Class 12 RD Sharma Solutions - Chapter 31 Probability - Exercise 31.1

Chapter 31 of RD Sharma's Class 12 Mathematics textbook delves into Probability, a crucial concept in statistics. Exercise 31.1 focuses on the fundamental principles of probability, including basic definitions, axioms, and elementary probability calculations. This exercise set introduces students to key concepts such as sample spaces, events, and probability measures, laying the groundwork for more advanced topics in subsequent exercises.

Important Formulas and Theorems in Probability

Probability Formula: P(E) = Number of favorable outcomes / Total number of possible outcomes

Complement Rule: P(A') = 1 - P(A)

Addition Rule for Mutually Exclusive Events: P(A ∪ B) = P(A) + P(B)

Multiplication Rule for Independent Events: P(A ∩ B) = P(A) * P(B)

Permutation Formula: nPr = n! / (n-r)!

Combination Formula: nCr = n! / [r! * (n-r)!]

Class 12 RD Sharma Mathematics Solutions - Exercise 31.1

Question 1: Ten cards numbered 1 through 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than 3, what is the probability that it is an even number?

Solution:

Here,
Sample space (S) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Let, A = the number on the drawn card is even
So, A = {2, 4, 6, 8, 10}
n(A) = 5
and B = the number on the drawn card is more than 3
So, B = {4, 5, 6, 7, 8, 9, 10}
n(B) = 7
Now, A ∩ B = {4, 6, 8, 10}
n(A ∩ B) = 4
Thus, the required probability is -
P(A/B) = n(A ∩ B)/ n(B) = 4/7

Question 2: Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that - 

(i) the youngest is a girl.

(ii) at least one is a girl.

Solution:

Let b and g represent the boy and the girl child respectively. If a family has two children, the sample space will be -
S = {(b, b), (b, g), (g, b), (g, g)}
n(S) = 4
Let A be the event that both children are girls.
So, A = {(g, g)}
n(A) = 1

(i) Let B be the event that the youngest child is a girl.
So, B = {(b, g), (g, g)}
n(B) = 2
Now, A ∩ B = {(g, g)}
n(A ∩ B) = 1
So, P(B) = n(B)/ n(S) = 2/4 = 1/2
and P(A ∩ B) = n(A ∩ B)/ n(S) = 1/4

Now, the conditional probability that both are girls, given that the youngest child is a girl, is -
P(A/B) = P(A ∩ B)/ P(B) = (1/4)/ (1/2) = 1/2
Thus, the required probability is 1/2.

(ii) Let C the event that at least one child is a girl.
So, C = {(b, g), (g, b), (g, g)}
n(C) = 3
Now, A ∩ C = {(g, g)}
n(A ∩ C) = 1
So, P(C) = n(C)/ n(S) = 3/4
and P(A ∩ C) = n(A ∩ C)/ n(S) = 1/4

Now, the conditional probability that both are girls, given that the youngest child is a girl, is -
P(A/C) = P(A ∩ C)/ P(C) = (1/4)/ (3/4) = 1/3
Thus, the required probability is 1/3.

Question 3: Given that the two numbers appearing on throwing two dice are different. Find the probability of the event 'the sum of numbers on the dice is 4'.

Solution:

Let A be the event of having two different numbers on the dice.
So, A = {(1, 2), (1, 3), (1, 4), (1, 5), (1, 6),(2, 1), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 4), (3, 5), (3, 6),
(4, 1), (4, 2), (4, 3), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5)}
n(A) = 30
and B be the event getting a sum of 4 on the dice.
So, B = {(1, 3), (2, 2), (3, 1)}
n(B) = 3
Now, A ∩ B = {(1, 3), (3, 1)}
n(A ∩ B) = 2
Thus, the required conditional probability is -
P(B/A) = n(A ∩ B)/ n(A) = 2/30 = 1/15

Question 4: A coin is tossed three times, if head occurs on first two tosses, find the probability of getting head on third toss.

Solution:

Let A be the event of a head appearing on the first two tosses.
So, A = {HHT, HHH}
n(A) = 2
and B be the event of getting a head on the third toss.
So, B = {HHH, HTH, THH, TTH}
n(B) = 4
Now, A ∩ B = {HHH}
n(A ∩ B) = 1
Thus, the required conditional probability is -
P(B/A) = n(A ∩ B)/ n(A) = 1/2

Question 5: A die is thrown three times, find the probability that 4 appears on the third toss if it is given that 6 and 5 appear respectively on first two tosses.

Solution:

Let A be the event of 4 appearing on the third toss, if a die is thrown three times.
So, A = {(1, 1, 4), (1, 2, 4), (1, 3, 4), (1, 4, 4), (1, 5, 4), (1, 6, 4),
(2, 1, 4), (2, 2, 4), (2, 3, 4), (2, 4, 4), (2, 5, 4), (2, 6, 4)
(3, 1, 4), (3, 2, 4), (3, 3, 4), (3, 4, 4), (3, 5, 4), (3, 6, 4)
(4, 1, 4), (4, 2, 4), (4, 3, 4), (4, 4, 4), (4, 5, 4), (4, 6, 4)
(5, 1, 4), (5, 2, 4), (5, 3, 4), (5, 4, 4), (5, 5, 4), (5, 6, 4)
(6, 1, 4), (6, 2, 4), (6, 3, 4), (6, 4, 4), (6, 5, 4), (6, 6, 4)}
n(A) = 36
and B be the event of 6 and 5 appearing respectively on first two tosses, if the die is tossed three times.
So, B = {(6, 5, 1), (6, 5, 2), (6, 5, 3), (6, 5, 4), (6, 5, 5), (6, 5, 6)}
n(B) = 6
Now, A ∩ B = {(6, 5, 4)}
n(A ∩ B) = 1
Thus, the required probability is -
P(A/B) = n(A ∩ B)/ n(B) = 1/6

Question 6: Compute P(A/B), if P(B) = 0.5 and P(A ∩ B) = 0.32

Solution:

Given, P(B) = 0.5 and P(A ∩ B) = 0.32
We know that, P(A/B) = P(A ∩ B)/ P(B) = 0.32/ 0.5 = 16/25
Thus, P(A/B) = 16/25

Question 7: If P(A) = 0.4, P(B) = 0.3 and P(B/A) = 0.5, Find P(A ∩ B) and P(A/B).

Solution:

Given, P(A) = 0.4, P(B) = 0.3 and P(B/A) = 0.5
We know that, 
P(B/A) = P(A ∩ B)/ P(A)
0.5 = P(A ∩ B)/ 0.4
P(A ∩ B) = 0.5 × 0.4 
Thus, P(A ∩ B) = 0.2

Now, P(A/B) = P(A ∩ B)/ P(B)= 0.2/ 0.3
Thus, P(A/B) = 2/3

Question 8: If A and B are two events such that P(A) = 1/3, P(B) = 1/5 and P(A ∪ B) = 11/30, find P(A/B) and P(B/A).

Solution: 

Given, P(A) = 1/3, P(B) = 1/5 and P(A ∪ B) = 11/30
We know that,
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
11/30 = 1/3 + 1/5 – P(A ∩ B)
P(A ∩ B) = 1/3 + 1/5 - 11/30
P(A ∩ B) = (10 + 6 - 11)/ 30
P(A ∩ B) = 5/30 = 1/6
Now,
P(A/B) = P(A ∩ B)/ P(B)
P(A/B) = (1/6)/ (1/5)
P(A/B)= 5/6

and P(B/A) = P(A ∩ B)/ P(A)
P(B/A) = (1/6)/ (1/3)
P(B/A) = 3/6 = 1/2

Thus, P(A/B) = 5/6 and P(B/A) = 1/2.

Question 9: A couple has two children. Find the probability that both the children are - 

(i) males, if it is known that at least one of the children is male.

(ii) females, if it is known that elder child is a female.

Solution: 

Let m and f represent the male and the female child respectively.

(i) Let A be the event that both are males. 
So, A = {(m, m)}
n(A) = 1
and B be the event that at least one is male. 
So, B = {(m, m), (m, f), (f, m)}
n(B) = 3
Now, A ∩ B = {(m, m)}
n(A ∩ B) = 1
Thus, the required probability is -
P(A/B) = n(A ∩ B) / n(B) = 1/3

(ii) Let C be the event that both are females. 
So, C = {(f, f)}
n(C) = 1
and D be the event that elder child is female. 
So, D = {(f, m), (f, f)}
n(D) = 2
Now, C ∩ D = {(f, f)}
n(C ∩ D) = 1
Thus, the required probability is -
P(C/D) = n(C ∩ D) / n(D) = 1/2

Conclusion

Exercise 31.1 in RD Sharma's Class 12 Probability chapter provides a solid foundation for understanding the basics of probability theory. Through a carefully curated set of problems, students learn to apply probability axioms, calculate simple and compound probabilities, and interpret probabilistic scenarios. This exercise set challenges students to think critically about chance events and develop a mathematical approach to quantifying uncertainty. The problems range from straightforward applications of probability formulas to more complex scenarios involving multiple events and outcomes. By mastering these fundamental concepts, students prepare themselves for more advanced topics in probability and statistics, such as conditional probability, Bayes' theorem, and probability distributions.

Practice Questions

1. A die is rolled once. What is the probability of getting an even number?

2. Two coins are tossed simultaneously. What is the probability of getting at least one head?

3. A card is drawn at random from a well-shuffled deck of 52 cards. What is the probability that it is a face card?

4. In a box, there are 5 red balls, 3 green balls, and 2 blue balls. If a ball is drawn at random, what is the probability that it is not blue?

5. A bag contains 4 white marbles, 5 black marbles, and 6 red marbles. If a marble is drawn at random, what is the probability of drawing a white marble?

6. Three dice are rolled simultaneously. What is the probability of getting a sum of 18?

7. In a class of 50 students, 20 are girls. If a student is selected at random, what is the probability that the selected student is a boy?

8. A letter is chosen at random from the word "PROBABILITY". What is the probability that it is a vowel?

9. Two dice are rolled. What is the probability of getting a sum less than 5?

10. From a standard deck of 52 cards, two cards are drawn without replacement. What is the probability that both are aces?