Chapter 7 of 8

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Updated 6 August 2026

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The Mathematics of Maybe : Introduction to Probability

CBSE · Class 9 · Mathematics

NCERT Solutions for The Mathematics of Maybe : Introduction to Probability — CBSE Class 9 Mathematics.

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EXERCISE SET 7.1

1(i)The next Monday will come after Sunday.Show solution

This is certain because Monday always comes after Sunday in the weekly sequence.

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1(ii)It will snow in Mumbai in July.Show solution

This is impossible or at least extremely unlikely in Mumbai in July, because the chapter’s scale places events like snow in Mumbai in July as impossible/very unlikely.

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1(iii)An elephant will walk through your classroom today.Show solution

This is impossible in ordinary classroom conditions, so it should be ranked at 0 on the probability scale.

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1(iv)You will greet at least one friend at school tomorrow.Show solution

This is more likely because in normal school life you are expected to meet and greet at least one friend tomorrow.

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EXERCISE SET 7.2

1(i)Calculate the probability that a randomly picked sweet from the sample is green.Show solution

From the sample of 30 sweets, 8 are green.

P(\text{green})=\frac{\text{number of green sweets}}{\text{total sweets}}=\frac{8}{30}=\frac{4}{15}\approx 0.2667

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1(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.Show solution

In the sample, 7 out of 30 sweets are yellow.

So the estimated number of yellow sweets in 600 is

\frac{7}{30}\times 600 = 7\times 20 = 140

So about 140 sweets are likely to be yellow.

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2(i)What is the probability that a randomly chosen student from the sample prefers the Arts Club?Show solution

From the sample of 40 students, 11 prefer the Arts Club.

P(\text{Arts Club})=\frac{11}{40}=0.275

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2(ii)Using the sample results, estimate how many students in the whole school are likely to prefer the Sports Club.Show solution

In the sample, 9 out of 40 students prefer Sports Club.

Estimated number in 800 students:

\frac{9}{40}\times 800 = 9\times 20 = 180

So about 180 students are likely to prefer the Sports Club.

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3(i)How many times did you get heads?Show solution

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3(ii)How many times did you get tails?Show solution

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3(iii)Calculate the experimental probability of getting heads.Show solution

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3(iv)If you toss the coin once more, what is the probability of getting tails?Show solution

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4Assign probabilities to the outcomes by using experimental probability.Show solution

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5What is the probability of getting an even number when rolling a fair 6-sided die?Show solution

A fair 6-sided die has 6 outcomes: 1, 2, 3, 4, 5, 6.

Even numbers are 2, 4, 6.

So,

P(\text{even})=\frac{3}{6}=\frac{1}{2}

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6(i)What is the experimental probability of rolling a '3'?Show solution

The die was rolled 12 times and showed 3 exactly 3 times.

Experimental probability:

P(3)=\frac{\text{number of times 3 occurred}}{\text{total trials}}=\frac{3}{12}=\frac{1}{4}=0.25

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6(ii)What is the theoretical probability of rolling a '3'?Show solution

For a fair 6-sided die, the favourable outcome for rolling a 3 is 1 and the total possible outcomes are 6.

So,

P(3)=\frac{1}{6}

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6(iii)Why might these probabilities be different? What would you expect to happen if you roll the die 60, 600, or 6000 times?Show solution

The two probabilities may be different because experimental probability is based on actual results from a limited number of trials, while theoretical probability is what we expect for a fair die. With a small number of rolls, the result may differ from the theoretical value by chance. As the number of rolls increases to 60, 600, or 6000, the experimental probability should generally get closer to the theoretical probability of **

\frac{1}{6}

**, though it may still not be exactly equal.

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4Toss 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.Show solution

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EXERCISE SET 7.3

1When a single 6-sided die is rolled, what is the total number of possible outcomes in the sample space?Show solution

A standard 6-sided die has outcomes 1, 2, 3, 4, 5, 6.

So the sample space has 6 possible outcomes.

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2(i)For the following experiments write down the sample space S.Show solution

The sample space for rolling a die and tossing a coin together is all ordered pairs:

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

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2(ii)Choosing a random integer between – 5 and + 5.Show solution

Choosing a random integer between

-5

and

+5

gives the sample space

S=\{-5,-4,-3,-2,-1,0,1,2,3,4,5\}

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2(iii)A box containing 5 green and 7 red balls. One ball is drawn at random.Show solution

For one ball drawn from a box with 5 green and 7 red balls, the possible outcomes are the colours:

S=\{\text{Green, Red}\}

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3(i)List the sample space of all possible snack and drink combinations a person could choose at the fair.Show solution

The snack and drink combinations are:

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

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3(ii)List the event ‘Selecting Samosa as a snack.’Show solution

The event ‘Selecting Samosa as a snack’ includes all outcomes where the snack is Samosa:

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

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EXERCISE SET 7.4

1(i)Draw a tree diagram showing all possible pairs of fruits.Show solution

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1(ii)List the sample space.Show solution

With basket A = {apple, orange, orange} and basket B = {banana, mango}, the possible pairs are:

S=\{(A,B), (A,M), (O,B), (O,M)\}

Using full fruit names:

S=\{(\text{Apple, Banana}), (\text{Apple, Mango}), (\text{Orange, Banana}), (\text{Orange, Mango})\}

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1(iii)What is the probability of picking one apple and one banana?Show solution

There are 4 equally likely outcomes in the sample space:

\{\text{Apple, Banana}, \text{Apple, Mango}, \text{Orange, Banana}, \text{Orange, Mango}\}

.

The favourable outcome for one apple and one banana is just 1 outcome.

So,

P(\text{one apple and one banana})=\frac{1}{4}

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2(i)What are the possible outcomes of the pen colours? Can you draw a tree diagram representing the possible outcomes?Show solution

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2(ii)Can you use the tree diagram to guess the probability that both you and your friend pick pens of the same colour?Show solution

Yes. If the pens are treated as fair and replacement is used, each colour is chosen according to its proportion in the box. The probability that both pick the same colour is the sum of the probabilities of both getting red, both getting black, or both getting green. So the tree diagram lets us estimate that probability by adding those branch probabilities.

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END-OF-CHAPTER EXERCISES

1(i)The probability of an impossible event is _____.Show solution

The probability of an impossible event is 0.

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1(ii)The set of all possible outcomes of a random experiment is called the _______.Show solution

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

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1(iii)The probability of an event that is certain to happen is _______.Show solution

The probability of an event that is certain to happen is 1.

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1(iv)Tossing a fair coin has a probability of _______ for getting heads.Show solution

A fair coin has two equally likely outcomes: heads and tails.

So the probability of getting heads is

\frac{1}{2}

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2In 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).Show solution

The number of students who like football is the frequency. So the relative frequency is

\frac{15}{50} = \frac{3}{10} = 0.3

So the blanks are relative frequency and 0.3.

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3(i)A driver attempts to start a car. The car starts or does not start.Show solution

The sample space is the set of possible outcomes:

- Car starts
- Car does not start

So,

S = \{\text{Starts}, \text{Does not start}\}

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3(ii)Tossing a fair coin once.Show solution

When a fair coin is tossed once, the possible outcomes are:

- Heads
- Tails

So, the sample space is

S = \{H, T\}

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3(iii)Rolling a fair 6-sided die.Show solution

For a fair 6-sided die, the possible outcomes are:

- 1, 2, 3, 4, 5, 6

So, the sample space is

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

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3(iv)Choosing a marble randomly from a bag that contains 3 red marbles and 7 blue marbles.Show solution

The bag has 3 red marbles and 7 blue marbles, so the possible outcomes are:

- Red marble
- Blue marble

Thus, the sample space is

S = \{\text{Red}, \text{Blue}\}

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3(v)A baby is born. It is a boy or a girl.Show solution

A baby can be born as either:

- Boy
- Girl

So the sample space is

S = \{\text{Boy}, \text{Girl}\}

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4(i)Two coins are tossed at the same time. What is the probability of getting at least one head?Show solution

For two tossed coins, the sample space is

S = \{HH, HT, TH, TT\}

The event at least one head is

E = \{HH, HT, TH\}

So,

P(E)=\frac{3}{4}

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4(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?

4(iii)A die is rolled once. What is the probability of getting a number greater than 4?

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?

4(v)Three coins are tossed simultaneously. What is the probability of getting exactly two heads?

5A bag has 3 candies: strawberry, lemon, and mint. One is picked at random. What is the probability of picking a strawberry candy?

6A 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.

7Find the probability that a randomly chosen tyre lasts:

7(i)Less than 4000 km.

7(ii)Between 4000 and 14000 km.

7(iii)More than 14000 km.

8(i)What is the probability that it is a P, E or C?

8(ii)What is the probability that it is not an E?

9(i)8?

9(ii)An odd number?

9(iii)A number greater than 2?

9(iv)A number less than 9?

9(v)A multiple of 3?

10(i)What is the probability of drawing a red ball and then a blue ball?

10(ii)What is the probability of drawing 2 blue balls?

11I 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.

12Write the sample space and calculate the probability based on the given information.

12(i)Two dice are rolled. What is the probability that the sum is a prime number greater than 5?

12(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?

12(iii)Three coins are tossed. What is the probability that the first coin shows heads and exactly two heads occur in total?

12(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?

12(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?

13A box contains 4 balls numbered 1 to 4. Record a sample space using a tree diagram for the following experiments:

13(i)A ball is drawn, and the number is recorded. Then the ball is returned, and a second ball is drawn and recorded.

13(ii)A ball is drawn and recorded. Without replacing the first ball, the experimenter draws and records a second ball.

13(iii)What are the sizes of these two sample spaces?

14List 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.

15Three 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?

15(i){1, 2, 3}

15(ii){0, 1, 2}

15(iii){0, 1, 2, 3, 4}

15(iv){0, 1, 2, 3}

16Suppose 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?

3Which of the following experiments have equally likely outcomes? Explain.

7A tyre company records distances before replacement in 1000 cases.

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EXERCISE SET 7.1

EXERCISE SET 7.2

EXERCISE SET 7.3

EXERCISE SET 7.4

END-OF-CHAPTER EXERCISES

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