# RD Sharma Solutions for CBSE Class 10 Mathematics chapter 16 - Probability

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## Chapter 16 - Probability Excercise Ex. 16.1

A die is thrown. Find the probability of getting:

(i) a prime number

(ii) 2 or 4

(iii) a multiple of 2 or 3

(iv) an even prime number

(v) a number greater than 5

(iv) a number lying between 2 and 6

A card is drawn at random from a pack of 52 cards. Find the probability that the card drawn is neither a king nor a queen.

If the probability of winning a game is 0.3, what is the probability of loosing it?

A bag contains 5 black, 7 red and 3 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is:

(i) red (ii) black or white (iii) not black

A bag contains 4 red, 5 black and 6 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is:

(i) white (ii) red (iii) not black (iv) red or white

One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting

(i) a king of red suit

(ii) a face card

(iii) a red face card

(iv) a queen of black suit

(v) a jack of hearts

(vi) a spade

Five cards-ten, jack, queen, king, and an ace of diamonds are shuffled face downwards. One card is picked at random.

(i) What is the probability that the card is a queen?

Five cards – ten, jack, queen, king, and an ace of diamonds are shuffled face downwards. One card is picked at random.

If a king is drawn first and put aside, what is the probability that the second card picked up is the

(i)ace? (ii)king?

A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is (i) red? (ii) black?

A coin has only two options-head and tail and both are equally likely events i.e. the probability of occurrence of both is same. Hence, a coin is a fair option to decide which team will choose ends in the game.

Harpreet tosses two different coins simultaneously (say, one is of Re 1 and other of Rs 2). What is the probability that gets at least one head?

Fill in the blanks:

(i) Probability of a sure event is ________.

(ii) Probability of an impossible event is _______.

(iii) The probability of an event (other than sure and impossible event) lies between ______.

(iv) Every elementary event associated to a random experiment has _______ probability.

(v) Probability of an event A + Probability of an event 'not A' = ___________.

(vi) Sum of the probabilities of each outcome in an experiment is ____________.

(i) 1

(ii) 0

(iii) 0 and 1

(iv) equal

(v) 1

(vi) 1

A box contains 100 red cards, 200 yellow cards and 50 blue cards. If a card is drawn at random from the box, then find the probability that it will be (i) a blue card (ii) not a yellow card (iii) neither yellow nor a blue card.

A box contains cards numbered 3,5,7,9,…,35,37. A card is drawn at random from the box. Find the probability that the number on the drawn card is a prime number.

A group consists of 12 persons, of which 3 are extremely patient, 6 are extremely honest and the rest are extremely kind. A person from the group is selected at random. Assuming that each person is equally likely to be selected, find the probability of selecting a person who is (i) extremely patient (ii) extremely kind or honest. Which of the above would you prefer more?

Cards numbered 1 to 30 are put in a bag. A card is drawn at random from this bag. Find the probability that the number on the drawn card is,

(i) Not divisible by 3

(ii) A prime number greater than 7

(iii) Not a perfect square number.

A piggy bank contains hundred 50 paise coins, fifty Rs. 1 coins, twenty Rs. 2 coins and ten Rs. 5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, find the probability that the coin which falls out

(i) is a 50 paise win

(ii) is of value more than Rs. 1

(iii) is of value less than Rs. 5

(iv) is a Rs. 1 or Rs. 2 coin

A bag contains cards numbered from 1 to 49. A card is drawn from the bag at random, after mixing the card thoroughly. Find the probability that the number on the drawn card is

(i) an odd number

(ii) a multiple of 5

(iii) a perfect square

(iv) an even prime number

A box contains 20 cards numbered from 1 to 20. A card is drawn at random from the box. Find the probability that the number on the drawn card is

- divisible 2 or 3
- a prime number

In a simultaneous throw of a pair of dice, find the probability that:

2 will come up at least once

In a simultaneous throw of a pair of dice, find the probability that:

2 will not come either time

What is the probability that an ordinary year has 53 Sundays?

(viii) that the product of numbers appearing on the top of the dice is less than 9.

(ix) that the difference of the numbers appearing on the top of the two dice is 2.

(x) that the numbers obtained have a product less than 16.

(iv) card (v) diamond

A number is selected at random from first 50 natural numbers. Find the probability that it is a multiple of 3 and 4.

A dice is rolled twice. Find the probability that

(i) 5 will not come up either time.

(ii) 5 will come up exactly once.

All the black face cards are removed from a pack of 52 cards. The remaining cards are well shuffled and then a card is drawn at random. Find the probability of getting a

(i) Face card

(ii) Red card

(iii) Black card

(iv) King

Cards numbered from 11 to 60 are kept in a box. If a card is drawn at random from the box, find the probability that the number on the drawn cards is

(i) an odd number

(ii) a perfect square number

(iii) divisible by 5

(iv) a prime number less than 20

All kings and queens are removed from a pack of 52 cards. The remaining cards are well-shuffled and then a card is randomly drawn from it. Find the probability that this card is

(i) a red face card

(ii) a black card

All jacks, queens and kings are removed from a pack of 52 cards. The remaining cards are well-shuffled and then a card is randomly drawn from it. Find the probability that this card is

(i) a black face card

(ii) a red card

Red queens and black jacks are removed from a pack of 52 playing is drawn at random from the remaining cards, after reshuffling them. Find the probability that the card drawn is

(i) a king

(ii) of red colour

(iii) a face card

(iv) a queen

In a bag there are 44 identical cards with figure of circle or square on them. There are 24 circles, of which 9 are blue and rest are green and 20 squares of which 11 are blue and rest are green. One card is drawn from the bag at random. Find the probability that it has the figure

- square
- green colour,
- blue circle and
- green square.

All red face cards are removed from a pack of playing cards. The remaining cards are well shuffled and then a card is drawn at random from them. Find the probability that the drawn card is

- a red card
- a face card and
- a card of clubs.

Two customers are visiting a particular shop in the same week (Monday to Saturday). Each is equally likely to visit the shop on any day as on another day. What is the probability that both will visit the shop on (i) the same day? (ii) different days? (iii) consecutive days?

Three coins are tossed together. Find the probability of getting :Exactly two heads

Sample space when three coins are tossed together is

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

n(S) = 8

Let A be the event ofgetting exactly two heads.

A = {HHT, THH, HHT}

n (A) = 3

P(A) =

Three coins are tossed together. Find the probability of getting :At most two heads

Sample space when three coins are tossed together is

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

n(S) = 8

Let A be the event of getting at most two heads.

A = {TTT, HTT, THT, TTH, HHT, THH, HHT}

n (A) = 7

P(A) =

Three coins are tossed together. Find the probability of getting :At least one head and one tail.

Sample space when three coins are tossed together is

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

n(S) = 8

Let A be the event of getting at least one head and one tail.

A = {HHT, HTH, THH, HTT, THT, TTH }

n (A) = 6

P(A) =

Three coins are tossed together. Find the probability of getting :No tails

Sample space when three coins are tossed together is

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

n(S) = 8

Let A be the event ofgetting no tails.

A = {HHH}

n (A) = 1

P(A) =

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that :

The sum of the two numbers that turn up is 8?

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(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)}

n(S) = 36

Let A be the event that the sum of two numbers which turn up is 8.

A = {(2,6),(3,5),(4,4),(5,3),(6,2)}

n(A) = 5

P(A) =

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that :

Of obtaining a total of 6?

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(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)}

n(S) = 36

Let A be the event that the sum of two numbers which turn up is 6.

A = {(1,5),(2,4),(3,3),(4,2),(5,1)}

n(A) = 5

P(A) =

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that :Of obtaining a total of 10?

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(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)}

n(S) = 36

Let A be the event that the sum of two numbers which turn up is 10.

A = {(4,6),(6,4),(5,5)}

n(A) = 3

P(A) =

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that :Of obtaining the same number on both dice?

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(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)}

n(S) = 36

Let A be the event of obtaining the same number on both dice.

A = {(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}

n(A) = 6

P(A) =

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that : Of obtaining a total more than 9?

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(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)}

n(S) = 36

Let A be the event of obtaining a total more than 9.

A = {(4,6),(5,5),(5,6),(6,4),(6,5),(6,6)}

n(A) = 6

P(A) =

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that :That the sum of the numbers appearing on the top of the dice is 13?

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(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)}

n(S) = 36

Let A be the event that the sum of the numbers appearing on the top of the dice is 13.

The maximum total of the numbers on the dice is 12.

Hence, n(A) = 0

P(A) =

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that :That the sum of the numbers appearing on the top of the dice is less than or equal to 12?

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(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)}

n(S) = 36

Let A be the event that the sum of the numbers appearing on the top of the dice is less than or equal to 12.

The maximum total of the numbers on the dice is 12.

Hence, n(A) = 36

P(A) =

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that :That the product of numbers appearing on the top of dice is less than 9.

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(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)}

n(S) = 36

Let A be the event that the product of numbers appearing on the top of the dice is less than 9.

A = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),

(2,1)(2,2),(2,3),(2,4), (3,1),(3,2),

(4,1),(4,2),(5,1),(6,1)}

n(A) = 16

P(A) =

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that :That the difference of the numbers appearing on the top of the two dice is 2.

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(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)}

n(S) = 36

Let A be the event that the difference of the numbers appearing on the top of the two dice is 2.

A = {(1,3),(2,4),(3,1),(3,5),(4,2),(4,6),(5,3),(6,4)}

}

n(A) = 8

P(A) =

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that : That the numbers obtained have a product less than 16.

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(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)}

n(S) = 36

Let A be the event that the numbers obtained have a product less than 16.

A = {{(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),

(4,1),(4,2),(4,3), (5,1),(5,2),

(5,3),(6,1),(6,2)}}

n(A) = 25

P(A) =

## Chapter 16 - Probability Excercise Ex. 16.2

A target shown in Fig. 16.11 consists of three concentric circles of radii 3, 7 and 9 cm respectively. A dart is thrown and lands on the target. What is the probability that the dart will land on the shaded region?

Assume first circle to be the circle with the smallest radius, that is 3. Similarly, second circle to be the circle with radius 7 and third circle to be the circle with radius 9.

In Fig. 16.12, points A, B, C and D are the centres of four circles that each have a radius of length one unit. If a point is selected at random from the interior of square ABCD. What is the probability that the point will be chosen from the shaded region?

In the fig., JKLM is a square with sides of length 6 units. Points A and B are the mid-points of sides KL and LM respectively. If a point is selected at random from the interior of the square. What is the probability that the point will be chosen from the interior of JAB?

In the fig., a square dart board is shown. The length of a side of the larger square is 1.5 times the length of a side of the smaller square. If a dart is thrown and lands on the larger square. What is the probability that it will land in the interior of the smaller square?

## Chapter 16 - Probability Excercise 16.35

If a digit is chosen at random from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, then the probability that it is odd, is

n(E) = total numbers

= 9

n(0) = odd numbers {1, 3, 5, 7, 9}

= 5

So, the correct option is (b).

In Q. No. 1, the probability that the digit is even, is

n(E) = 9

n(4) = no. is even {2, 4, 6, 8}

= 4

So, the correct option is (a).

## Chapter 16 - Probability Excercise 16.36

In Q. No. 1, the probability that the digit is a multiple of 3 is

n(E) = 9

n(A) = no. is multiple of 3 {3, 6, 9}

= 3

So, the correct option is (a).

If three coins are tossed simultaneously, then the probability of getting at least two heads, is

3 coins are tossed simultaneously.

Hence sample space = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

Event (E) = at least two Heads

= {HHH, HHT, HTH, THH}

n(s) = 8

n(E) = 4

So, the correct option is (c).

In a single throw of a die, the probability of getting a multiple of 3 is

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

n(s) = 6

Event (E) = getting a multiple of 3

= {3, 6}

n(E) = 2

So, the correct option is (b).

A bag contains three green marbles, four blue marbles, and two orange marbles. If a marble is picked at random, then the probability that it is not an orange marble is

A number is selected at random from the numbers 3, 5, 5, 7, 7, 7, 9, 9, 9, 9 The probability that the selected number is their average is

The probability of throwing a number greater than 2 with a fair dice is

Sample space (S) = {1, 2, 3, 4, 5, 6}

n(S) = 6

Event (E) = getting number greater than 2

= {3, 4, 5, 6}

n(E) = 4

So, the correct option is (c).

A card is accidently dropped from a pack of 52 playing cards. The probability that it is an ace is

n(S) = 52

no. of ace in a pack of 52 cards = 4

n(E) = 4

So, the correct option is (b).

A number is selected from numbers 1 to 25. The probability that it is prime is

n(S) = 25

Event (E) = prime numbers between 1 to 25

= {2, 3, 5, 7, 11, 13, 17, 19, 23}

n(E) = 9

Note: The answer does not match the options in the question.

Which of the following cannot be the probability of an event?

We know probability P(E) of an event lies between 0 < P(E) < 1 ......(1)

(a), (c), (d) satisfies the (1) but (b) is a negative number. It can't be the probability of an event.

So, the correct option is (b).

If P(E) = 0.05, then P(not E) =

(a) - 0.05

(b) 0.5

(c) 0.9

(d) 0.95

We know

P(E) + P(not E) = 1

given P(E) = 0.05

so P(not E) = 1 - 0.05

= 0.95

So, the correct option is (d).

Which of the following cannot be the probability of occurrence of an event?

(a) 0.2

(b) 0.4

(c) 0.8

(d) 1.6

We know 0 < P(E) < 1

(a), (b), (c) fullfill the condition. But (d) doesn't

Hence (d) is correct option.

So, the correct option is (d).

The probability of a certain event is

(a) 0

(b) 1

(c) 1/2

(d) no existent

An event that is certain to occur is called Certain event.

Probability of certain event is 1.

Ex: If it is Monday, the probability that tomorrow is Tuesday is certain and therefore probability is 1.

So, the correct option is (b).

## Chapter 16 - Probability Excercise 16.37

The probability of an impossible event is

(a) 0

(b) 1

(c) 1/2

(d) non-existent

Events that are not possible are impossible event.

Probability of impossible event is 0.

So, the correct option is (a).

Aarushi sold 100 lottery tickets in which 5 tickets carry prizes. If Priya purchased a ticket, what is the probability of Priya winning a prize?

A number is selected from first 50 natural numbers. What is the probability that it is a multiple of 3 or 5 ?

## Chapter 16 - Probability Excercise 16.38

A number x is chosen at random from the numbers -3, -2, -1, 0, 1, 2, 3 the probability that |x| < 2 is

Sample space (s) = {-3, -2, -1, 0, 1, 2, 3}

n(s) = 7

Event (E) = |x| < 2

= {-1, 0, 1}

n(E) = 3

So, the correct option is (c).

If a number x is chosen from the numbers 1, 2, 3, and a number y is selected from the number 1, 4, 9. Then, P(xy < 9)

The probability that a non-leap year has 53 Sundays, is

There are 365 days in a non-leap year.

52 complete weeks and 1 spare day.

so This day can be any out of 7 day of week.

Hence n(s) = 7

Now, year already have 52 Sundays. so for a total of 53 Sundays in a calendar year, this spare day must be a Sunday.

Hence n(E) = 1

So, the correct option is (d).

In a single throw of a pair of dice, the probability of getting the sum a perfect sqaure is

We know on throwing a pair of die there are a total of 36 possible outcomes.

n(S) = sum is a perfect square

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

n(E) = 7

So, the correct option is (b).

What is the probability that a non-leap year has 53 Sundays ?

There are 365 days in a non-leap year.

52 complete weeks and 1 spare day.

So this day can be any out of 7 day of a week.

Hence n(s) = 7

Now, a non-leap year already has 52 Sundays. So for a total of 53 Sundays in a calendar year, this spare day must be a Sunday.

Hence n(E) = 1

So, the correct option is (b).

Two dice are rolled simultaneously. The probability that they show different faces is

What is the probability that a leap year has 52 Mondays?

## Chapter 16 - Probability Excercise 16.39

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