Class 11 Maths Chapter 14 – Probability (NCERT Solutions with Formulas & Examples)

Introduction

Probability is the branch of mathematics that deals with measuring uncertainty. It quantifies the chances of occurrence of events in random experiments. In this chapter, we study events, axiomatic approach to probability, equally likely outcomes, and rules for combining probabilities. These concepts are essential for statistics, data science, and real‑life decision making.

Key Formulas

• Probability of an event:

$$P(E)=\frac{\text{Number of favourable outcomes}}{\text{Total possible outcomes}}$$

• Complementary event:

$$P(\text{not A})=1-P(A)$$

• Addition rule:

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

• Mutually exclusive events: If $A\cap B=\mathrm{\varnothing }$, then

$$P(A\cup B)=P(A)+P(B)$$

 Solved NCERT Examples (Step by Step)

Example 1

Rolling a die: A = prime number, B = odd number. Find (i) A ∪ B, (ii) A ∩ B, (iii) A – B, (iv) not A. Solution: S = {1,2,3,4,5,6}, A = {2,3,5}, B = {1,3,5}. (i) A ∪ B = {1,2,3,5} (ii) A ∩ B = {3,5} (iii) A – B = {2} (iv) not A = {1,4,6}

Example 5

One card drawn from 52 cards. Find probability of (i) diamond, (ii) not ace, (iii) black card, (iv) not diamond, (v) not black card. Solution: (i) Diamonds = 13 → P = 13/52 = 1/4 (ii) Aces = 4 → P(not ace) = 1 – 4/52 = 12/13 (iii) Black cards = 26 → P = 26/52 = 1/2 (iv) Not diamond = 1 – 1/4 = 3/4 (v) Not black = 1 – 1/2 = 1/2

Example 6

Bag with 9 discs: 4 red, 3 blue, 2 yellow. Find probability of (i) red, (ii) yellow, (iii) blue, (iv) not blue, (v) red or blue. Solution: (i) Red = 4/9 (ii) Yellow = 2/9 (iii) Blue = 3/9 = 1/3 (iv) Not blue = 1 – 1/3 = 2/3 (v) Red or blue = 4/9 + 1/3 = 7/9

Example 7

Anil qualifies exam with P=0.05, Ashima with P=0.10, both with P=0.02. Find (a) both not qualify, (b) at least one not qualify, (c) only one qualifies. Solution: (a) P(E′ ∩ F′) = 1 – P(E ∪ F) = 1 – (0.05+0.10–0.02) = 0.87 (b) 1 – P(both qualify) = 1 – 0.02 = 0.98 (c) P(only one) = P(E) – P(E∩F) + P(F) – P(E∩F) = 0.05–0.02 + 0.10–0.02 = 0.11

(Continue similarly for all NCERT solved examples from Exercises 14.1, 14.2, and Miscellaneous.)

 15 FAQs with Step‑by‑Step Solutions 

Q1. Are events E: “die shows 4” and F: “die shows even number” mutually exclusive? Answer: No. F includes {2,4,6}. Since 4 ∈ E and 4 ∈ F, they are not mutually exclusive.

Q2. A die is thrown. Find probability of A: number less than 7. Answer: All outcomes {1–6}. P(A) = 6/6 = 1.

Q3. A die is thrown. Find probability of B: number greater than 7. Answer: Impossible event. P(B) = 0.

Q4. A die is thrown. Find probability of C: multiple of 3. Answer: Outcomes {3,6}. P(C) = 2/6 = 1/3.

Q5. A die is thrown. Find probability of D: number less than 4. Answer: Outcomes {1,2,3}. P(D) = 3/6 = 1/2.

Q6. A die is thrown. Find probability of E: even number greater than 4. Answer: Outcomes {6}. P(E) = 1/6.

Q7. A die is thrown. Find probability of F: number not less than 3. Answer: Outcomes {3,4,5,6}. P(F) = 4/6 = 2/3.

Q8. Two dice thrown. Event A: first die even. Probability? Answer: First die outcomes {2,4,6}. P(A) = 3/6 = 1/2.

Q9. Two dice thrown. Event B: first die odd. Probability? Answer: Outcomes {1,3,5}. P(B) = 3/6 = 1/2.

Q10. Two dice thrown. Event C: sum ≤ 5. Probability? Answer: Favourable outcomes = 4+3+2 = 9. Total = 36. P(C) = 9/36 = 1/4.

Q11. Are A and B mutually exclusive? Answer: Yes. First die cannot be both odd and even. P(A∩B)=0.

Q12. Are A and B exhaustive? Answer: Yes. Every outcome is either odd or even. P(A∪B)=1.

Q13. Find probability of drawing a diamond from 52 cards. Answer: 13/52 = 1/4.

Q14. Find probability of drawing not an ace. Answer: 1 – 4/52 = 12/13.

Q15. A coin tossed twice. Probability of at least one tail? Answer: Outcomes {HT, TH, TT}. 3/4.

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