26. Chapter 26. Introduction to Probability NIOS Class 10 Mathematics Textbook Solutions

Chapter 26. Introduction to Probability NIOS Class 10 Mathematics Textbook Solutions for Exam Preparation

Chapter 26. INTRODUCTION TO PROBABILITY

CHECK YOUR PROGRESS 26.1

1. Which of the following is a random experiment?

(i) Suppose you guess the answer to a multiple choice question having four options A, B, C, and D, in which only one is correct.

(ii) The natural numbers 1 to 20 are written on separate slips (one number on one slip) and put in a bag. You draw one slip without looking into the bag.

(iii) You drop a stone from a height

(iv) Each of Hari and John chooses one of the numbers 1, 2, 3, independently.

Solution: The random experiments are:

(i) Guessing the answer to a multiple-choice question .

(ii) Drawing one slip from a bag containing numbers 1 to 20.

(iii) Dropping a stone from a height .

2. What are the possible outcomes of random experiments in Q. 1 above?

Solution: The possible outcomes of the given experiments are:

(i) Guessing the answer to a multiple-choice question:

Possible outcomes = {A, B, C, D}

(ii) Drawing one slip from a bag containing numbers 1 to 20:

Possible outcomes = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}

(iii) Dropping a stone from a height:

Possible outcomes = { Stone falls to the ground , Stone does not fall to the ground }

(iv) Each of Hari and John chooses one of the numbers 1, 2, and 3 independently:

Possible outcomes = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)} .

CHECK YOUR PROGRESS 26.2

1. Find the probability of getting a number 5 in a single throw of a die.

Solution: The number of possible outcomes = 6 [1,2,3,4,5,6]

The number of outcomes favourable to “a number 5”= 1 .

2. A die is tossed once. What is the probability that it shows:

(i) a number 7? (ii) a number less than 5?

Solution: The number of possible outcomes = 6 [1,2,3,4,5,6]

The number of outcomes favourable to “a number 7”= 0

(ii) The number of outcomes favourable to “a number less than 5” = 4 [1,2,3,4]

3. From a pack of 52 cards, a card is drawn at random. What is the probability of this card to be a king?

Solution: The number of possible outcomes = 52

Number of outcomes favourable = 4

4. An integer is chosen between 0 and 20. What is the probability that this chosen integer is a prime number?

Solution: The number of possible outcomes = 19

Number of outcomes favourable = 8 [2,3,5,7,11,13,17,19]

5. A bag contains 3 red and 5 white balls. A ball is drawn from the bag without looking into it. What is the probability of this ball to be of (i) red colour (ii) white colour?

Solution: Total number of ball = 3 + 5 = 8 .

(i) The number of red balls = 3

(ii) The number of white balls = 5

6. 3 males and 4 females appear for an interview, of which one candidate is to be selected. Find the probability of selection of a (i) male candidate (ii) female candidate.

Solution: Total number of candidate = 3 + 4 = 7 .

(i) The number of male candidate = 3

(ii) The number of male candidate = 4

CHECK YOUR PROGRESS 26.3

1. Complete the following statements by filling in blank spaces:

(a) The probability of an event is always greater than or equal to _______ but less than or equal to _______.

(b) The probability of an event that is certain to occur is ________. Such an event is called ________

(d) The sum of probabilities of two complementary events is _________

(e) The sum of probabilities of all the elementary events of an experiment is ______.

Solution: (a) The probability of an event is always greater than or equal to 0 but less than or equal to 1 .

(b) The probability of an event that is certain to occur is 1 . Such an event is called sure or certain event .

(d) The sum of probabilities of two complementary events is 1.

(e) The sum of probabilities of all the elementary events of an experiment is 1 .

2. A die is thrown once. What is the probability of getting

(a) an even number (b) an odd number (c) a prime number

Solution: The number of all possible outcomes = 6 [1,2,3,4,5,6]

(a) The number of outcomes favourable to the Event E “an even number” = 3 [2,4,6]

(b) The number of outcomes favourable to the Event E “an odd number” = 3 [1,3,5]

3. In Question 2 above, verify: P(an even number) + P(an odd number) = 1

Solution: We have,

and

Therefore,

Hence,

4. A die is thrown once. Find the probability of getting

(i) a number less than 4 (ii) a number greater than or equal to 4 (iii) a composite number (iv) a number which is not composite

Solution: The number of all possible outcomes = 6 [1,2,3,4,5,6]

(i) The number of outcomes favourable to the Event E “a number less than 4” = 3 [1,2,3]

(ii) The number of outcomes favourable to the Event E “a number greater than or equal to 4” = 3 [4,5,6]

(iii) The number of outcomes favourable to the Event E “a composite number” = 2 [4,6]

(iii) The number of outcomes favourable to the Event E “a number which is not composite” = 4 [1,2,3,5]

5. If P(E) = 0.88, what is the probability of ‘not E’?

Solution: Here, P(E) = 0.88

We have,

6. If , find P(E).

Solution:

We have,

7. A card is drawn from a well shuffled deck of 52 playing cards. Find the probability that this card will be

(i) a red card (ii) a black card (iii) a red queen (iv) an ace of black colour (v) a jack of spade (vi) a king of club (vii) not a face card (viii) not a jack of diamonds

Solution: The number of all possible outcomes = 52

(i) The number of outcomes favourable to the Event E “a red card” = 26

(ii) The number of outcomes favourable to the Event E “a black card” = 26

(iii) The number of outcomes favourable to the Event E “a red queen” = 2

(iv) The number of outcomes favourable to the Event E “an ace of black colour” = 2

(v) The number of outcomes favourable to the Event E “a jack of spade” = 1

(vi) The number of outcomes favourable to the Event E “ a king of club” = 1

(vii) The number of outcomes favourable to the Event E “not a face card” = 52 – 12 =40

(viii) The number of outcomes favourable to the Event E “ not a jack of diamonds” = 52 – 1 = 51

8. A bag contains 15 white balls and 10 blue balls. A ball is drawn at random from the bag. What is the probability of drawing

(i) a ball of not blue colour (ii) a ball not of white colour

Solution: Total number of balls = 15 + 10 = 25

(i) The number of outcomes favourable to the Event E “a ball of not blue colour” = 25 – 10 = 15 .

(ii) The number of outcomes favourable to the Event E “a ball not of white colour” = 25 – 15 = 10 .

9. In a bag there are 3 red, 4 green and 2 blue marbles. If a marble is picked up at random what is the probability that it is

(i) not green ? (ii) not red ? (iii) not blue ?

Solution: Total number of balls = 3 + 4 + 2 = 9

(i) The number of outcomes favourable to the Event E “not green” = 9 – 4 = 5 .

(ii) The number of outcomes favourable to the Event E “not red” = 9 – 3 = 6 .

(iii) The number of outcomes favourable to the Event E “not blue” = 9 – 2 = 7 .

10. Two different coins are tossed at the same time. Write down all possible outcomes. What is the probability of getting head on one and tail on the other coin?

Solution: All possible outcomes are: HH,HT,TH,TT

The total number of possible outcomes = 4.

The number of outcomes favourable to the Event E “head on one and tail on the other coin” = 4 – 2 = 2 . [HT,TH]

11. In Question 10 above, what is the probability that both the coins show tails?

Solution: The total number of possible outcomes = 4.

The number of outcomes favourable to the Event E “both the coins show tails” = 1 [TT] .

12. Two dice are thrown simultaneously and the sum of the numbers appearing on them is noted. What is the probability that the sum is

(i) 7 (ii) 8 (iii) 9 (iv) 10 (v) 12

Solution: The total number of possible outcomes = 6 × 6 = 36.

The sample space = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

(i) The number of outcomes favourable to the Event E “the sum of 7” = 6 [(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)] .

(ii) The number of outcomes favourable to the Event E “the sum of 8” = 5 [(2,6) ,(3,5),(4,4),(5,3),(6,2)] .

(iii) The number of outcomes favourable to the Event E “the sum of 9” = 4 [(3,6) ,(4,5),(5,4),(6,3)] .

(iv) The number of outcomes favourable to the Event E “the sum of 10” = 3 [(4,6),(5,5),(6,4)] .

(v) The number of outcomes favourable to the Event E “the sum of 12” = 1 [(6,6)] .

13. 8 defective toys are accidentally mixed with 92 good ones in a lot of identical toys. One toy is drawn at random from this lot. What is the probability that this toy is defective?

Solution: Total number of toys = 92 + 8 = 100 .

(v) The number of outcomes favourable to the Event E “defective toys” = 8 .

TERMINAL EXERCISE

1. Which of the following statements are True (T) and which are False (F):

(i) Probability of an event can be 1.01

(ii) If P(E) = 0.08, then P( )= 0.02

(iii) Probability of an impossible event is 1

(iv) For an event E, 0 ≤ P(E) ≤ 1

(v) P( ) = 1 + P(E)

Solution: (i) Probability of an event can be 1.01. → False (F)
[ Probability always lies between 0 and 1. ]

(ii) If P(E) = 0.08, then P( )= 0.02 → False (F)
[ We have, ]

(iii) Probability of an impossible event is 1. → False (F)
[ The probability of an impossible event is 0. ]

(iv) For an event E, . → True (T)
[ We have, ]

(v) P( ) = 1 + P(E) → False (F)
[ We have, ]

2. A card is drawn from a well shuffled deck of 52 cards. What is the probability that this card is a face card of red colour?

Solution: The number of all possible outcomes = 52

(i) The number of outcomes favourable to the Event E “a face card of red colour ” = 6

3. Two coins are tossed at the same time. What is the probability of getting atleast one head?

[Hint: P(atleast one head) = 1 – P(no head)]

Solution: The number of all possible outcomes = 4 [ HH ,HT ,TH ,TT ]

(i) The number of outcomes favourable to the Event E “ no head ” = 1 [ TT ]

And

4. A die is tossed two times and the number appearing on the die is noted each time. What is the probability that the sum of two numbers so obtained is

(i) greater than 12? (ii) less than 12? (iii) greater than 11? (iv) greater than 2?

Solution: The total number of possible outcomes = 6 × 6 = 36.

(i) The number of outcomes favourable to the Event E “greater than 12” = 0

(ii) The number of outcomes favourable to the Event E “less than 12” = 36 – 1 = 35

(iii) The number of outcomes favourable to the Event E “greater than 11” = 1 [(6,6)]

(iv) The number of outcomes favourable to the Event E “greater than 2” = 36 – 1 = 35 [(1,1)]

5. Refer to Question 4 above. What is the probability that the product of two number is 12?

Solution: The total number of possible outcomes = 6 × 6 = 36.

(i) The number of outcomes favourable to the Event E “the product of two number is 12” = 4 [(2,6),(3,4),(4,3) ,(6,2)]

6. Refer to Question 4 above. What is the probability that the difference of two numbers is 2?

Solution: The total number of possible outcomes = 6 × 6 = 36.

(i) The number of outcomes favourable to the Event E “the difference of two numbers is 2” = 8

[(1,3), (2,4), (3,5), (4,6), (3,1), (4,2), (5,3), (6,4) ]

7. A bag contains 15 red balls and some green balls. If the probability of drawing a green ball is 1/6 , find the number of green balls.

Solution: Let x the number of green balls .

Total number of balls

A/Q,

Therefore, the number of green ball is 3 .

8. Which of the following can not be the probability of an event?

(A) 2/3 (B) – 1.01 (C) 12% (D) 0.3

Answer: (B) – 1.01

[ A probability always lies between 0 and 1, i.e., ]

9. In a single throw of two dice, the probability of getting the sum 2 is

(A) (B) (C) (D)

Answer: (C)

[ Total number of possible outcomes = 6 × 6 = 36

The number of favourable outcomes = 1 [(1,1)

]

10. In a simultaneous toss of two coins, the probability of getting one head and one tail is

(A) (B) (C) (D)

Answer: (C)

[ Total number of possible outcomes = 4 [ HH , HT , TH , TT ]

The number of favourable outcomes = 2 [HT,TH]

Therefore, ]

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26. Chapter 26. Introduction to Probability NIOS Class 10 Mathematics Textbook Solutions

Chapter 26. Introduction to Probability NIOS Class 10 Mathematics Textbook Solutions for Exam Preparation

Chapter 26. INTRODUCTION TO PROBABILITY

CHECK YOUR PROGRESS 26.1

CHECK YOUR PROGRESS 26.2

CHECK YOUR PROGRESS 26.3

TERMINAL EXERCISE

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