multi-step experiments.

Experimental Probability = Number of times the event occurred / Total number of trials

Theoretical Probability P(E) = Number of favourable outcomes / Total number of possible outcomes