Unit 13 Class 9 Math Solutions – Probability
Unit 13 Class 9 Math Solutions contain complete exercise-wise PDF solutions of the chapter Probability. Students can find the solutions of Exercise 13.1, Exercise 13.2, and Review Exercise 13 on this page.
This unit introduces experiments, outcomes, sample spaces, events, favourable outcomes, theoretical probability, relative frequency, experimental probability, expected frequency, and practical applications of probability.
The solution PDFs will include complete working, organized sample spaces, probability calculations, frequency tables, expected-frequency questions, real-life applications, and clearly stated final answers for the Punjab Board Class 9 Mathematics book.
Students can view the PDFs online and use the save or download option in the PDF viewer to keep the solutions for offline study.
These Unit 13 solutions are useful for homework, revision, class tests, annual examinations, and board exam preparation.
Solutions of Exercise 13.1 Unit 13 Class 9 Math Notes
Exercise 13.1 of Unit 13 Class 9 Math Notes focuses on the basic language and calculations of probability. Students identify experiments, outcomes, sample spaces, events, and favourable outcomes before calculating the probability of simple events.
[Embed the Exercise 13.1 PDF here]
What Is Probability?
Probability measures the chance that an event will occur.
A probability is written as a fraction, decimal, or percentage between zero and one.
\[0\leq P(E)\leq1\]
Random Experiment
A random experiment is a process whose exact result cannot be predicted with certainty before it is performed.
Examples include tossing a coin, rolling a die, selecting a card, spinning a spinner, or choosing an object from a bag.
Trial
One performance of a random experiment is called a trial.
If a coin is tossed 50 times, the experiment has 50 trials.
Outcome
An outcome is one possible result of an experiment.
For a coin toss, the possible outcomes are head and tail.
For a standard die, the possible outcomes are the numbers from 1 to 6.
Sample Space
The sample space is the set of all possible outcomes of an experiment.
It is usually represented by the letter S.
For one coin toss:
\[S=\{H,T\}\]
For one roll of a standard die:
\[S=\{1,2,3,4,5,6\}\]
Sample Points
Each individual outcome in a sample space is called a sample point.
The number of sample points is written as:
\[n(S)\]
Event
An event is a collection of one or more outcomes from the sample space.
For a die roll, the event of obtaining an even number is:
\[E=\{2,4,6\}\]
Favourable Outcomes
Favourable outcomes are the outcomes that satisfy the condition stated in the event.
For the event of obtaining an even number on a die, there are three favourable outcomes.
\[n(E)=3\]
Theoretical Probability
When all outcomes are equally likely, the probability of an event is:
\[P(E)=\frac{n(E)}{n(S)}\]
Count the favourable outcomes, count all possible outcomes, and simplify the fraction.
Probability of an Impossible Event
An impossible event cannot occur.
Its probability is:
\[P(E)=0\]
For example, obtaining 7 on a standard six-sided die is impossible.
Probability of a Certain Event
A certain event must occur.
Its probability is:
\[P(E)=1\]
For example, obtaining a number less than 7 on a standard die is certain.
Likely and Unlikely Events
A likely event has a probability greater than one-half but less than one.
\[\frac{1}{2}<P(E)<1\]
An unlikely event has a probability greater than zero but less than one-half.
\[0<P(E)<\frac{1}{2}\]
Equally Likely Events
Events are equally likely when they have the same chance of occurring.
For a fair coin, head and tail are equally likely.
\[P(H)=P(T)=\frac{1}{2}\]
Probability Scale
The probability scale runs from zero to one.
Zero represents an impossible event, one-half represents an even chance, and one represents a certain event.
Decimals and percentages may also be placed on the same scale.
Complement of an Event
The complement of an event consists of all outcomes in the sample space that are not in the event.
If the event is represented by E, its complement may be written as:
\[E’\]
The probability rule is:
\[P(E’)=1-P(E)\]
Checking Complementary Probabilities
An event and its complement cover the complete sample space.
Therefore:
\[P(E)+P(E’)=1\]
This rule is useful when the probability of an event not occurring is easier to calculate.
Probability of a Coin Outcome
A fair coin has two equally likely outcomes.
The probability of a head is:
\[P(H)=\frac{1}{2}\]
The probability of a tail is:
\[P(T)=\frac{1}{2}\]
Tossing Two Coins
When two coins are tossed, list the ordered outcomes carefully.
\[S=\{HH,HT,TH,TT\}\]
The outcomes HT and TH are different because the results occur in a different order.
Probability with a Die
A standard die has six equally likely outcomes.
For example, the probability of obtaining a prime number uses the favourable outcomes 2, 3, and 5.
\[P(\text{prime})=\frac{3}{6}=\frac{1}{2}\]
Even, Odd, Prime, and Composite Outcomes
Students should list the relevant numbers before calculating probability.
On a standard die:
\[\text{even outcomes}=\{2,4,6\}\]
\[\text{odd outcomes}=\{1,3,5\}\]
\[\text{prime outcomes}=\{2,3,5\}\]
Probability with Cards
In a card-selection question, identify the total number of cards and the number satisfying the required condition.
If every card is equally likely to be selected, use the basic probability formula.
Read the question carefully to determine whether the cards are numbered, coloured, or labelled.
Probability with Coloured Objects
For counters, marbles, balls, or other coloured objects, add the number of all objects to find the sample-space size.
The number of objects with the required colour gives the favourable outcomes.
\[P(\text{required colour})=\frac{\text{number of required-colour objects}}{\text{total number of objects}}\]
Probability with a Spinner
A spinner has equally likely outcomes only when its sectors have equal areas.
For an equally divided spinner:
\[P(E)=\frac{\text{number of favourable sectors}}{\text{total number of sectors}}\]
If the sectors are not equal, the areas must be considered.
Probability from Letters or Digits
Some questions select one letter from a word or one digit from a collection.
Repeated letters or digits must be counted separately because each position represents a possible selection.
The total number of positions gives the number of possible outcomes.
Systematic Listing of Outcomes
For a compound experiment, use an organized list or table so that no outcome is missed or counted twice.
Calculate the total number of outcomes only after the complete sample space has been written.
Simplifying Probability Answers
A probability fraction should be reduced to its simplest form.
It may also be converted into a decimal or percentage when the question requires it.
\[\text{percentage probability}=P(E)\times100\%\]
Solutions of Exercise 13.2 Unit 13 Class 9 Math Notes
Exercise 13.2 focuses on relative frequency, experimental probability, expected frequency, and real-life applications. Students use recorded results to estimate probabilities and use probabilities to predict how often events may occur.
[Embed the Exercise 13.2 PDF here]
Relative Frequency
Relative frequency shows how often an outcome occurs compared with the total number of trials.
\[\text{relative frequency}=\frac{\text{frequency of the event}}{\text{total number of trials}}\]
Relative frequency is an estimate based on observed data.
Experimental Probability
Experimental probability is calculated from the results of an experiment.
\[P_{\text{experimental}}(E)=\frac{\text{number of times }E\text{ occurs}}{\text{total number of trials}}\]
It is also called empirical probability or relative-frequency probability.
Theoretical and Experimental Probability
Theoretical probability is based on equally likely outcomes and mathematical reasoning.
Experimental probability is based on recorded results.
The two values may not be exactly equal in a small experiment.
Effect of Increasing the Number of Trials
When an experiment is repeated many times, the relative frequency usually becomes more stable.
It often moves closer to the theoretical probability, although an exact match is not guaranteed.
Relative-Frequency Table
A relative-frequency table contains the outcomes, observed frequencies, and relative frequencies.
Divide every frequency by the same total number of trials.
The relative frequencies should add to approximately one.
Checking Relative Frequencies
The check is:
\[\sum \text{relative frequencies}=1\]
A small difference may occur when decimal values have been rounded.
Finding a Missing Frequency
If the total number of trials is known, subtract the known frequencies from the total.
\[\text{missing frequency}=\text{total frequency}-\text{sum of known frequencies}\]
The missing value can then be used to calculate its relative frequency.
Finding Frequency from Relative Frequency
Rearrange the relative-frequency formula.
\[\text{frequency}=\text{relative frequency}\times\text{total number of trials}\]
The final frequency should normally be a whole number.
Expected Frequency
Expected frequency estimates the number of times an event should occur in a stated number of trials.
\[\text{expected frequency}=P(E)\times N\]
Here, N is the total number of trials.
Expected Frequency from a Fraction
Multiply the total number of trials by the probability fraction.
For example, if an event has probability one-quarter in 200 trials:
\[\text{expected frequency}=\frac{1}{4}\times200=50\]
Expected Frequency from a Decimal
A decimal probability is multiplied directly by the number of trials.
For example:
\[0.3\times500=150\]
Expected Frequency from a Percentage
Convert the percentage into a fraction or decimal before multiplying.
For example:
\[15\%=0.15\]
\[0.15\times400=60\]
Expected and Observed Frequency
Expected frequency is a prediction based on probability.
Observed frequency is what actually happens in an experiment.
The two values may differ because random variation occurs.
Estimating Probability from Data
When a large set of experimental results is given, estimate probability by dividing the observed event frequency by the total frequency.
\[\text{estimated probability}=\frac{\text{observed frequency}}{\text{total frequency}}\]
Expected Frequency in Coin Experiments
For a fair coin, the expected frequency of heads in N tosses is:
\[\frac{1}{2}N\]
The same expected frequency applies to tails.
Expected Frequency in Die Experiments
For a fair die, each individual face has probability one-sixth.
The expected frequency of one specified face in N rolls is:
\[\frac{N}{6}\]
For an event containing several faces, multiply N by the probability of the complete event.
Expected Frequency with Colours
If the probability of selecting a colour is known, multiply that probability by the total number of selections.
This method is used in questions involving counters, balls, cards, spinners, and survey categories.
Real-Life Applications of Expected Frequency
Expected frequency can be used to estimate future results in quality control, games, surveys, school data, product testing, weather records, and repeated selections.
The result is an estimate rather than a guaranteed exact count.
Interpreting an Expected Value
An expected frequency describes a long-run prediction.
For example, an expected frequency of 75 means that the event is predicted to occur about 75 times, not that it must occur exactly 75 times.
Rounding Expected Frequency
If a calculation gives a non-integer expected frequency, follow the instruction in the question.
For a practical count, the result may be rounded to a suitable whole number.
Do not round intermediate values too early.
Comparing Two Events
An event with the greater probability has the greater expected frequency when both events are repeated the same number of times.
The ratio of expected frequencies matches the ratio of their probabilities.
Working Backward from Expected Frequency
If the expected frequency and total number of trials are known, divide to find the probability.
\[P(E)=\frac{\text{expected frequency}}{N}\]
Working Backward to Find the Number of Trials
If probability and expected frequency are known, divide to find the total number of trials.
\[N=\frac{\text{expected frequency}}{P(E)}\]
Solutions of Review Exercise 13 Unit 13 Class 9 Math Notes
Review Exercise 13 revises the complete Probability unit. It combines definitions, sample spaces, simple probabilities, complements, relative frequency, experimental probability, and expected-frequency applications.
[Embed the Review Exercise 13 PDF here]
Multiple-Choice Questions
The multiple-choice section checks probability terminology, probability values, sample spaces, events, relative frequency, and expected frequency.
Students should identify the correct definition or formula before selecting an option.
Definitions and Short Questions
The review may ask students to define experiment, outcome, sample space, event, favourable outcome, relative frequency, and expected frequency.
Definitions should be brief, accurate, and supported with a simple example where required.
Writing Sample Spaces
Students should list every possible outcome without repetition.
For compound experiments, a systematic table or organized list helps prevent missing outcomes.
Finding Simple Probabilities
Use the number of favourable outcomes and the total number of equally likely outcomes.
\[P(E)=\frac{n(E)}{n(S)}\]
Simplify the final fraction.
Complementary Probability
When a question asks for an event not occurring, use:
\[P(E’)=1-P(E)\]
This may be faster than listing every unfavourable outcome.
Probability from Tables
Some review questions present frequencies in a table.
Add the complete frequency column to find the total and divide the required frequency by that total.
Relative Frequency Revision
Relative frequency is calculated from observed data.
\[\text{relative frequency}=\frac{\text{event frequency}}{\text{total frequency}}\]
Check that all category relative frequencies add to one.
Expected Frequency Revision
Multiply the probability by the number of trials.
\[\text{expected frequency}=P(E)\times N\]
State the result in the context of the question.
Real-Life Probability Problems
The review applies probability to repeated experiments, school data, surveys, games, selections, and other practical situations.
Students should identify whether the question gives theoretical probability or experimental data before choosing the method.
Important Rules and Formulas of Unit 13 Class 9 Math
Probability of an Event
\[P(E)=\frac{n(E)}{n(S)}\]
Range of Probability
\[0\leq P(E)\leq1\]
Probability of an Impossible Event
\[P(E)=0\]
Probability of a Certain Event
\[P(E)=1\]
Complement Rule
\[P(E’)=1-P(E)\]
Complementary Probabilities
\[P(E)+P(E’)=1\]
Relative Frequency
\[\text{relative frequency}=\frac{\text{event frequency}}{\text{total number of trials}}\]
Experimental Probability
\[P_{\text{experimental}}(E)=\frac{\text{number of successful trials}}{\text{total number of trials}}\]
Expected Frequency
\[\text{expected frequency}=P(E)\times N\]
Probability from Expected Frequency
\[P(E)=\frac{\text{expected frequency}}{N}\]
Common Mistakes in Unit 13 Class 9 Math Notes
Students may confuse an outcome with an event.
The sample space may be written incompletely or with repeated outcomes.
In a two-coin experiment, HT and TH may be incorrectly treated as the same outcome.
The number of favourable outcomes may be used as the denominator instead of the total number of outcomes.
The basic theoretical formula should be used only when the outcomes are equally likely.
Students sometimes write a probability greater than one or less than zero.
The complement rule may be applied with the wrong sign.
Repeated letters or objects may be counted only once even though each position or object is a separate possible selection.
Relative frequency may be confused with raw frequency.
The frequency should be divided by the total number of trials, not by the number of categories.
Expected frequency may be found by dividing instead of multiplying.
Experimental probability should not be expected to match theoretical probability exactly in a small number of trials.
Relative frequencies may fail to add to one because of arithmetic errors.
Expected frequency should be interpreted as an estimate, not a guaranteed result.
Intermediate decimals should not be rounded too early.
Final probabilities should be simplified and written in the requested form.
Exam Preparation Tips for Unit 13 Class 9 Math Notes
Memorize the definitions of experiment, trial, outcome, sample space, event, and favourable outcome.
Learn the probability, complement, relative-frequency, and expected-frequency formulas.
Practise writing sample spaces for coins, dice, cards, spinners, and selections.
Count repeated outcomes carefully.
Check that every theoretical probability lies between zero and one.
Simplify probability fractions before writing the final answer.
Practise converting probabilities among fractions, decimals, and percentages.
Learn the difference between theoretical and experimental probability.
Check that relative frequencies add to approximately one.
Multiply probability by the number of trials to find expected frequency.
Read practical questions carefully to identify the total number of trials.
Attempt Review Exercise 13 independently before checking the solution PDF.
Use the save or download option in the PDF viewer to keep the solutions for offline revision.
Why Unit 13 Class 9 Math Solutions Are Important
Unit 13 Class 9 Math Solutions are important because probability helps students measure uncertainty and make predictions.
Probability is used in statistics, science, weather forecasting, insurance, business, medicine, quality control, games, surveys, and data analysis.
The chapter also strengthens students’ understanding of fractions, ratios, percentages, sets, and information handling.
A strong understanding of probability prepares students for higher mathematics, statistics, computer science, economics, and scientific research.
FAQs About Unit 13 Class 9 Math Solutions
What is the topic of Unit 13 Class 9 Math?
The topic of Unit 13 Class 9 Math is Probability.
How many exercises are included in Unit 13?
Unit 13 includes Exercise 13.1, Exercise 13.2, and Review Exercise 13.
What is covered in Exercise 13.1?
Exercise 13.1 covers experiments, outcomes, sample spaces, events, favourable outcomes, theoretical probability, complements, probability scales, and simple probability questions.
What is covered in Exercise 13.2?
Exercise 13.2 covers relative frequency, experimental probability, expected frequency, frequency tables, and practical applications.
What is covered in Review Exercise 13?
Review Exercise 13 revises definitions, sample spaces, simple probability, complementary events, relative frequency, experimental probability, and expected frequency.
What is the formula for probability?
The formula is \[P(E)=\frac{n(E)}{n(S)}\], when all outcomes are equally likely.
What is a sample space?
A sample space is the set of all possible outcomes of an experiment.
What is an event?
An event is a collection of one or more outcomes from the sample space.
What is the probability of an impossible event?
The probability of an impossible event is zero.
What is the probability of a certain event?
The probability of a certain event is one.
What is relative frequency?
Relative frequency is the frequency of an event divided by the total number of trials.
What is expected frequency?
Expected frequency is the predicted number of times an event will occur and is found by multiplying probability by the number of trials.
Can students download the Unit 13 solutions PDFs?
Yes. Students can view the exercise-wise PDFs on this page and use the save or download option provided by the PDF viewer.
Are these Unit 13 Class 9 Math Solutions useful for exam preparation?
Yes. The solutions are useful for homework, revision, class tests, annual examinations, and board exam preparation.
Disclaimer
These Unit 13 Class 9 Math Solutions are provided for educational help. Students should use them to understand probability methods, check their work, and study alongside the official textbook and their teacher’s instructions.
The article has been prepared in advance so that the Exercise 13.1, Exercise 13.2, and Review Exercise 13 solution PDFs can be embedded when they are ready.
Final Words
Unit 13 Class 9 Math Solutions help students understand Probability in a clear and organized way.
Exercise 13.1 develops the basic concepts of probability, Exercise 13.2 explains relative and expected frequency, and Review Exercise 13 revises the complete chapter.
List outcomes systematically, use the correct denominator, and distinguish carefully between theoretical probability, experimental probability, and expected frequency.