Probability: Important Questions and Step-by-Step Solutions for Class 10

Probability is an important chapter in Class 10 Mathematics that helps students understand the likelihood of events occurring in different situations. The chapter introduces fundamental concepts such as experimental probability, theoretical probability, and outcomes, which are essential for solving real-world and examination-based problems. In this guide, students will find important questions for Class 10 Maths Probability according to the latest syllabus and exam pattern. These exercises help students become familiar with different question types commonly asked in assessments. 

Our subject experts have provided detailed solutions for these problems based on the latest CBSE syllabus and the NCERT textbook. This material helps students revise the chapter easily and perform well in the final examination.

Table of Contents

• What Is Probability?
• Key Terms in Probability
• Experimental vs Theoretical Probability
• SECTION A: MCQ Questions (1 Mark Each)
• SECTION B: 1-Mark / Short Objective Questions
• SECTION C: 2-Mark Questions
• SECTION D: 3-Mark Questions
• SECTION E: 4/5-Mark Long Answer & Case Study Questions
• Frequently Tested Previous Year Board Questions

What Is Probability?

Probability is the branch of mathematics that measures the likelihood of an event occurring. It is expressed as a number between 0 and 1:

P = 0 ⇒  Impossible event (e.g., rolling a 7 on a standard die)

P = 1 ⇒  Sure/certain event (e.g., getting a number less than 7 when rolling a die)

0 < P < 1 ⇒  All other events

Probability Formula:
P(E) = Number of outcomes favourable to E / Total number of equally likely outcomes

Key Terms in Probability

Experimental vs Theoretical Probability

Experimental (Empirical) Probability:

P(E) = Number of trials where the event occurred / Total number of trials

Theoretical Probability:

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

SECTION A: MCQ Questions (1 Mark Each)

MCQs from Probability are a guaranteed feature of the Class 10 board paper. Here are the most frequently asked types.

Q1. A die is thrown once. The probability of getting an even prime number is:

(a) 1/6    (b) 1/3    (c) 2/3    (d) 1/2

Answer: (a) 1/6

The only even prime number is 2.

Favourable outcomes = {2}

n(E) = 1

P(E) = 1/6

Q2. The probability of an impossible event is

(a) 1    (b) 0    (c) 1/2    (d) not defined

Answer: (b) 0

Q3. A bag contains 3 red balls and 5 black balls. A ball is drawn at random. The probability of drawing a red ball is:

(a) 3/8    (b) 5/8    (c) 1/8    (d) 3/5

Answer: (a) 3/8

Total balls = 3 + 5 = 8. Favourable outcomes = 3.

P(red) = 3/8

Q4. Which of the following cannot be the probability of an event?

(a) 0.7    (b) 2/3    (c) 15%    (d) 7/5

Answer: (d) 7/5

Probability must lie between 0 and 1 (inclusive). 7/5 = 1.4 > 1, so it is not a valid probability.

Q5. A number is chosen at random from 1 to 20. The probability of it being a prime number is:

(a) 2/5    (b) 1/5    (c) 3/10    (d) 1/2

Answer: (a) 2/5

Primes from 1 to 20: 2, 3, 5, 7, 11, 13, 17, 19

Number of favourable outcomes = 8 primes

P = 8/20 = 2/5

Q6. If P(E) = 0.05, then P(not E) is:

(a) 0.05    (b) 0.95    (c) 0.9    (d) 1.05

Answer: (b) 0.95

P(not E) = 1 − P(E) = 1 − 0.05 = 0.95

Q7. A card is drawn from a well-shuffled deck of 52 cards. The probability of getting a face card is:

(a) 1/13    (b) 3/13    (c) 4/13    (d) 1/4

Answer: (b) 3/13

Face cards = 12 (Jack, Queen, King × 4 suits)

P = 12/52 = 3/13

 

SECTION B: 1-Mark / Short Objective Questions

Q8. A bag contains a red ball, a blue ball, and a yellow ball (all the same size). Kritika takes out a ball without looking into the bag. What is the probability that she takes out the yellow ball?

Solution: Total balls = 3. Favourable outcomes = 1 (yellow ball)

P(yellow) = 1/3

Similarly, P(red) = 1/3 and P(blue) = 1/3.

Q9. If P(E) = 0.65, what is P(not E)?

Solution: P(not E) = 1 − P(E)  = 1 - 0.65 = 0.35

Q10. An integer is chosen at random from 1 to 100. Find the probability that it is divisible by 8.

Solution: Numbers from 1 to 100 divisible by 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96

⇒ Number of favourable outcomes = 12

P = 12/100 = 3/25

SECTION C: 2-Mark Questions

Q11. A game of chance consists of spinning an arrow that comes to rest at any one of the numbers 1, 2, 3, 4, 5, 6, 7, or 8, all equally likely. Find the probability that it points at:

(i) the number 8

(ii) a number greater than 2

(iii) an odd number

Solution: Total outcomes n(S) = 8

(i) P(8) = 1/8

(ii) Numbers greater than 2 = {3, 4, 5, 6, 7, 8

Favourable outcomes = 6 outcomes

P = 6/8 = 3/4

(iii) Odd numbers = {1, 3, 5, 7}

Favourable outcomes = 4 outcomes

P = 4/8 = 1/2

Q12. A bag contains 9 black and 12 white balls. One ball is drawn at random. Find the probability that the ball is black.

Solution: Total balls = 9 + 12 = 21

Favourable outcomes (black) = 9

P(black) = 9/21 = 3/7

Q13. A bag contains 10 red, 5 blue, and 7 green balls. A ball is drawn at random. Find the probability that it is NOT a blue ball.

Solution: Total balls = 10 + 5 + 7 = 22

Blue balls = 5

Not blue = 22 − 5 = 17

P(not blue) = 17/22

Alternatively: P(not blue) = 1 − P(blue) = 1 − 5/22 = 17/22

Q14. An integer is chosen between 0 and 100 (exclusive). Find the probability that it is:

(i) divisible by 7

(ii) NOT divisible by 7

Solution: Integers between 0 and 100 (exclusive), n(S) = 99

(i) Multiples of 7 up to 99: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98

Number of favourable outcomes = 14 numbers

P(divisible by 7) = 14/99

(ii) P(not divisible by 7) = 1 − 14/99 = 85/99

Q15. A letter is chosen at random from the letters of the word ‘ASSASSINATION’. Find the probability that it is a vowel.

Solution: Word: ASSASSINATION

Total letters = 13

Vowels in the word: A, A, I, A, I, O = 6 vowels

P(vowel) = 6/13

SECTION D: 3-Mark Questions

Q16. Cards marked with numbers 5 to 50 are placed in a box and mixed thoroughly. A card is drawn at random. Find the probability that the number on the card is:

(i) a prime number less than 10

(ii) a perfect square

Solution: Cards range from 5 to 50

n(S) = 46

(i) Primes less than 10 in the range 5 to 50: 5, 7

Number of favourable outcomes = 2 

P = 2/46 = 1/23

(ii) Perfect squares from 5 to 50: 9, 16, 25, 36, 49

Number of favourable outcomes = 5 

P = 5/46

Q17. Two dice are thrown simultaneously. Find the probability that:

(i) 5 will not come up on either die

(ii) 5 will come up on at least one die

(iii) 5 will come up on both dice

Solution: Total outcomes when two dice are thrown = 6 × 6 = 36

(i)Outcomes where 5 does NOT appear on either die:

Each die has 5 options (1, 2, 3, 4, 6), so favourable outcomes = 5 × 5 = 25

P = 25/36

(ii) P(5 on at least one) = 1 − P(no 5 on either) = 1 − 25/36 = 11/36

(iii) 5 on both dice = (5, 5)

Favourable outcomes = 1 

P = 1/36

Q18. Two different dice are tossed together. Find the probability that:

(i)The number appearing on each die is even

(ii) the sum of the numbers appearing on the two dice is 5

Solution: Total outcomes = 36

(i) Even numbers on each die: each die can show 2, 4, or 6 (3 options each)

Favourable outcomes = 3 × 3 = 9

P = 9/36 = 1/4

(ii) Pairs that give sum = 5: (1,4), (2,3), (3,2), (4,1)

Favourable outcomes = 4 

P = 4/36 = 1/9

Q19. From a well-shuffled deck of 52 cards, black Jacks, black Kings, and black Aces are removed. A card is drawn from the remaining deck. Find the probability of getting:

(i) a red card

(ii) not a diamond card

Solution: Cards removed: 2 black Jacks + 2 black Kings + 2 black Aces = 6 cards

Remaining cards = 52 − 6 = 46

(i) Red cards are untouched = 26 (all hearts + all diamonds)

P(red card) = 26/46 = 13/23

(ii) Diamond cards = 13 (untouched)

Not a diamond card = 46 − 13 = 33

P(not a diamond) = 33/46

Q20. Cards numbered 1 to 30 are put in a bag. A card is drawn at random. Find the probability that the number on the card is:

(i) not divisible by 3

(ii) a prime number greater than 7

(iii) not a perfect square

Solution: n(S) = 30

(i) Multiples of 3 from 1 to 30: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 

Number of multiples of  3 = 10 numbers

Not divisible by 3: 30 − 10 = 20

P = 20/30 = 2/3

(ii) Primes greater than 7 and ≤ 30: 11, 13, 17, 19, 23, 29

Number of primes greater than 7 and ≤ 30 = 6 numbers

P = 6/30 = 1/5

(iii) Perfect squares from 1 to 30: 1, 4, 9, 16, 25 

Number of perfect squares from 1 to 30 = 5 numbers

Not a perfect square: 30 − 5 = 25

P = 25/30 = 5/6

Q21. A bag contains cards numbered 2 to 90. A card is drawn at random. Find the probability that it bears:

(i) a two-digit number

(ii) a perfect square number

Solution: Cards: 2 to 90 

n(S) = 89

(i) Two-digit numbers from 2 to 90: 10 to 90 

Favourable outcomes = 81 

P = 81/89

(ii) Perfect squares from 2 to 90: 4, 9, 16, 25, 36, 49, 64, 81 

Favourable outcomes = 8 

P = 8/89

SECTION E: 4/5-Mark Long Answer & Case Study Questions

Q22: A card is drawn at random from a well-shuffled deck of 52 playing cards. Find the probability of getting:

(i) a king of red colour

(ii) a face card

(iii) a red face card

(iv) the jack of hearts

(v) a spade

(vi) the queen of diamonds

Solution: n(S) = 52

(i) Red kings = King of Hearts + King of Diamonds = 2

P = 2/52 = 1/26

(ii) Face cards = 3 per suit × 4 suits = 12

P = 12/52 = 3/13

(iii) Red face cards = 3 (Hearts) + 3 (Diamonds) = 6

P = 6/52 = 3/26

(iv) Jack of Hearts = 1 card

P = 1/52

(v) Spades = 13 cards

P = 13/52 = 1/4

(vi) Queen of Diamonds = 1 card

P = 1/52

Q23: Two dice are thrown simultaneously, and the product of the numbers appearing on them is noted. Find the probability that the product is:

(i) less than 9

(ii) a perfect square

(iii) a prime number

Solution: Total outcomes = 36

(i)Products less than 9:

List all pairs (a, b) where a × b < 9:

Favourable pairs: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (4,1), (4,2), (5,1), (6,1)

Number of favourable outcomes = 16 pairs

P = 16/36 = 4/9

(ii) Perfect square products: 1, 4, 9, 16, 25, 36

Pairs: (1,1)=1, (1,4), (4,1)=4, (2,2)=4, (3,3)=9, (4,4)=16, (5,5)=25, (6,6)=36

Valid:  (1,1), (1,4), (4,1), (2,2), (3,3), (4,4), (5,5), (6,6)

Favourable number of outcomes = 8 pairs

P = 8/36 = 2/9

(iii) Prime products: 2, 3, 5

Pairs: (1,2), (2,1) = 2; (1,3), (3,1) = 3; (1,5), (5,1) = 5

Fourable outcomes = 6 pairs

P = 6/36 = 1/6

Q24: A shopkeeper has 100 shirts in his shop. 45 are white, 30 are blue, and 25 are black. A customer walks in and picks a shirt at random.

(i)What is the probability that the customer picks a white shirt?

(ii) What is the probability of NOT picking a blue shirt?

(iii) What is the probability of picking either a black or white shirt?

Solution: n(S) = 100

(i) P(white) = 45/100 = 9/20

(ii) P(not blue) = 1 − P(blue) = 1 − 30/100 = 70/100 = 7/10

(iii) Black or white = 25 + 45 = 70

P = 70/100 = 7/10

Frequently Tested Previous Year Board Questions

PYQ 1 (Board 2020): A die is thrown twice. Find the probability that:

(i) 5 will come up at least once

(ii) 5 will come up exactly once

PYQ 2 (Board 2019): All the red face cards are removed from a pack of 52 playing cards. A card is drawn at random from the remaining deck. Find the probability of getting:

(i) a face card (ii) a non-face card (iii) a black king

PYQ 3 (Board 2018): Tickets numbered 1 to 20 are mixed up and one ticket is drawn at random. What is the probability that the ticket drawn bears a number which is a multiple of 3 or 5?