Class 12 Maths Chapter 13 Probability – Exercise 13.3 NCERT Solutions

Introduction

Exercise 13.3 focuses on conditional probability. Students learn how to calculate probability of an event given another event has occurred, apply multiplication theorem, and solve practical problems. This exercise is crucial for understanding advanced probability concepts like independence of events and Bayes’ theorem.

Key Concepts

1. Conditional Probability: Probability of event $A$ given $B$ has occurred:

$$P(A\mathrm{\mid }B)=\frac{P(A\cap B)}{P(B)},P(B)\mathrm{\ne }0$$

2. Multiplication Theorem:

$$P(A\cap B)=P(A\mathrm{\mid }B)\cdot P(B)=P(B\mathrm{\mid }A)\cdot P(A)$$

3. Independent Events: If $P(A\mathrm{\mid }B)=P(A)$, then $A$ and $B$ are independent.

Students Frequently Make Mistakes

• Forgetting denominator $P(B)$ in conditional probability.

• Confusing $P(A\mathrm{\mid }B)$ with $P(B\mathrm{\mid }A)$.

• Errors in multiplication theorem application.

• Misinterpreting independence condition.

• Skipping requirement $P(B)\mathrm{\ne }0$.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. A card is drawn from 52 cards. Find probability of getting king given it is red. Red cards = 26, red kings = 2.

$$P(K\mathrm{\mid }R)=\frac{2}{26}=\frac{1}{13}$$

Q2. A die is thrown. Find probability of getting even number given number > 3. Numbers > 3: $\{4,5,6\}$. Even among them: $\{4,6\}$.

$$P(E\mathrm{\mid }>3)=\frac{2}{3}$$

Q3. A coin is tossed twice. Find probability of getting two heads given first toss is head. Sample space with first toss head: $\{HH,HT\}$. Favorable: $\{HH\}$.

$$P(2H\mathrm{\mid }1H)=\frac{1}{2}$$

Q4. A card is drawn. Find probability of ace given card is spade. Spades = 13, ace of spades = 1.

$$P(A\mathrm{\mid }S)=\frac{1}{13}$$

Q5. A die is thrown. Find probability of prime number given number is odd. Odd numbers: $\{1,3,5\}$. Prime among them: $\{3,5\}$.

$$P(P\mathrm{\mid }O)=\frac{2}{3}$$

Q6. A card is drawn. Find probability of face card given card is black. Black cards = 26, black face cards = 6.

$$P(F\mathrm{\mid }B)=\frac{6}{26}=\frac{3}{13}$$

Q7. A die is thrown. Find probability of number divisible by 3 given number > 2. Numbers > 2: $\{3,4,5,6\}$. Divisible by 3: $\{3,6\}$.

$$P(D\mathrm{\mid }>2)=\frac{2}{4}=\frac{1}{2}$$

Q8. A card is drawn. Find probability of queen given card is red. Red cards = 26, red queens = 2.

$$P(Q\mathrm{\mid }R)=\frac{2}{26}=\frac{1}{13}$$

Q9. A die is thrown. Find probability of even number given number $\le 4$. Numbers $\le 4$: $\{1,2,3,4\}$. Even: $\{2,4\}$.

$$P(E\mathrm{\mid }\le 4)=\frac{2}{4}=\frac{1}{2}$$

Q10. A card is drawn. Find probability of king given card is face card. Face cards = 12, kings = 4.

$$P(K\mathrm{\mid }F)=\frac{4}{12}=\frac{1}{3}$$

FAQs (10)

FAQ1. What is conditional probability? Probability of event given another event has occurred.

FAQ2. What is formula for conditional probability? $P(A\mathrm{\mid }B)=\frac{P(A\cap B)}{P(B)}$.

FAQ3. What is multiplication theorem? $P(A\cap B)=P(A\mathrm{\mid }B)\cdot P(B)$.

FAQ4. What are independent events? Events where $P(A\mathrm{\mid }B)=P(A)$.

FAQ5. What is probability of king given red card? $\frac{1}{13}$.

FAQ6. What is probability of even number given number > 3? $\frac{2}{3}$.

FAQ7. What is probability of two heads given first toss head? $\frac{1}{2}$.

FAQ8. What is probability of ace given spade card? $\frac{1}{13}$.

FAQ9. What is probability of queen given red card? $\frac{1}{13}$.

FAQ10. Why is Exercise 13.3 important? It builds mastery of conditional probability and independence of events.

Conclusion

Exercise 13.3 has 10 solved questions and 10 FAQs that strengthen your understanding of conditional probability. This builds the foundation for Bayes’ theorem and advanced probability concepts in Class 12 Maths.

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