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Class 9 Maths Part 1 Chapter 7 – The Mathematics of Maybe: Introduction to Probability | Ganita Manjari Notes & Book Solutions

Class 9 Maths Part 1 Chapter 7 – The Mathematics of Maybe: Introduction to Probability from Ganita Manjari with easy Notes and complete Book Solutions. Get clear explanations, important concepts, questions and answers, and revision material for Chapter 7 to help Class 9 students learn, practise and prepare effectively for examinations.

Grade 9 Maths – Chapter 7: The Mathematics of Maybe
📘 Grade 9 Maths – Chapter 7100%
Ganita Manjari • Grade 9 • Chapter 7

The Mathematics of Maybe: Introduction to Probability

Easy notes, concepts and book solutions for Chapter 7.

Easy NotesExercise 7.1Exercise 7.2Exercise 7.3Exercise 7.4End Exercises

🌟 Easy Notes & Concepts

1. Probability

Probability tells us how likely an event is to happen.

0 ≤ P(E) ≤ 1

0 means impossible and 1 means certain.

2. Random Experiment

A random experiment is an experiment whose exact result cannot be predicted in advance.

Examples: tossing a coin and rolling a die.

3. Experimental Probability

Experimental Probability = Number of times event occurs ÷ Total number of trials

4. Theoretical Probability

P(E) = Favourable outcomes ÷ Total possible outcomes

This is used when outcomes are equally likely.

5. Sample Space

The sample space is the set of all possible outcomes.

For a coin:

S = {H,T}

For a die:

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

6. Event

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

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

E = {5,6}

7. Tree Diagram

A tree diagram shows the different possible paths in a multi-step experiment.

8. Gambler’s Fallacy

In independent random experiments, earlier results do not change the probability of the next independent result.

📘 Exercise Set 7.1

Q1Rank 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.
(ii) It will snow in Mumbai in July.
(iii) An elephant will walk through your classroom today.
(iv) You will greet at least one friend at school tomorrow.

(i) Certain: Monday always comes after Sunday.

(ii) Less likely: Snow in Mumbai in July is extremely unlikely.

(iii) Less likely: An elephant entering a classroom is extremely unlikely.

(iv) More likely: Under normal school circumstances, greeting a friend is likely.

📘 Exercise Set 7.2

Q1A 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.
(ii) If there are 600 sweets in total in the large bag, estimate how many are likely to be yellow, based on the sample results.

Green sweets = 8.

P(green) = 8/30 = 4/15

Yellow fraction:

7/30

Estimated yellow sweets:

600 × 7/30 = 140
Answer: (i) 4/15    (ii) 140 sweets
Q2A 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?
(ii) Using the sample results, estimate how many students in the whole school are likely to prefer the Sports Club.

(i) Arts Club = 11 out of 40.

P(Arts) = 11/40

(ii) Sports Club = 9 out of 40.

800 × 9/40 = 180
Answer: (i) 11/40    (ii) 180 students
Q3Toss a coin 20 times and record the result each time (heads or tails).

(i) How many times did you get heads?
(ii) How many times did you get tails?
(iii) Calculate the experimental probability of getting heads.
(iv) If you toss the coin once more, what is the probability of getting tails?

The answers to (i), (ii) and (iii) depend on the actual 20 tosses performed.

If heads occur H times:

Experimental P(heads) = H/20

For a fair coin:

P(tails) = 1/2
Q4Toss 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.

Let B, T and S be the numbers of times the cup lands on its bottom, top and side.

P(bottom)=B/100
P(top)=T/100
P(side)=S/100

Since B + T + S = 100, the three probabilities add to 1.

Q5What is the probability of getting an even number when rolling a fair 6-sided die?

Even numbers are 2, 4 and 6.

P(even)=3/6=1/2
Answer: 1/2
Q6Suppose you roll a 6-sided die 12 times and get a ‘3’ three times.

(i) What is the experimental probability of rolling a ‘3’?
(ii) What is the theoretical probability of rolling a ‘3’?
(iii) Why might these probabilities be different? What would you expect to happen if you roll the die 60, 600, or 6000 times?

(i)

Experimental probability = 3/12 = 1/4

(ii)

Theoretical probability = 1/6

(iii) A small number of trials can give results different from the theoretical probability because of random variation. With many trials, the experimental probability generally gets closer to 1/6.

📘 Exercise Set 7.3

Q1When a single 6-sided die is rolled, what is the total number of possible outcomes in the sample space?
S={1,2,3,4,5,6}

Therefore, there are 6 possible outcomes.

Q2For the following experiments write down the sample space S.

(i) Rolling a die and tossing a coin together.
(ii) Choosing a random integer between – 5 and + 5.
(iii) A box containing 5 green and 7 red balls. One ball is drawn at random.

(i)

S={(1,H),(1,T),(2,H),(2,T),(3,H),(3,T),(4,H),(4,T),(5,H),(5,T),(6,H),(6,T)}

(ii)

S={−5,−4,−3,−2,−1,0,1,2,3,4,5}

(iii)

S={Green,Red}
Q3In 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.
(ii) List the event ‘Selecting Samosa as a snack.’

(i)

S={(Samosa,Chai),(Samosa,Lassi), (Pakora,Chai),(Pakora,Lassi), (Bhaji,Chai),(Bhaji,Lassi)}

(ii)

E={(Samosa,Chai),(Samosa,Lassi)}

📘 Exercise Set 7.4

Q1There 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.
(ii) List the sample space.
(iii) What is the probability of picking one apple and one banana?
Basket A├── Apple │ ├── Banana │ └── Mango │ └── Orange ├── Banana └── Mango

Probability of Apple from Basket A:

1/3

Probability of Banana from Basket B:

1/2
P(Apple and Banana) =1/3 × 1/2 =1/6
Answer: 1/6
Q2Let 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?
(ii) Can you use the tree diagram to guess the probability that both you and your friend pick pens of the same colour?
First pick R → R, B, GFirst pick B → R, B, GFirst pick G → R, B, G

Possible colour pairs:

RR,RB,RG,BR,BB,BG,GR,GB,GG

Because the pen is replaced after the first pick:

P(R)=3/9
P(B)=4/9
P(G)=2/9

Therefore:

P(same) = (3/9×3/9) + (4/9×4/9) + (2/9×2/9) = 29/81
Answer: 29/81

📕 End-of-Chapter Exercises

Q1Fill in the blanks.

(i) The probability of an impossible event is _______.
(ii) The set of all possible outcomes of a random experiment is called the __________.
(iii) The probability of an event that is certain to happen is _______.
(iv) Tossing a fair coin has a probability of ______ for getting heads.

(i) 0

(ii) Sample space

(iii) 1

(iv) 1/2

Q2In 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).

15 is the frequency.

Relative frequency = 15/50 = 3/10 = 0.3
Q3Which of the following experiments have equally likely outcomes? Explain.

(i) A driver attempts to start a car. The car starts or does not start.
(ii) Tossing a fair coin once.
(iii) Rolling a fair 6-sided die.
(iv) Choosing a marble randomly from a bag that contains 3 red marbles and 7 blue marbles.
(v) A baby is born. It is a boy or a girl.

(ii) Yes. A fair coin has equal chances of heads and tails.

(iii) Yes. Each face of a fair die is equally likely.

(i) Not necessarily.

(iv) No. Red has probability 3/10 and blue has probability 7/10.

(v) The information given does not establish equal probabilities.

Q4Write 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?
(ii) Ten identical cards numbered 1 to 10 are placed in a box. One card is drawn at random. What is the probability of drawing a card with an even number?
(iii) A die is rolled once. What is the probability of getting a number greater than 4?
(iv) A bag contains 3 red balls, 2 blue balls, and 1 green ball. One ball is picked at random. What is the probability that it is not red?
(v) Three coins are tossed simultaneously. What is the probability of getting exactly two heads?

(i) S={HH,HT,TH,TT}.

At least one head = HH, HT, TH.

P=3/4

(ii) Even cards = 2,4,6,8,10.

P=5/10=1/2

(iii) Numbers greater than 4 are 5 and 6.

P=2/6=1/3

(iv) Not red means blue or green.

P=3/6=1/2

(v) Exactly two heads: HHT, HTH, THH.

P=3/8
Q5A bag has 3 candies: strawberry, lemon, and mint. One is picked at random. What is the probability of picking a strawberry candy?
P(strawberry)=1/3
Answer: 1/3
Q6A 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 answer in a table format.
ShirtPantsOutfit
RedJeansRed + Jeans
RedKhakisRed + Khakis
RedShortsRed + Shorts
BlueJeansBlue + Jeans
BlueKhakisBlue + Khakis
BlueShortsBlue + Shorts
There are 6 possible outfits.
Q7A tyre company records distances before replacement in 1000 cases.
Distance (km)Less than 40004001 to 90009001 to 14000More than 14000
Number of cases20210325445
Find the probability that a randomly chosen tyre lasts:

(i) Less than 4000 km.
(ii) Between 4000 and 14000 km.
(iii) More than 14000 km.

(i)

20/1000=0.02

(ii)

(210+325)/1000 =535/1000 =0.535

(iii)

445/1000=0.445
Q8The letters of the word ‘PEACE’ are placed on cards. Leela draws a card without looking.

(i) What is the probability that it is a P, E or C?
(ii) What is the probability that it is not an E?

There are 5 cards: P, E, A, C, E.

(i) P, E or C gives 4 favourable cards.

P=4/5

(ii) Not E means P, A or C = 3 cards.

P=3/5
Q9A 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?
(ii) An odd number?
(iii) A number greater than 2?
(iv) A number less than 9?
(v) A multiple of 3?

Total outcomes = 8.

(i) 8 → 1/8

(ii) 1,3,5,7 → 4/8=1/2

(iii) 3,4,5,6,7,8 → 6/8=3/4

(iv) All 8 outcomes → 1

(v) 3,6 → 2/8=1/4

Q10A 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?
First draw:R (4/9) ├── B (5/8) → RB = 5/18 └── R (3/8) → RR = 1/6B (5/9) ├── R (4/8) → BR = 5/18 └── B (4/8) → BB = 5/18

(i)

P(R then B)=4/9×5/8=5/18

(ii)

P(B then B)=5/9×4/8=5/18
Q11I 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.

An impossible event is getting a sum of 13.

P(sum=13)=0

A certain event is getting a sum between 2 and 12.

P(2≤sum≤12)=1
Q12Write 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?
(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?
(iii) Three coins are tossed. What is the probability that the first coin shows heads and exactly two heads occur in total?
(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?
(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?

(i) Prime sums greater than 5 are 7 and 11.

There are 6 outcomes giving sum 7 and 2 giving sum 11.

P=8/36=2/9

(ii) Total ways to choose 2 balls from 9:

C(9,2)=36

Same-colour pairs:

C(4,2)+C(3,2)+C(2,2)=10

Different-colour pairs = 36−10=26.

P=26/36=13/18

(iii) Favourable outcomes are HHT and HTH.

P=2/8=1/4

(iv) There are 24 arrangements. An even number ends in 2 or 4, giving 12 arrangements.

P=12/24=1/2

(v)

P=C(3,2)(1/4)²(3/4) =9/64
Q13A 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.
(ii) A ball is drawn and recorded. Without replacing the first ball, the experimenter draws and records a second ball.
(iii) What are the sizes of these two sample spaces?

(i) With replacement:

n(S)=4×4=16

All ordered pairs (1,1) through (4,4) are possible.

(ii) Without replacement:

n(S)=4×3=12

The second number cannot be the same as the first.

Q14List 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.
S={(H,1),(H,2),(H,3),(H,4),(H,5),(H,6), (T,1),(T,2),(T,3),(T,4),(T,5),(T,6)}
There are 12 outcomes.
Q15Three 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?

(i) {1, 2, 3}
(ii) {0, 1, 2}
(iii) {0, 1, 2, 3, 4}
(iv) {0, 1, 2, 3}

With three coins, the number of heads can be 0, 1, 2 or 3.

Correct sample space = {0,1,2,3}

Therefore, (iv) is correct.

(i) misses 0.

(ii) misses 3.

(iii) contains the impossible outcome 4.

Q16Suppose 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?

Rectangle area = 2 × 3 = 6 m².

Circle diameter = 1 m.

Therefore radius = 0.5 m.

Area of circle =π(0.5)² =π/4
Probability =(π/4)/6 =π/24
Answer: π/24 ≈ 0.131

🧠 Quick Revision

  • Probability always lies between 0 and 1.
  • Experimental probability uses actual results.
  • Theoretical probability uses equally likely outcomes.
  • Sample space = all possible outcomes.
  • Event = one or more outcomes from the sample space.
  • Tree diagrams help list outcomes of multi-step experiments.
  • For a fair coin: P(H)=P(T)=1/2.
  • For a fair six-sided die: P(any one number)=1/6.

prepare more with the given Worksheet of chapter 7, which will help you to improve your knowledge .

CLASS 9 • MATHEMATICS

Chapter 7 Worksheet

Practice Questions & Revision Worksheet

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