Probability — Class 11 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 11 Mathematics chapter "Probability" — 8 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 11 Mathematics chapter "Probability" — 8 important questions with detailed answers for CBSE board exam pre…
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NCERT Solutions for Class 11 Mathematics — Probability. Step-by-step answers to all 8 textbook questions from this chapter, written for CBSE board preparation and free to use.
Probability — All 8 Questions Solved
Q1. A die is thrown once. Find the probability of getting (a) an even number, (b) a number greater than 4.
When a die is thrown once, possible outcomes: {1, 2, 3, 4, 5, 6}
Total possible outcomes = 6
(a) Find probability of getting an even number:
Favourable outcomes: {2, 4, 6}
Number of favourable outcomes = 3
P(even) = 3/6 = 1/2
(b) Find probability of getting a number greater than 4:
Favourable outcomes: {5, 6}
Number of favourable outcomes = 2
P(greater than 4) = 2/6 = 1/3
Final Answer: (a) P(even) = 1/2, (b) P(>4) = 1/3
Q2. Two coins are tossed simultaneously. Find the probability of getting (a) both heads, (b) at least one tail.
When two coins are tossed, possible outcomes: {HH, HT, TH, TT}
Total possible outcomes = 4
(a) Find probability of both heads:
Favourable outcomes: {HH}
Number of favourable outcomes = 1
P(both heads) = 1/4
(b) Find probability of at least one tail:
Favourable outcomes: {HT, TH, TT}
Number of favourable outcomes = 3
P(at least one tail) = 3/4
Alternatively:
P(at least one tail) = 1 - P(both heads) = 1 - 1/4 = 3/4
Final Answer: (a) P(both heads) = 1/4, (b) P(at least one tail) = 3/4
Q3. A card is drawn from a standard deck of 52 cards. Find the probability of drawing (a) a king, (b) a red card, (c) a diamond.
A standard deck has 52 cards with 4 kings, 26 red cards, and 13 diamonds.
(a) Probability of drawing a king:
Number of kings = 4
P(king) = 4/52 = 1/13
(b) Probability of drawing a red card:
Number of red cards = 26
P(red) = 26/52 = 1/2
(c) Probability of drawing a diamond:
Number of diamonds = 13
P(diamond) = 13/52 = 1/4
Final Answer: (a) P(king) = 1/13, (b) P(red) = 1/2, (c) P(diamond) = 1/4
Q4. Two dice are thrown simultaneously. Find the probability that the sum of numbers is (a) 7, (b) greater than 10.
When two dice are thrown, total possible outcomes = 6 × 6 = 36
(a) Find probability that sum = 7:
Favourable outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
Number of favourable outcomes = 6
P(sum = 7) = 6/36 = 1/6
(b) Find probability that sum > 10:
For sum > 10, we need sum = 11 or 12
Sum = 11: (5,6), (6,5) → 2 outcomes
Sum = 12: (6,6) → 1 outcome
Number of favourable outcomes = 2 + 1 = 3
P(sum > 10) = 3/36 = 1/12
Final Answer: (a) P(sum = 7) = 1/6, (b) P(sum > 10) = 1/12
Q5. In a group of 100 students, 60 like Mathematics, 50 like Physics, and 30 like both subjects. A student is selected at random. Find the probability that the student likes (a) Mathematics or Physics, (b) neither subject.
Given: Total students = 100
M = students who like Mathematics = 60
P = students who like Physics = 50
M ∩ P = students who like both = 30
Step 1: Use the formula:
M ∪ P = M + P - (M ∩ P)
M ∪ P = 60 + 50 - 30 = 80
(a) Probability of liking Mathematics or Physics:
P(M ∪ P) = 80/100 = 4/5
(b) Probability of liking neither subject:
Students who like neither = Total - (M ∪ P)
= 100 - 80 = 20
P(neither) = 20/100 = 1/5
Alternatively:
P(neither) = 1 - P(M ∪ P) = 1 - 4/5 = 1/5
Final Answer: (a) P(M or P) = 4/5, (b) P(neither) = 1/5
Q6. The probability that a student passes in Mathematics is 3/5 and in Physics is 2/5. Assuming independence, find the probability that the student passes (a) both subjects, (b) at least one subject.
Given: P(M) = 3/5 and P(P) = 2/5 (independent events)
(a) Find probability of passing both subjects:
Since events are independent:
P(M ∩ P) = P(M) × P(P)
P(M ∩ P) = (3/5) × (2/5) = 6/25
(b) Find probability of passing at least one subject:
P(at least one) = P(M ∪ P)
= P(M) + P(P) - P(M ∩ P)
= 3/5 + 2/5 - 6/25
= 15/25 + 10/25 - 6/25
= 19/25
Alternatively:
P(at least one) = 1 - P(fail both)
= 1 - [P(not M) × P(not P)]
= 1 - [(2/5) × (3/5)]
= 1 - 6/25
= 19/25
Final Answer: (a) P(both) = 6/25, (b) P(at least one) = 19/25
Q7. A bag contains 5 red balls, 3 blue balls, and 2 green balls. Two balls are drawn without replacement. Find the probability that both balls are red.
Given: 5 red, 3 blue, 2 green balls
Total balls = 5 + 3 + 2 = 10
Step 1: For first draw:
P(1st ball is red) = 5/10 = 1/2
Step 2: For second draw (without replacement):
After removing one red ball, remaining balls = 9
Remaining red balls = 4
P(2nd ball is red | 1st was red) = 4/9
Step 3: Apply multiplication rule:
P(both red) = P(1st red) × P(2nd red | 1st red)
P(both red) = (5/10) × (4/9)
P(both red) = 20/90
P(both red) = 2/9
Final Answer: The probability that both balls are red is 2/9
Q8. A number is selected at random from the integers 1 to 50. Find the probability that the selected number is (a) divisible by 5, (b) divisible by 5 but not by 10.
Total integers from 1 to 50: 50
(a) Find probability of selecting a number divisible by 5:
Numbers divisible by 5: {5, 10, 15, 20, 25, 30, 35, 40, 45, 50}
Count = 50/5 = 10
P(divisible by 5) = 10/50 = 1/5
(b) Find probability of selecting a number divisible by 5 but not by 10:
Numbers divisible by 5: {5, 10, 15, 20, 25, 30, 35, 40, 45, 50}
Numbers divisible by 10 (and hence by 5): {10, 20, 30, 40, 50}
Count = 5
Numbers divisible by 5 but not by 10:
{5, 15, 25, 35, 45}
Count = 5
P(divisible by 5 but not by 10) = 5/50 = 1/10
Final Answer: (a) P(div by 5) = 1/5, (b) P(div by 5 but not 10) = 1/10
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 getting an even number when a fair die is rolled once?
- A. 1/2
- B. 1/3
- C. 1/6
- D. 2/3
Answer: A. 1/2 — The sample space when rolling a die is {1, 2, 3, 4, 5, 6}. The favorable outcomes (even numbers) are {2, 4, 6}. Therefore, the probability is 3/6 = 1/2.
MCQ 2. A coin is tossed. What is the sample space for this experiment?
- A. {H}
- B. {T}
- C. {H, T}
- D. {H, H}
Answer: C. {H, T} — The sample space is the set of all possible outcomes of an experiment. When a coin is tossed, the possible outcomes are Head (H) or Tail (T).
MCQ 3. If P(E) = 0.45, what is the probability of the event 'not E' (P(E'))?
- A. 0.55
- B. 0.45
- C. 0.65
- D. 0.75
Answer: A. 0.55 — The probability of an event not occurring is 1 minus the probability of the event occurring. So, P(E') = 1 - P(E) = 1 - 0.45 = 0.55.
MCQ 4. Two coins are tossed simultaneously. What is the probability of getting exactly one head?
- A. 1/4
- B. 1/2
- C. 3/4
- D. 1/3
Answer: B. 1/2 — The sample space for tossing two coins is {HH, HT, TH, TT}. The favorable outcomes for exactly one head are {HT, TH}. Thus, the probability is 2/4 = 1/2.
MCQ 5. Which of the following values cannot be the probability of an event?
- A. 0.7
- B. 1/2
- C. 1.5
- D. 0
Answer: C. 1.5 — The probability of any event must be a real number between 0 and 1, inclusive. A value like 1.5 is greater than 1, so it cannot be a probability.
MCQ 6. A bag contains 3 red balls and 5 black balls. A ball is drawn at random. What is the probability that it is a red ball?
- A. 3/5
- B. 5/8
- C. 3/8
- D. 1/3
Answer: C. 3/8 — The total number of balls in the bag is 3 (red) + 5 (black) = 8. There are 3 red balls. So, the probability of drawing a red ball is 3/8.
MCQ 7. If A and B are two mutually exclusive events, and P(A) = 0.3, P(B) = 0.5, then P(A or B) is:
- A. 0.8
- B. 0.2
- C. 0.15
- D. 0.7
Answer: A. 0.8 — For mutually exclusive events, the probability of A or B occurring is the sum of their individual probabilities. P(A U B) = P(A) + P(B) = 0.3 + 0.5 = 0.8.
MCQ 8. From a well-shuffled deck of 52 cards, a card is drawn at random. What is the probability that it is a face card or a red card?
- A. 8/13
- B. 19/26
- C. 7/13
- D. 11/26
Answer: A. 8/13 — There are 12 face cards and 26 red cards. There are 6 red face cards (King, Queen, Jack of Hearts and Diamonds). Using P(A U B) = P(A) + P(B) - P(A ∩ B), we get P(Face Card U Red Card) = (12/52) + (26/52) - (6/52) = 32/52 = 8/13.
MCQ 9. In a group of 30 students, 18 play football and 15 play cricket. If 8 students play both, how many students play neither football nor cricket?
- A. 5
- B. 8
- C. 10
- D. 12
Answer: A. 5 — Students playing football or cricket = P(F U C) = P(F) + P(C) - P(F ∩ C) = 18 + 15 - 8 = 25. Students playing neither = Total students - Students playing (F or C) = 30 - 25 = 5.
MCQ 10. Which of the following describes a sure event?
- A. Getting a 7 when a die is rolled.
- B. Getting a number less than 7 when a die is rolled.
- C. Getting heads when a coin is tossed.
- D. Drawing a red card from a deck of cards.
Answer: B. Getting a number less than 7 when a die is rolled. — A sure event is an event that is certain to occur, meaning its probability is 1. When a die is rolled, the outcome will always be a number less than 7 (1, 2, 3, 4, 5, or 6).
MCQ 11. A letter is chosen at random from the word 'MATHEMATICS'. What is the probability that it is a vowel?
- A. 4/11
- B. 2/11
- C. 5/11
- D. 3/11
Answer: A. 4/11 — The word 'MATHEMATICS' has 11 letters in total. The vowels in the word are A, E, A, I. There are 4 vowels. So, the probability of choosing a vowel is 4/11.
MCQ 12. Three distinct numbers are chosen randomly from the first 20 natural numbers. What is the probability that their sum is odd?
- A. 9/38
- B. 1/2
- C. 19/38
- D. 1/4
Answer: B. 1/2 — Total ways to choose 3 numbers from 20 is C(20,3) = 1140. For the sum to be odd, either all three are odd (O+O+O) or one is odd and two are even (O+E+E). There are 10 odd and 10 even numbers. Ways (3 odd) = C(10,3) = 120. Ways (1 odd, 2 even) = C(10,1) * C(10,2) = 10 * 45 = 450. Total favorable ways = 120 + 450 = 570. Probability = 570/1140 = 1/2.
MCQ 13. If A is an event, which of the following statements correctly describes the range of its probability P(A)?
- A. P(A) < 0
- B. P(A) > 1
- C. 0 <= P(A) <= 1
- D. P(A) = 1.5
Answer: C. 0 <= P(A) <= 1 — The probability of any event must be a non-negative value that is less than or equal to 1. So, P(A) must lie between 0 and 1, inclusive.
MCQ 14. A pair of dice is rolled. What is the probability of getting a sum of 7?
- A. 1/6
- B. 1/12
- C. 5/36
- D. 1/9
Answer: A. 1/6 — The total number of outcomes when rolling two dice is 36 (6x6). The outcomes that sum to 7 are {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}, which are 6 outcomes. The probability is 6/36 = 1/6.
MCQ 15. From a group of 5 men and 4 women, a committee of 3 people is to be formed. What is the probability that the committee consists of exactly 2 men and 1 woman?
- A. 10/21
- B. 5/14
- C. 1/7
- D. 15/21
Answer: A. 10/21 — Total ways to form a committee of 3 from 9 people is C(9,3) = 84. Ways to choose 2 men from 5 is C(5,2) = 10. Ways to choose 1 woman from 4 is C(4,1) = 4. Favorable outcomes = 10 * 4 = 40. Probability = 40/84 = 10/21.
MCQ 16. A card is drawn from a well-shuffled deck of 52 cards. What is the probability of drawing a black king?
- A. 1/13
- B. 1/26
- C. 1/52
- D. 2/13
Answer: B. 1/26 — There are two black kings in a standard deck of 52 cards: the King of Spades and the King of Clubs. So, the probability is 2/52 = 1/26.
MCQ 17. Which of the following events are mutually exclusive when a single die is rolled?
- A. A: getting an even number; B: getting a number greater than 3.
- B. A: getting an odd number; B: getting an even number.
- C. A: getting a number less than 3; B: getting a prime number.
- D. A: getting a multiple of 2; B: getting a multiple of 3.
Answer: B. A: getting an odd number; B: getting an even number. — Mutually exclusive events are events that cannot occur simultaneously. An odd number and an even number cannot be the outcome of a single die roll at the same time.
MCQ 18. If P(A) = 0.5, P(B) = 0.6, and P(A ∩ B) = 0.3, find P(A' ∩ B'), where A' and B' are complements of A and B.
- A. 0.2
- B. 0.4
- C. 0.7
- D. 0.8
Answer: A. 0.2 — Using De Morgan's Law, P(A' ∩ B') = P((A U B)'). First, P(A U B) = P(A) + P(B) - P(A ∩ B) = 0.5 + 0.6 - 0.3 = 0.8. Then, P((A U B)') = 1 - P(A U B) = 1 - 0.8 = 0.2.
MCQ 19. The sum of the probabilities of all elementary events of an experiment is:
- A. 0
- B. 1
- C. 0.5
- D. depends on the experiment
Answer: B. 1 — According to the axioms of probability, the sum of the probabilities of all possible elementary outcomes in a sample space must always be equal to 1.
MCQ 20. A bag contains 4 white, 5 red, and 6 blue balls. One ball is drawn at random. What is the probability that it is not red?
- A. 1/3
- B. 2/3
- C. 4/15
- D. 9/15
Answer: B. 2/3 — Total number of balls = 4 + 5 + 6 = 15. The number of red balls is 5. So, P(red) = 5/15 = 1/3. The probability of not drawing a red ball is 1 - P(red) = 1 - 1/3 = 2/3.
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.