The Mathematics of Maybe: Introduction to Probability Class 9 [Ganita]
Master The Mathematics of Maybe: Introduction to Probability Class 9 [Ganita] with comprehensive NCERT Solutions, Practice Questions, MCQs, Sample Papers, Case Based Questions, and Video lessons.
NCERT Solutions
The Mathematics of Maybe: Introduction to Probability Class 9 [Ganita] β NCERT Solutions
Each question below opens its complete step-by-step Teachoo solution.
Exercise Set 7.1
1 questionEx 7.1, 1
(i) Rank the following events on a scale from 0 (Impossible) to 1 (Certain). Label each event: Impossible, less likely, equally likely (even chance), more likely, certain. Give reasons why you gave each event its ranking.
(i) The next Monday will come after Sunday.
The days of the week follow a permanent cycle
Monday is defined as the day that immediately follows Sunday.
Therefore, there is a 100% chance of this happening.
Exercise Set 7.2
6 questionsEx 7.2, 1
(i) A teacher mixes a large bag of sweets of different colours and randomly selects a sample of 30 sweets. She counts the number of
sweets of each colour:
10 red sweets | 8 green sweets | 7 yellow sweets | 5 blue sweets
(i) Calculate the probability that a randomly picked sweet from the sample is green.
Now,
Number of green sweets = 8
Total Number of sweets = 30
Ex 7.2, 2
(i)
A survey is conducted at a school where a random sample of 40
students is asked about their favourite club. The responses are:
14 students: Science Club | 11 students: Arts Club |
9 students: Sports Club | 6 students: Debate Club
Assume there are 800 students in the whole school.
(i) What is the probability that a randomly chosen student from the sample prefers the Arts Club?
Given Data:
Total students in the sample = 40
Science = 14, Arts = 11, Sports = 9, Debate = 6
Total students in the school (population) = 800
Ex 7.2, 3
(i)
Toss a coin 20 times and record the result each time (heads or tails).
(i) How many times did you get heads?
Let's pretend I flipped a coin 20 times and got
Heads 12 times and Tails 8 times
Ex 7.2, 4
Toss a paper cup into the air 100 times. After each toss record whether the cup lands on its bottom, upside down on its top or on its side (See Fig. 7.5). Assign probabilities to the outcomes by using experimental probability.
After tossing a cup 100 times (which took so long!), here are the results
Letβs find Probability of each
P(landed on bottom) = (ππ’ππππ ππ π‘ππππ ππ‘ ππππππ ππ πππ‘π‘ππ)/(πππ‘ππ ππ’ππππ ππ π‘ππ π ππ )
= 44/100
= 0.44
Ex 7.2, 5
What is the probability of getting an even number when rolling a fair 6-sided die?
On a 6-sided die, our outcomes are 1, 2, 3, 4, 5, 6
And, here even numbers are 2, 4, 6
Ex 7.2, 6
(i)
Suppose you roll a 6-sided die 12 times and get a β3β three times.
(i) What is the experimental probability of rolling a β3β?
Given
Total rolls = 12
Number of times '3' was rolled = 3
Exercise Set 7.3
3 questionsEx 7.3, 1
When a single 6-sided die is rolled, what is the total number of
possible outcomes in the sample space?
Letβs roll a 6-sided die
Ex 7.3, 2
(i) For the following experiments write down the sample space S.
(i) Rolling a die and tossing a coin together.
Letβs roll a 6-sided die & coin together
Ex 7.3, 3
(i) In a village fair, there are 3 popular snacks available: Samosa, Pakora, and Bhaji. For drinks, villagers can choose either Chai or Lassi.
(i) List the sample space of all possible snack and drink combinations a person could choose at the fair.
Here, we need to match every single snack with every single drink to find all the different combo meals available
Thus, our Sample Space is
S = { (Samosa, Chai),
(Samosa, Lassi),
(Pakora, Chai),
(Pakora, Lassi),
(Bhaji, Chai),
(Bhaji, Lassi) }
Ex 7.3, 3 (ii) In a village fair, there are 3 popular snacks available: Samosa, Pakora, and Bhaji. For drinks, villagers can choose either Chai or Lassi.
(ii) List the event βSelecting Samosa as a snack.β
Our sample Space is
S = {(Samosa, Chai), (Samosa, Lassi),(Pakora, Chai),(Pakora, Lassi), (Bhaji, Chai), (Bhaji, Lassi) }
Exercise Set 7.4
2 questionsEx 7.4, 1
(i) There are two fruit baskets A and B. Basket A has one apple and two
oranges. Basket B has one banana and one mango. You randomly
pick one fruit from each basket.
(i) Draw a tree diagram showing all possible pairs of fruits.
Since we pick fruits from basket one after another
Letβs start with Basket A, and then B
Note: Because there are 2 oranges and 1 apple,
Probability of getting apple = (ππ’ππππ ππ ππππππ )/(πππ‘ππ πππ’ππ‘π )
= π/π
Ex 7.4, 2
(i) Let us say that you have a box containing 3 red pens, 4 black pens and 2 green pens. You pick a pen (without looking) from the box and put it back. Then your friend does the same.
(i) What are the possible outcomes of the pen colours? Can you draw a tree diagram representing the possible outcomes?
Letβs draw our tree diagram
Note: Because there are 3 red pens, 4 black pens and 2 green pens
Probability of getting red = (ππ’ππππ ππ πππ ππππ )/(πππ‘ππ ππππ )
= 3/(3 + 4 + 2)=π/π
End-of-Chapter Exercises
20 questionsQuestion 1
(i) Fill in the blanks.
(i) The probability of an impossible event is _______.
The probability of an impossible event is 0
Question 1 (ii) Fill in the blanks.
(ii) The set of all possible outcomes of a random experiment is called the __________.
The set of all possible outcomes of a random experiment is called the Sample Space.
Question 1 (iii) Fill in the blanks.
(iii) The probability of an event that is certain to happen is _______.
The probability of an event that is certain to happen is 1.
Question 1 (iv) Fill in the blanks.
(iv) Tossing a fair coin has a probability of ______ for getting heads.
Tossing a fair coin has a probability of 0.5 (or Β½) for getting heads.
Question 2
In a survey of 50 students, 15 students said they liked football.
The number of students who like football is 15, and the ________
(frequency/relative frequency) is __________ (fill in the fraction or
decimal).
Now,
Relative frequency = (ππ’ππππ ππ π π‘π’ππππ‘π π€βπ ππππ ππππ‘ππππ)/(πππ‘ππ π π‘π’ππππ‘π )
= 15/50
= π/ππ
Question 3
(i) Which of the following experiments have equally likely outcomes?
Explain.
(i) A driver attempts to start a car. The car starts or does not start.
The question is asked about equally likely outcomes β i.e. all outcomes have the same probability
Question 4
(i) Write the sample space and calculate the probability based on the given information.
(i) Two coins are tossed at the same time. What is the probability of getting at least one head?
Since two coints are tossed
Our Sample Space would be
S = {HH, HT, TH, TT}
Question 5
A bag has 3 candies: strawberry, lemon, and mint. One is picked at
random. What is the probability of picking a strawberry candy?
Now,
Number of strawberry candy = 1
Total candies = 3
Question 6
A child has 2 shirts (one red and one blue) and 3 types of pants
(jeans, khakis, and shorts). List all the possible combinations of outfits consisting of one shirt and one pair of pants. Display your
Question 7
(i) A tyre company records distances before replacement in 1000 cases.
View solutionQuestion 8
(i) The letters of the word βPEACEβ are placed on cards. Leela draws a
card without looking.
(i) What is the probability that it is a P, E or C?
Now,
Total letters = 5
Question 9
(i) A game of chance consists of spinning an arrow (see Fig. 7.7.) which
comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8, and these are equally likely outcomes. What is the probability that it will point at
(i) 8?
Total numbers = 8
Number of times we can get 8 = 1
Question 10
A basket contains 4 red balls and 5 blue balls. One ball is drawn
and laid aside, and a second ball is drawn. Draw a tree diagram
to represent the possible outcomes and probabilities. Use the tree
diagram to answer the following questions.
(i) What is the probability of drawing a red ball and then a blue ball?
(ii) What is the probability of drawing 2 blue balls?
Question 11
I throw a pair of 6-sided dice. Write down an event that has a probability of 0 and an outcome that has a probability of 1.
Here, we throw pair of 6-sided die (not 1 die, 2 dies)
Question 12 (i)
Write the sample space and calculate the probability based on the
given information.
(i) Two dice are rolled. What is the probability that the sum is a prime number greater than 5?
Letβs look at our Sample Space
Second Die
First Die
Now,
Prime number greater than 5 are 7, 11, 13, 17, β¦
Question 12 (ii)
Write the sample space and calculate the probability based on the
given information.
(ii) A bag contains 4 red, 3 green, and 2 blue balls. Two balls are drawn without replacement. What is the probability that both are of different colours?
Here, the ball is taken out without replacement
So, our probabilities when picking second ball needs to be carefully calculated
Now,
Probability that both are of different colours
= Probability it is (Red, Green)
+ Probability it is (Red, Blue)
+ Probability it is (Green, Red)
+ Probability it is (Green, Blue)
+ Probability it is (Blue, Red)
+ Probability it is (Blue, Green)
= π/π Γπ/π+π/π Γπ/π+π/π Γπ/π+π/π Γπ/π+π/π Γπ/π+π/π Γπ/π
= (12 + 8 + 12 + 6 + 8 + 6)/72
= 52/72
= ππ/ππ
Question 12 (iii)
Write the sample space and calculate the probability based on the
given information.
(iii) Three coins are tossed. What is the probability that the first coin shows heads and exactly two heads occur in total?
Since 3 coins are tossed
Our Sample Space would be
S = {HHH, HHT, HTH, HTT, TTT, THT, TTH, THH}
Question 12 (iv)
Write the sample space and calculate the probability based on the
given information.
(iv) A four-digit number is formed using the digits 1, 2, 3, and 4 with no repetition. What is the probability that the number is even?
For a number to be even, it must end in 2 or 4
Since there are 4 digits, and 2 of them are even
Question 12 (v)
Write the sample space and calculate the probability based on the
given information.
(v) A student takes a multiple-choice test with 3 questions, each having 4 options (A, B, C, D), with only one correct answer.
What is the probability that the student guesses and gets exactly 2 answers correct?
Now,
Probability of guessing right (R) is 1/4
Probability of guessing wrong (W) is 3/4
Question 13
(i)
A box contains 4 balls numbered 1 to 4. Record a sample space using a tree diagram for the following experiments:
(i) A ball is drawn, and the number is recorded. Then the ball is returned, and a second ball is drawn and recorded.
Here, the ball is taken out with replacement
So, our probabilities when picking second ball remains same for all cases
Question 14
List the elements of a sample space for the simultaneous tossing
of a coin and drawing of a card from a set of 6 cards numbered 1
through 6.
Letβs toss a coin and select a card numbered 1 to 6
Question 15
Three coins are tossed, and the number of heads is recorded.
Which of the following lists is a sample space for this experiment?
Why do the other lists fail to qualify as a sample space?
{1, 2, 3} (ii) {0, 1, 2}
(iii) {0, 1, 2, 3, 4} (iv) {0, 1, 2, 3}
Question 16
Suppose you drop a dye at random on the rectangular region shown in Fig. 7.8. What is the probability that it will land inside the circle with a diameter of 1 m?
Here, we use area to find probability
Why Learn This With Teachoo?
The Mathematics of Maybe: Introduction to Probability is Chapter 7 of NCERT Class 9 Ganita Manjari Part 1. It introduces the language of chance, compares experimental and theoretical probability, connects probability with statistical data and uses sample spaces, events and tree diagrams. Teachoo explains each idea step by step and provides solutions for Exercise Sets 7.1 to 7.4 and the end-of-chapter exercises.
What is probability?
Probability measures how likely an event is. A probability lies from 0 to 1. An impossible event has probability 0, a certain event has probability 1 and other events lie between them. Probability may also be expressed as a fraction, decimal or percentage.
A random experiment is a repeatable process whose exact outcome cannot be predicted with certainty. The sample space is the set of all possible outcomes, and an event is a selected subset of that sample space. For a fair experiment with equally likely outcomes,
P(event) = number of favourable outcomes ÷ total number of possible outcomes.
The complement of event A contains outcomes where A does not occur, so P(not A) = 1 − P(A).
Experimental and theoretical probability
Experimental probability is based on observed relative frequency:
P(E) ≈ number of times E occurs ÷ total trials.
Theoretical probability is derived from a mathematical model of possible outcomes. Experimental results may differ from the theoretical value in a small number of trials, but they often stabilise closer to it as the number of trials grows.
Students analyse statistical data using probability and examine whether assumptions such as fairness or equal likelihood are justified. A probability calculation is only as sound as its model.
Elements of probability and tree diagrams
The chapter develops outcomes, events, sample spaces and favourable cases through coins, dice, spinners, cards and contextual experiments. Tree diagrams display multi-stage experiments by branching at each stage. Every complete path represents one combined outcome.
For independent stages, path probabilities are multiplied, while probabilities of mutually exclusive target paths are added. Even before formal rules are used, a tree helps list outcomes without omission or duplication.
Topics covered on Teachoo
Teachoo includes:
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definition and interpretation of probability;
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Exercise Set 7.1;
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experimental and theoretical probability;
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analysis of statistical data using probability;
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Exercise Set 7.2;
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sample spaces, outcomes and events;
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Exercise Set 7.3;
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tree diagrams for multi-stage experiments;
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Exercise Set 7.4; and
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end-of-chapter exercise solutions.
Learning outcomes
Students should be able to identify a random experiment, construct its sample space and calculate probabilities for equally likely outcomes. They should distinguish experimental from theoretical probability, interpret relative frequency and use complements. They should draw a complete tree diagram for successive stages, combine appropriate paths and judge whether an equal-likelihood assumption is valid.
Why is this chapter important?
Probability supports statistics, risk assessment, genetics, finance, science and decision-making. The chapter teaches students to quantify uncertainty without pretending that one result is guaranteed. It also develops disciplined counting: missing or duplicating outcomes produces an incorrect model.
How Teachoo helps you study
Teachoo organises probability from definitions to experiments and trees. Begin each question by writing the experiment and sample space. Confirm that outcomes are equally likely before using favourable ÷ total. In experimental questions, use the observed number of trials and describe the result as an estimate.
For trees, label every branch and follow complete paths from start to finish. Add only paths that represent alternative ways for the target event to occur. Use Teachoo’s solution to verify the sample space before checking the arithmetic.
Common mistakes to avoid
Do not confuse an outcome with an event: an event may contain several outcomes. A result that has not appeared in a small experiment is not automatically impossible. Do not assume physical coins, dice or spinners are fair unless stated or supported. In tree diagrams, include all branches at every stage and avoid counting the same path twice.
Quick revision checklist
Write complete sample spaces for one coin, die and spinner experiment, then identify several events and complements. Compare a theoretical probability with results from a simulated or recorded experiment and explain why they may differ. Draw a tree for a two-stage process, label branch probabilities and calculate a target event from complete paths. Finally, examine a supposedly fair game and state which assumptions must be true for its probability model to be valid.
Approach to competency-based questions
Case studies often contain extra narrative. Extract the experiment, possible outcomes and event before using numbers. If outcomes are not equally likely, favourable outcomes divided by total outcome types is invalid. Use frequencies or branch probabilities instead. End by interpreting the result as likelihood, not as a guarantee of what the next trial will produce.
Deeper reasoning and concept connections
In The Mathematics of Maybe: Introduction to Probability (Ganita Manjari Part 1), fluency means more than repeating a procedure. Students should be able to recognise the underlying structure when the numbers, diagram, wording or orientation changes. A useful routine is: identify the mathematical objects, list the known and unknown quantities, state the governing property, carry out the steps and verify that every condition has been used.
Look for connections within the chapter as well. A definition usually leads to a representation; the representation reveals a pattern; and the pattern supports a rule or calculation. Explaining this chain improves retention and helps with case-based questions. It also prevents the common mistake of selecting a formula simply because its symbols resemble the numbers in the question.
How to solve unfamiliar and competency-based questions
Use a five-step response: interpret, represent, select, solve and verify. Interpret the wording; represent the information; select a definition, property or formula; solve without skipping the logical step; and verify through substitution, estimation, measurement or an alternative representation. This routine works for direct exercises as well as case-based questions.
If information appears unnecessary, ask whether it establishes a hidden condition. If information is missing, state what cannot be determined instead of inventing a value. In written answers, name the rule being used. Clear reasoning helps a teacher award method marks and also makes the page easier for a student—or an AI answer system—to retrieve for the precise doubt being asked.
What complete mastery looks like
For The Mathematics of Maybe: Introduction to Probability (Ganita Manjari Part 1), a student should be able to define the central ideas in simple language, recognise them in different representations, solve routine questions accurately and explain the method used. They should also be able to correct a flawed solution, create an example satisfying given conditions and combine two ideas from the chapter in one problem. A reliable mastery test is to solve one direct question, one application question and one reasoning question without looking at notes, then explain all three aloud or in writing.
Keep a compact error log with four labels: concept, interpretation, calculation and presentation. Reattempt each error after a gap instead of rereading the answer immediately. Improvement comes from correcting the decision that caused the mistake, not from repeating questions whose method is already known.
Additional frequently asked questions
What should a student know before starting The Mathematics of Maybe: Introduction to Probability (Ganita Manjari Part 1)?
Revise the definitions, number operations, diagrams or notation used at the beginning of the chapter. The prerequisite list should be short: if an earlier skill blocks progress, repair that skill with two or three focused questions and return to the chapter.
How can a student check an answer in The Mathematics of Maybe: Introduction to Probability (Ganita Manjari Part 1)?
Use an independent check whenever possible: substitute the result, reverse the operation, estimate its size, compare it with the figure, test a simpler case or solve using another representation. A check should examine the mathematical condition, not merely repeat the same arithmetic.
How many questions are enough for strong preparation?
There is no fixed number. Stop counting questions and track coverage: every concept, every standard method, at least one mixed problem, one competency-based problem and every previously incorrect type should be solved independently. Ten varied, analysed questions are more valuable than fifty copied solutions.
How should Teachoo solutions be used without becoming dependent on them?
Attempt the question first and mark the exact step where progress stops. Read only enough of the solution to repair that step, close it and restart the question. Finally, solve a similar problem without help. This turns a solution into feedback rather than a substitute for thinking.
Frequently asked questions
What is the difference between experimental and theoretical probability?
Experimental probability comes from observed trials; theoretical probability comes from a mathematical model of possible outcomes.
Can probability be greater than 1?
No. Every probability lies between 0 and 1 inclusive.
Why are tree diagrams useful?
They display all paths in a multi-stage experiment systematically, reducing omissions and duplicate counting.
List the outcomes first and calculate second. A correct probability begins with a complete and justified sample space.