Probability — Class 10 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Probability" — 9 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Probability" — 9 important questions with detailed answers for CBSE board exam pre…
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
NCERT Solutions for Class 10 Mathematics — Probability. Step-by-step answers to all 9 textbook questions from this chapter, written for CBSE board preparation and free to use.
Probability — All 9 Questions Solved
Q1. A fair die is thrown once. What is the probability of getting a number greater than 4?
Step 1: Total possible outcomes when a die is thrown = {1, 2, 3, 4, 5, 6} = 6
Step 2: Favorable outcomes (number > 4) = {5, 6} = 2
Step 3: Probability = Favorable outcomes / Total outcomes
Step 4: P(number > 4) = 2/6 = 1/3
Final Answer: Probability = 1/3 or approximately 0.333
Q2. Two fair dice are thrown simultaneously. Find the probability of getting a sum of 7.
Step 1: Total possible outcomes when two dice are thrown = 6 × 6 = 36
Step 2: Favorable outcomes (sum = 7): (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
Step 3: Number of favorable outcomes = 6
Step 4: Probability = 6/36 = 1/6
Final Answer: Probability = 1/6 or approximately 0.167
Q3. A bag contains 5 red balls, 3 blue balls, and 2 green balls. A ball is drawn at random. What is the probability of drawing a blue ball?
Step 1: Total number of balls = 5 + 3 + 2 = 10
Step 2: Number of blue balls = 3
Step 3: Probability = Number of blue balls / Total number of balls
Step 4: P(blue) = 3/10
Final Answer: Probability = 3/10 or 0.3
Q4. A card is drawn from a standard deck of 52 playing cards. What is the probability of drawing a face card (Jack, Queen, or King)?
Step 1: Total number of cards = 52
Step 2: Number of face cards in a deck:
Jacks: 4 (one of each suit)
Queens: 4 (one of each suit)
Kings: 4 (one of each suit)
Total face cards = 4 + 4 + 4 = 12
Step 3: Probability = Favorable outcomes / Total outcomes
Step 4: P(face card) = 12/52 = 3/13
Final Answer: Probability = 3/13 or approximately 0.231
Q5. A bag contains 4 red balls and 6 white balls. Two balls are drawn one after the other without replacement. Find the probability that both are red.
Step 1: Total number of balls = 4 + 6 = 10
Step 2: Probability of first ball being red = 4/10
Step 3: After drawing one red ball, remaining balls = 9
Step 4: Remaining red balls = 3
Step 5: Probability of second ball being red = 3/9
Step 6: Since both events are dependent:
P(both red) = P(first red) × P(second red | first red)
P(both red) = (4/10) × (3/9)
Step 7: P(both red) = 12/90 = 2/15
Final Answer: Probability = 2/15 or approximately 0.133
Q6. Three coins are tossed simultaneously. What is the probability of getting at least one tail?
Step 1: Total possible outcomes = 2³ = 8
Step 2: All possible outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
Step 3: Method 1 (Direct counting):
Favorable outcomes (at least one tail): HHT, HTH, HTT, THH, THT, TTH, TTT = 7
P(at least one tail) = 7/8
Step 4: Method 2 (Complement):
P(at least one tail) = 1 - P(no tail) = 1 - P(all heads)
P(all heads) = 1/8
P(at least one tail) = 1 - 1/8 = 7/8
Final Answer: Probability = 7/8 or 0.875
Q7. In a lottery, there are 10 tickets out of which 3 are winning tickets. If you buy 2 tickets, what is the probability that both are winning tickets?
Step 1: Total number of tickets = 10
Step 2: Winning tickets = 3
Step 3: We need to find P(both tickets are winning)
Step 4: Number of ways to choose 2 winning tickets from 3 = C(3,2) = 3
Step 5: Total number of ways to choose 2 tickets from 10 = C(10,2) = 10!/(2!×8!) = 45
Step 6: Probability = Favorable outcomes / Total outcomes
Step 7: P(both winning) = 3/45 = 1/15
Step 8: Alternatively using conditional probability:
P(first winning) = 3/10
P(second winning | first winning) = 2/9
P(both winning) = (3/10) × (2/9) = 6/90 = 1/15
Final Answer: Probability = 1/15 or approximately 0.067
Q8. A number is chosen at random from the numbers 1 to 30. What is the probability that the chosen number is a multiple of 5?
Step 1: Total numbers = 30
Step 2: Multiples of 5 from 1 to 30: 5, 10, 15, 20, 25, 30
Step 3: Count of multiples of 5 = 6
Step 4: Probability = Favorable outcomes / Total outcomes
Step 5: P(multiple of 5) = 6/30 = 1/5
Final Answer: Probability = 1/5 or 0.2
Q9. A box contains 6 balls numbered 1 to 6. Two balls are drawn one after the other with replacement. Find the probability that the sum of the numbers on the balls is 8.
Step 1: Total possible outcomes = 6 × 6 = 36 (with replacement)
Step 2: Favorable outcomes (sum = 8):
(2,6): sum = 8
(3,5): sum = 8
(4,4): sum = 8
(5,3): sum = 8
(6,2): sum = 8
Total favorable outcomes = 5
Step 3: Probability = 5/36
Final Answer: Probability = 5/36 or approximately 0.139
Probability — Practice MCQs with Answers
Attempt these 20 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. What is the probability of an impossible event?
- A. 0
- B. 1
- C. 0.5
- D. 0.25
Answer: A. 0 — An impossible event is an event that cannot occur, so its probability is 0.
MCQ 2. What is the probability of a sure event?
- A. 0
- B. 1
- C. 0.5
- D. 0.25
Answer: B. 1 — A sure event is an event that is certain to occur, so its probability is 1.
MCQ 3. Which of the following cannot be the probability of an event?
- A. 0.7
- B. 2/3
- C. -1.5
- D. 15%
Answer: C. -1.5 — The probability of an event must be a value between 0 and 1, inclusive. It cannot be negative.
MCQ 4. A coin is tossed once. What is the probability of getting a head?
- A. 1/2
- B. 1/3
- C. 1/4
- D. 1
Answer: A. 1/2 — There are two equally likely outcomes (Head or Tail). The favorable outcome is 1 (Head). So, P(Head) = 1/2.
MCQ 5. A die is thrown once. What is the probability of getting an even number?
- A. 1/6
- B. 1/3
- C. 1/2
- D. 2/3
Answer: C. 1/2 — The possible outcomes are {1, 2, 3, 4, 5, 6}. Even numbers are {2, 4, 6}, which are 3 outcomes. So, P(even number) = 3/6 = 1/2.
MCQ 6. A die is thrown once. What is the probability of getting a prime number?
- A. 1/6
- B. 1/3
- C. 1/2
- D. 2/3
Answer: C. 1/2 — The possible outcomes are {1, 2, 3, 4, 5, 6}. Prime numbers are {2, 3, 5}, which are 3 outcomes. So, P(prime number) = 3/6 = 1/2.
MCQ 7. A card is drawn from a well-shuffled deck of 52 playing cards. What is the probability of drawing a red card?
- A. 1/4
- B. 1/2
- C. 1/13
- D. 1/26
Answer: B. 1/2 — There are 26 red cards (13 hearts and 13 diamonds) in a deck of 52 cards. So, P(red card) = 26/52 = 1/2.
MCQ 8. A card is drawn from a well-shuffled deck of 52 playing cards. What is the probability of drawing a King?
- A. 1/13
- B. 1/26
- C. 1/52
- D. 4/13
Answer: A. 1/13 — There are 4 Kings in a deck of 52 cards. So, P(King) = 4/52 = 1/13.
MCQ 9. If P(E) = 0.35, what is the probability of 'not E'?
- A. 0.35
- B. 0.65
- C. 0.5
- D. 0.7
Answer: B. 0.65 — The sum of the probability of an event and the probability of its complement is 1. So, P(not E) = 1 - P(E) = 1 - 0.35 = 0.65.
MCQ 10. An experiment for which the outcome cannot be predicted with certainty is called a/an _________.
- A. event
- B. trial
- C. random experiment
- D. outcome
Answer: C. random experiment — A random experiment is an experiment whose outcome cannot be predicted with certainty, but all possible outcomes are known.
MCQ 11. The sum of the probabilities of all the elementary events of an experiment is:
- A. 0
- B. 1
- C. 0.5
- D. depends on the experiment
Answer: B. 1 — The sum of the probabilities of all elementary events in any experiment is always equal to 1.
MCQ 12. The theoretical probability P(E) of an event E is defined as:
- A. Number of outcomes favorable to E / Total number of outcomes
- B. Total number of outcomes / Number of outcomes favorable to E
- C. Number of trials in which E happened / Total number of trials
- D. Always 0.5
Answer: A. Number of outcomes favorable to E / Total number of outcomes — The theoretical probability of an event is the ratio of the number of outcomes favorable to the event to the total number of equally likely outcomes.
MCQ 13. Two coins are tossed simultaneously. What is the probability of getting exactly one head?
- A. 1/4
- B. 1/2
- C. 3/4
- D. 1
Answer: B. 1/2 — The possible outcomes are {HH, HT, TH, TT}. Favorable outcomes for exactly one head are {HT, TH}, which are 2. Total outcomes are 4. So, P(exactly one head) = 2/4 = 1/2.
MCQ 14. Two dice are thrown simultaneously. What is the probability that the sum of the numbers appearing on the dice is an even number?
- A. 1/4
- B. 1/3
- C. 1/2
- D. 2/3
Answer: C. 1/2 — Total outcomes are 36. An even sum occurs if (even+even) or (odd+odd). This gives 3*3 + 3*3 = 18 favorable outcomes. P(even sum) = 18/36 = 1/2.
MCQ 15. A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is red?
- A. 3/8
- B. 5/8
- C. 3/5
- D. 1/2
Answer: A. 3/8 — Total number of balls = 3 (red) + 5 (black) = 8. Number of red balls = 3. So, P(red ball) = 3/8.
MCQ 16. A bag contains 4 blue, 6 red, and 5 green balls. A ball is drawn at random. What is the probability that the ball drawn is not black?
- A. 1/3
- B. 1/15
- C. 1
- D. 0
Answer: C. 1 — The bag contains only blue, red, and green balls. There are no black balls. Therefore, drawing a ball that is not black is a sure event, and its probability is 1.
MCQ 17. In a non-leap year, what is the probability that a randomly chosen person has their birthday on a Sunday?
- A. 1/7
- B. 1/365
- C. 52/365
- D. 53/365
Answer: A. 1/7 — A non-leap year has 365 days, which means 52 full weeks and 1 odd day. This odd day can be any of the 7 days of the week with equal probability. So, the probability of it being a Sunday is 1/7.
MCQ 18. A card is drawn from a well-shuffled deck of 52 playing cards. What is the probability of drawing a picture card?
- A. 3/13
- B. 1/13
- C. 4/13
- D. 1/4
Answer: A. 3/13 — Picture cards (Jack, Queen, King) are 3 per suit, so 3 * 4 = 12 picture cards in total. P(picture card) = 12/52 = 3/13.
MCQ 19. A card is drawn from a well-shuffled deck of 52 playing cards. What is the probability of drawing a red King?
- A. 1/13
- B. 1/26
- C. 1/52
- D. 2/13
Answer: B. 1/26 — There are two red suits (Hearts and Diamonds), each having one King. So, there are 2 red Kings. P(red King) = 2/52 = 1/26.
MCQ 20. A number is selected at random from the numbers 1 to 20. What is the probability that it is a multiple of 3?
- A. 1/20
- B. 3/20
- C. 3/10
- D. 7/20
Answer: C. 3/10 — Total numbers from 1 to 20 are 20. Multiples of 3 are {3, 6, 9, 12, 15, 18}, which are 6 numbers. P(multiple of 3) = 6/20 = 3/10.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.