NCERT solutions for class 12 maths chapter 13 probability-Exercise: 13.1

Question:1 Given that and are events such

that and find and

Answer:

It is given that and

Question:2 Compute if and

Answer:

It is given that and

Question:3 If and find

(i)

Answer:

It is given that P(A)=0.8, P(B)=0.5 and P(B|A)=0.4

Question:3 If and find

(ii)

Answer:

It is given that and

Question:3 If and find

(iii)

Answer:

It is given that

Question:4 Evaluate if and

Answer:

Given in the question and

We know that:

Use,

Question:5 If and , find

(i)

Answer:

Given in the question

and

By using formula:

Question:5 If and find

(ii)

Answer:

It is given that -

We know that:

Question:5 If and find

(iii)

Answer:

Given in the question-

and

Use formula

Question:6 A coin is tossed three times, where

(i)E : head on third toss ,F : heads on first two tosses

Answer:

The sample space S when a coin is tossed three times is

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

It can be seen that the sample space (S) has 8 elements.

Total number of outcomes

According to question

E: head on third toss, F: heads on first two tosses

Question:6 A coin is tossed three times, where

(ii)E : at least two heads ,F : at most two heads

Answer:

The sample space S when a coin is tossed three times is

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

It can be seen that the sample space (S) has 8 elements.

Total number of outcomes

According to question

E : at least two heads , F : at most two heads

Question:6 A coin is tossed three times, where

(iii)E : at most two tails ,F : at least one tail

Answer:

The sample space S when a coin is tossed three times is

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

It can be seen that the sample space (S) has 8 elements.

Total number of outcomes

According to question

E: at most two tails, F: at least one tail

Question:7 Two coins are tossed once, where

(i) E : tail appears on one coin, F : one coin shows head

Answer:

E : tail appears on one coin, F : one coin shows head

Total outcomes =4

Question:7 Two coins are tossed once, where

(ii)E : no tail appears,F : no head appears

Answer:

E : no tail appears, F : no head appears

Total outcomes =4

Question:8 A die is thrown three times,

E : 4 appears on the third toss, F : 6 and 5 appears respectively on first two tosses

Answer:

E : 4 appears on the third toss, F : 6 and 5 appears respectively on first two tosses

Total outcomes

Question:9 Mother, father and son line up at random for a family picture

E : son on one end, F : father in middle

Answer:

E : son on one end, F : father in middle

Total outcomes

Let S be son, M be mother and F be father.

Then we have,

Question:10 A black and a red dice are rolled.

(a) Find the conditional probability of obtaining a sum greater than , given that the

black die resulted in a

Answer:

A black and a red dice are rolled.

Total outcomes

Let the A be event obtaining a sum greater than and B be a event that the black die

resulted in a

Question:10 A black and a red dice are rolled.

(b) Find the conditional probability of obtaining the sum , given that the red die

resulted in a number less than .

Answer:

A black and a red dice are rolled.

Total outcomes

Let the A be event obtaining a sum 8 and B be a event thatthat the red die resulted in a

number less than .

Red dice is rolled after black dice.

Question:11 A fair die is rolled. Consider

events and Find

(i) and

Answer:

A fair die is rolled.

Total oucomes

and

Question:11 A fair die is rolled. Consider

events and Find

(ii) and

Answer:

A fair die is rolled.

Total oucomes

,

Question:11 A fair die is rolled. Consider

events and Find

(iii) and

Answer:

A fair die is rolled.

Total oucomes

and

,

Question:12 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,

Answer:

Assume that each born child is equally likely to be a boy or a girl.

Let first and second girl are denoted by respectively also first and second

boy are denoted by

If a family has two children, then total

outcomes

Let A= both are girls

and B= the youngest is a girl =

Therefore, the required probability is 1/2

Question:12 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

(ii) at least one is a girl?

Answer:

Assume that each born child is equally likely to be a boy or a girl.

Let first and second girl are denoted by respectively also first and second

boy are denoted by

If a family has two children, then total

outcomes

Let A= both are girls

and C= at least one is a girl =

Question:13 An instructor has a question bank consisting of 300 easy True / False

questions, 200 difficult True / False questions, 500 easy multiple choice questions and

400 difficult multiple choice questions. If a question is selected at random from the

question bank, what is the probability that it will be an easy question given that it is a

multiple choice question?

Answer:

An instructor has a question bank consisting of 300 easy True / False questions, 200

difficult True / False questions, 500 easy multiple choice questions and 400 difficult

multiple choice questions.

Total number of questions

Let A = question be easy.

Let B = multiple choice question

easy multiple questions

Therefore, the required probability is 5/9

Question:14 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’.

Answer:

Two dice are thrown.

Total outcomes

Let A be the event ‘the sum of numbers on the dice is 4.

Let B be the event that two numbers appearing on throwing two dice are different.

Therefore, the required probability is 1/15

Question:15 Consider the experiment of throwing a die, if a multiple of 3 comes up,

throw the die again and if any other number comes, toss a coin. Find the conditional

probability of the event ‘the coin shows a tail’, given that ‘at least one die shows a 3’.

Answer:

Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die

again and if any other number comes, toss a coin.

Total outcomes

Total number of outcomes =20

Let A be a event when coin shows a tail.

Let B be a event that ‘at least one die shows a 3’.

Question:16 In the following Exercise 16 choose the correct answer:

If then is

(A)

(B)

(C)

(D)

Answer:

It is given that

Hence, is not defined .

Thus, correct option is C.

Question:17 In the following Exercise 17 choose the correct answer:

If and are events such that then

(A) but

(B)

(C)

(D)

Answer:

It is given that

Hence, option D is correct.

NCERT solutions for class 12 maths chapter 13 probability-Exercise: 13.2

Question:1 If and find if and are independent

events.

Answer:

and

Given : and are independent events.

So we have,

Question: 2 Two cards are drawn at random and without replacement from a pack of

52 playing cards. Find the probability that both the cards are black.

Answer:

Two cards are drawn at random and without replacement from a pack of 52 playing

cards.

There are 26 black cards in a pack of 52.

Let be the probability that first cards is black.

Then, we have

Let be the probability that second cards is black.

Then, we have

The probability that both the cards are black

Question:3 A box of oranges is inspected by examining three randomly selected

oranges drawn without replacement. If all the three oranges are good, the box is

approved for sale, otherwise, it is rejected. Find the probability that a box

containing oranges out of which are good and are bad ones will be approved for

sale.

Answer:

Total oranges = 15

Good oranges = 12

Bad oranges = 3

Let be the probability that first orange is good.

The, we have

Let be the probability that second orange is good.

Let be the probability that third orange is good.

The probability that a box will be approved for sale

Question:4 A fair coin and an unbiased die are tossed. Let A be the event ‘head

appears on the coin’ and B be the event ‘3 on the die’. Check whether A and B are

independent events or not.

Answer:

A fair coin and an unbiased die are tossed,then total outputs are:

A is the event ‘head appears on the coin’ .

Total outcomes of A are :

B is the event ‘3 on the die’.

Total outcomes of B are :

Also,

Hence, A and B are independent events.

Question:5 A die marked in red and in green is tossed. Let be the event,

‘the number is even,’ and be the event, ‘the number is red’.

Are and independent?

Answer:

Total outcomes .

is the event, ‘the number is even,’

Outcomes of A

is the event, ‘the number is red’.

Outcomes of B

Also,

Thus, both the events A and B are not independent.

Question:6 Let and be events

with and Are E and F independent?

Answer:

Given :

and

For events E and F to be independent , we need

Hence, E and F are not indepent events.

Question:7 Given that the events and are such

that and Find if they are

(i) mutually exclusive

Answer:

Given,

Also, A and B are mutually exclusive means .

Question:7 Given that the events and are such

that and Find p if they are

(ii) independent

Answer:

Given,

Also, A and B are independent events means

. Also

Question:8 Let A and B be independent events with and Find

(i)

Answer:

and

Given : A and B be independent events

So, we have

Question:8 Let and be independent events with and Find

(ii)

Answer:

and

Given : A and B be independent events

So, we have

We have,

Question:8 Let and be independent events with and Find

(iii)

Answer:

and

Given : A and B be independent events

So, we have

Question:8 Let A and B be independent events

with and Find

(iv)

Answer:

and

Given : A and B be independent events

So, we have

Question:9 If and are two events such

that and find

Answer:

If and are two events such that and

use,

Question:10 Events A and B are such

that and State whether and are

independent ?

Answer:

If and are two events such

that and

As we can see

Hence, A and B are not independent.

Question:11 Given two independent events and such

that Find

(i)

Answer:

Given two independent events and .

Also , we know

Question:11 Given two independent events A and B such

that Find

(ii)

Answer:

Given two independent events and .

Question:11 Given two independent events A and B such

that Find

(iii)

Answer:

Question:11 Given two independent events and such

that Find

(iv)

Answer:

Question:12 A die is tossed thrice. Find the probability of getting an odd number at

least once.

Answer:

A die is tossed thrice.

Outcomes

Odd numbers

The probability of getting an odd number at first throw

The probability of getting an even number

Probability of getting even number three times

The probability of getting an odd number at least once = 1 - the probability of getting an

odd number in none of throw

= 1 - probability of getting even number three times

Question:13 Two balls are drawn at random with replacement from a box containing 10

black and 8 red balls. Find the probability that

(i) both balls are red.

Answer:

Two balls are drawn at random with replacement from a box containing 10 black and 8

red balls.

Total balls =18

Black balls = 10

Red balls = 8

The probability of getting a red ball in first draw

The ball is repleced after drawing first ball.

The probability of getting a red ball in second draw

the probability that both balls are red

Question:13 Two balls are drawn at random with replacement from a box

containing black and red balls. Find the probability that

(ii) first ball is black and second is red.

Answer:

Two balls are drawn at random with replacement from a box containing 10 black and 8

red balls.

Total balls =18

Black balls = 10

Red balls = 8

The probability of getting a black ball in the first draw

The ball is replaced after drawing the first ball.

The probability of getting a red ball in the second draw

the probability that the first ball is black and the second is red

Question:13 Two balls are drawn at random with replacement from a box

containing black and red balls. Find the probability that

(iii) one of them is black and other is red.

Answer:

Two balls are drawn at random with replacement from a box containing 10 black and 8

red balls.

Total balls =18

Black balls = 10

Red balls = 8

Let the first ball is black and the second ball is red.

The probability of getting a black ball in the first draw

The ball is replaced after drawing the first ball.

The probability of getting a red ball in the second draw

the probability that the first ball is black and the second is red

Let the first ball is red and the second ball is black.

The probability of getting a red ball in the first draw

The probability of getting a black ball in the second draw

the probability that the first ball is red and the second is black

Thus,

The probability that one of them is black and the other is red = the probability that the

first ball is black and the second is red + the probability that the first ball is red and the

second is black

Question:14 Probability of solving specific problem independently by A and B

are and respectively. If both try to solve the problem independently, find the

probability that

(i) the problem is solved

Answer:

and

Since, problem is solved independently by A and B,

probability that the problem is solved

Question:14 Probability of solving specific problem independently by A and B

are and respectively. If both try to solve the problem independently, find the

probability that

(ii) exactly one of them solves the problem

Answer:

and

,

,

probability that exactly one of them solves the problem

probability that exactly one of them solves the problem

Question:15 One card is drawn at random from a well shuffled deck of cards. In

which of the following cases are the events and independent ?

(i) E : ‘the card drawn is a spade’

F : ‘the card drawn is an ace’

Answer:

One card is drawn at random from a well shuffled deck of cards

Total ace = 4

total spades =13

E : ‘the card drawn is a spade

F : ‘the card drawn is an ace’

a card which is spade and ace = 1

Hence, E and F are indepentdent events .

Question:15 One card is drawn at random from a well shuffled deck of 52 cards. In

which of the following cases are the events E and F independent ?

(ii) E : ‘the card drawn is

F : ‘the card drawn is a king’

Answer:

One card is drawn at random from a well shuffled deck of cards

Total black card = 26

total king =4

E : ‘the card drawn is black’

F : ‘the card drawn is a king’

a card which is black and king = 2

Hence, E and F are indepentdent events .

Question:15 One card is drawn at random from a well shuffled deck of 52 cards. In

which of the following cases are the events E and F independent ?

(iii) E : ‘the card drawn is a king or queen’

F : ‘the card drawn is a queen or jack’.

Answer:

One card is drawn at random from a well shuffled deck of cards

Total king or queen = 8

total queen or jack = 8

E : ‘the card drawn is a king or queen’

F : ‘the card drawn is a queen or jack’.

a card which is queen = 4

Hence, E and F are not indepentdent events

Question:16 In a hostel, of the students read Hindi newspaper, read

English newspaper and read both Hindi and English newspapers. A student is

selected at random.

(a) Find the probability that she reads neither Hindi nor English newspapers

Answer:

H : of the students read Hindi newspaper,

E : read English newspaper and

read both Hindi and English newspapers.

the probability that she reads neither Hindi nor English newspapers

Question:16 In a hostel, of the students read Hindi newspaper, read

English newspaper and read both Hindi and English newspapers. A student is

selected at random.

(b) If she reads Hindi newspaper, find the probability that she reads English newspaper.

Answer:

H : of the students read Hindi newspaper,

E : read English newspaper and

read both Hindi and English newspapers.

The probability that she reads English newspape if she reads Hindi

newspaper

Question:16 In a hostel, of the students read Hindi newspaper, read

English newspaper and read both Hindi and English newspapers. A student is

selected at random.

(c) If she reads English newspaper, find the probability that she reads Hindi newspaper.

Answer:

H : of the students read Hindi newspaper,

E : read English newspaper and

read both Hindi and English newspapers.

the probability that she reads Hindi newspaper if she reads English

newspaper

Question:17 The probability of obtaining an even prime number on each die, when a

pair of dice is rolled is

(A)

(B)

(C)

(D)

Answer:

when a pair of dice is rolled, total outcomes

Even prime number

The probability of obtaining an even prime number on each die

Option D is correct.

Question:18 Two events A and B will be independent, if

(A) and are mutually exclusive

(B)

(C)

(D)

Answer:

Two events A and B will be independent, if

Or

Option B is correct.

NCERT solutions for class 12 maths chapter 13 probability-Exercise: 13.3

Question:1 An urn contains red and black balls. A ball is drawn at random, its

colour is noted and is returned to the urn. Moreover, additional balls of the colour

drawn are put in the urn and then a ball is drawn at random. What is the probability that

the second ball is red?

Answer:

Black balls = 5

Red balls = 5

Total balls = 10

CASE 1 Let red ball be drawn in first attempt.

Now two red balls are added in urn .

Now red balls = 7, black balls = 5

Total balls = 12

CASE 2

Let black ball be drawn in first attempt.

Now two black balls are added in urn .

Now red balls = 5, black balls = 7

Total balls = 12

the probability that the second ball is red =

Question:2 A bag contains red and black balls, another bag contains red

and black balls. One of the two bags is selected at random and a ball is drawn frome

bag which is found to be red. Find the probability that the ball is drawn from the first

bag.

Answer:

BAG 1 : Red balls =4 Black balls=4 Total balls = 8

BAG 2 : Red balls = 2 Black balls = 6 Total balls = 8

B1 : selecting bag 1

B2 : selecting bag 2

Let R be a event of getting red ball

probability that the ball is drawn from the first bag,

given that it is red is .

Using Baye's theorem, we have

Question:3 Of the students in a college, it is known that reside in hostel

and are day scholars (not residing in hostel). Previous year results report

that of all students who reside in hostel attain grade and of day scholars

attain grade in their annual examination. At the end of the year, one student is chosen

at random from the college and he has an grade, what is the probability that the

student is a hostlier?

Answer:

H : reside in hostel

D : day scholars

A : students who attain grade A

By Bayes theorem :

Question:4 In answering a question on a multiple choice test, a student either knows

the answer or guesses. Let be the probability that he knows the answer and be the

probability that he guesses. Assuming that a student who guesses at the answer will be

correct with probability . What is the probability that the student knows the answer

given that he answered it correctly?

Answer:

A : Student knows answer.

B : Student guess the answer

C : Answer is correct

By Bayes theorem :

Question:5 A laboratory blood test is effective in detecting a certain disease

when it is in fact, present. However, the test also yields a false positive result

for of the healthy person tested (i.e. if a healthy person is tested, then, with

probability the test will imply he has the disease). If of the population

actually has the disease, what is the probability that a person has the disease given that

his test result is positive ?

Answer:

A : Person selected is having the disease

B : Person selected is not having the disease.

C :Blood result is positive.

By Bayes theorem :

Question:6 There are three coins. One is a two headed coin (having head on both

faces), another is a biased coin that comes up heads of the time and third is an

unbiased coin. One of the three coins is chosen at random and tossed, it shows heads,

what is the probability that it was the two headed coin?

Answer:

Given : A : chossing a two headed coin

B : chossing a biased coin

C : chossing a unbiased coin

D : event that coin tossed show head.

Biased coin that comes up heads of the time.

Question:7 An insurance company insured 2000 scooter drivers, 4000 car drivers and

6000 truck drivers. The probability of an accidents are , and respectively.

One of the insured persons meets with an accident. What is the probability that he is a

scooter driver?

Answer:

Let A : scooter drivers = 2000

B : car drivers = 4000

C : truck drivers = 6000

Total drivers = 12000

D : the event that person meets with an accident.

Question:8 A factory has two machines and Past record shows that

machine produced of the items of output and machine B produced of the

items. Further, of the items produced by machine and produced by

machine were defective. All the items are put into one stockpile and then one item is

chosen at random from this and is found to be defective. What is the probability that it

was produced by machine ?

Answer:

A : Items produced by machine A

B : Items produced by machine B

X : Produced item found to be defective.

Hence, the probability that defective item was produced by machine =

.

Question:9 Two groups are competing for the position on the Board of directors of a

corporation. The probabilities that the first and the second groups will win

are and respectively. Further, if the first group wins, the probability of introducing

a new product is and the corresponding probability is if the second group wins.

Find the probability that the new product introduced was by the second group.

Answer:

A: the first groups will win

B: the second groups will win

X: Event of introducing a new product.

Probability of introducing a new product if the first group wins :

Probability of introducing a new product if the second group wins :

Hence, the probability that the new product introduced was by the second group :

Question:10 Suppose a girl throws a die. If she gets a or , she tosses a coin three

times and notes the number of heads. If she gets or , she tosses a coin once

and notes whether a head or tail is obtained. If she obtained exactly one head, what is

the probability that she threw or with the die?

Answer:

Let, A: Outcome on die is 5 or 6.

B: Outcome on die is 1,2,3,4

X: Event of getting exactly one head.

Probability of getting exactly one head when she tosses a coin three times

:

Probability of getting exactly one head when she tosses a coin one time :

Hence, the probability that she threw or with the die =

Question:11 A manufacturer has three machine operators and The first

operator produces defective items, where as the other two operators B and C

produce and defective items respectively. is on the job for of the

time, is on the job for of the time and is on the job for of the time. A

defective item is produced, what is the probability that it was produced by ?

Answer:

Let A: time consumed by machine A

B: time consumed by machine B

C: time consumed by machine C

Total drivers = 12000

D: Event of producing defective items

Hence, the probability that defective item was produced by =

Question:12 A card from a pack of cards is lost. From the remaining cards of the

pack, two cards are drawn and are found to be both diamonds. Find the probability of

the lost card being a diamond.

Answer:

Let A : Event of choosing a diamond card.

B : Event of not choosing a diamond card.

X : The lost card.

If lost card is diamond then 12 diamond cards are left out of 51 cards.

Two diamond cards are drawn out of 12 diamond cards in ways.

Similarly, two cards are drawn out of 51 cards in ways.

Probablity of getting two diamond cards when one diamond is lost :

If lost card is not diamond then 13 diamond cards are left out of 51 cards.

Two diamond cards are drawn out of 13 diamond cards in ways.

Similarly, two cards are drawn out of 51 cards in ways.

Probablity of getting two diamond cards when one diamond is not lost

:

The probability of the lost card being a diamond :

Hence, the probability of the lost card being a diamond :

Question:13 Probability that A speaks truth is . A coin is tossed. A reports that a head

appears. The probability that actually there was head is

(A)

(B)

C)

(D)

Answer:

Let A : A speaks truth

B : A speaks false

X : Event that head appears.

A coin is tossed , outcomes are head or tail.

Probability of getting head whether A speaks thruth or not is

The probability that actually there was head is

Hence, option A is correct.

Question:14 If and are two events such that and then which of

the following is correct?

(A)

(B)

(C)

(D) None of these

Answer:

If and then

Also,

We know that

Hence, we can see option C is correct.

NCERT solutions for class 12 maths chapter 13 probability-Exercise: 13.4

Question:1(i) State which the following are not the probability distributions of a random

variable. Give reasons for your answer.

Answer:

As we know the sum of probabilities of a probability distribution is 1.

Sum of probabilities

The given table is the probability distributions of a random variable.

Question:1(ii) State which of the following are not the probability distributions of a

random variable. Give reasons for your answer.

Answer:

As we know probabilities cannot be negative for a probability distribution .

The given table is not a the probability distributions of a random variable.

Question:1(iii) State which of the following are not the probability distributions of a

random variable. Give reasons for your answer.

Answer:

As we know sum of probabilities of a probability distribution is 1.

Sum of probablities

The given table is not a the probability distributions of a random variable because

sum of probabilities is not 1.

Question:1(iv) State which of the following are not the probability distributions of a

random variable. Give reasons for your answer.

Answer:

As we know sum of probabilities of a probability distribution is 1.

Sum of probablities

The given table is not a the probability distributions of a random variable because

sum of probabilities is not 1.

Question:2 An urn contains red and black balls. Two balls are randomly drawn.

Let represent the number of black balls. What are the possible values of Is a

random variable ?

Answer:

B = black balls

R = red balls

The two balls can be selected as BR,BB,RB,RR.

X = number of black balls.

Hence, possible values of X can be 0, 1 and 2.

Yes, X is a random variable.

Question:3 Let represent the difference between the number of heads and the

number of tails obtained when a coin is tossed times. What are possibl valuess of ?

Answer:

The difference between the number of heads and the number of tails obtained when a

coin is tossed times are :

Thus, possible values of X are 0, 2, 4 and 6.

Question:4(i) Find the probability distribution of

number of heads in two tosses of a coin.

Answer:

When coin is tossed twice then sample space

Let X be number of heads.

X can take values of 0,1,2.

Table is as shown :

X 0 1 2

P(X)

Question:4(ii) Find the probability distribution of

number of tails in the simultaneous tosses of three coins.

Answer:

When 3 coins are simultaneous tossed then sample

space

Let X be number of tails.

X can be 0,1,2,3

X can take values of 0,1,2.

Table is as shown :

X 0 1 2 3

P(X)

Question:4(iii) Find the probability distribution of

number of heads in four tosses of a coin.

Answer:

When coin is tossed 4 times then sample

space

Let X be number of heads.

X can be 0,1,2,3,4

Table is as shown :

X 0 1 2 3 4

P(X)

Question:5(i) Find the probability distribution of the number of successes in two tosses

of a die, where a success is defined as

number greater than 4

Answer:

When a die is tossed twice , total outcomes = 36

Number less than or equal to 4 in both toss :

Number less than or equal to 4 in first toss and number more than or equal to 4 in

second toss + Number less than or equal to 4 in second toss and number more than or

equal to 4 in first toss:

Number less than 4 in both tosses :

Probability distribution is as :

X 0 1 2

P(X)

Question:5(ii) Find the probability distribution of the number of successes in two tosses

of a die, where a success is defined as

six appears on at least one die .

Answer:

When a die is tossed twice , total outcomes = 36

Six does not appear on any of the die :

Six appear on atleast one die :

Probability distribution is as :

X 0 1

P(X)

Question:6 From a lot of bulbs which include defectives, a sample of bulbs is

drawn at random with replacement. Find the probability distribution of the number of

defective bulbs.

Answer:

Total bulbs = 30

defective bulbs = 6

Non defective bulbs

bulbs is drawn at random with replacement.

Let X : number of defective bulbs

4 Non defective bulbs and 0 defective bulbs :

3 Non defective bulbs and 1 defective bulbs :

2 Non defective bulbs and 2 defective bulbs :

1 Non defective bulbs and 3 defective bulbs :

0 Non defective bulbs and 4 defective bulbs :

the probability distribution of the number of defective bulbs is as :

X 0 1 2 3 4

P(X)

Question:7 A coin is biased so that the head is times as likely to occur as tail. If the

coin is tossed twice, find the probability distribution of number of tails.

Answer:

the coin is tossed twice, total outcomes =4

probability of getting a tail be x.

i.e.

Then

and

Let X : number of tails

No tail :

1 tail :

2 tail :

the probability distribution of number of tails are

X 0 1 2

P(X)

Question:8(i) A random variable X has the following probability distribution:

Answer:

Sum of probabilities of probability distribution of random variable is 1.

Question:8(ii) A random variable has the following probability distribution:

Answer:

Question:8(iii) A random variable has the following probability distribution:

Answer:

Question:8(iv) A random variable X has the following probability distribution:

Answer:

Question:9(a) The random variable X has a probability distribution P(X) of the following

form, where k is some number :

Determine the value of

Answer:

Sum of probabilities of probability distribution of random variable is 1.

Question:9(b) The random variable has a probability distribution of the

following form, where k is some number :

Find

Answer:

Question:10 Find the mean number of heads in three tosses of a fair coin.

Answer:

Let X be the success of getting head.

When 3 coins are tossed then sample

space

X can be 0,1,2,3

The probability distribution is as

X 0 1 2 3

P(X)

mean number of heads :

Question:11 Two dice are thrown simultaneously. If denotes the number of sixes,

find the expectation of .

Answer:

denotes the number of sixes, when two dice are thrown simultaneously.

X can be 0,1,2.

Not getting six on dice

Getting six on one time when thrown twice :

Getting six on both dice :

X 0 1 2

P(X)

Expectation of X = E(X)

Question:12 Two numbers are selected at random (without replacement) from the first

six positive integers. Let denote the larger of the two numbers obtained. Find

Answer:

Two numbers are selected at random (without replacement) from the first six positive

integers in ways.

denote the larger of the two numbers obtained.

X can be 2,3,4,5,6.

X=2, obsevations :

X=3, obsevations :

X=4, obsevations :

X=5, obsevations :

X=6, obsevations :

Probability distribution is as follows:

X 2 3 4 5 6

P(X)

Question:13 Let denote the sum of the numbers obtained when two fair dice are

rolled. Find the variance and standard deviation of .

Answer:

denote the sum of the numbers obtained when two fair dice are rolled.

Total observations = 36

X can be 2,3,4,5,6,7,8,9,10,11,12

Probability distribution is as follows :

X 2 3 4 5 6 7 8 9 10 11 12

P(X)

Standard deviation =

Question:14 A class has students whose ages

are and years. One student is selected

in such a manner that each has the same chance of being chosen and the age X of the

selected student is recorded. What is the probability distribution of the random

variable Find mean, variance and standard deviation of .

Answer:

Total students = 15

probability of selecting a student :

The information given can be represented as frequency table :

X 14 15 16 17 18 19 20 21

f 2 1 2 3 1 2 3 1

Probability distribution is as :

X 14 15 16 17 18 19 20 21

P(X)

Question:15 In a meeting, of the members favour and oppose a certain

proposal. A member is selected at random and we take . if he opposed,

and if he is in favour. Find and Var .

Answer:

Given :

Probability distribution is as :

X 0 1

P(X) 0.3 0.7

Question:16 The mean of the numbers obtained on throwing a die having written 1 on

three faces, on two faces and on one face is, Choose the correct answer in the

following:

(A)

(B)

(C)

(D)

Answer:

X is number representing on die.

Total observations = 6

X 1 2 5

P(X)

Option B is correct.

Question:17 Suppose that two cards are drawn at random from a deck of cards.

Let be the number of aces obtained. Then the value of is Choose the correct

answer in the following:

(A)

(B)

(C)

(D)

Answer:

X be number od aces obtained.

X can be 0,1,2

There 52 cards and 4 aces, 48 are non-ace cards.

The probability distribution is as :

X 0 1 2

P(X)

Option D is correct.

NCERT solutions for class 12 maths chapter 13 probability-Exercise: 13.5

Question:1(i) A die is thrown 6 times. If ‘getting an odd number’ is a success, what is

the probability of

5 successes?

Answer:

X be the number of success of getting an odd number.

X has a binomial distribution.

Question:1(ii) A die is thrown 6 times. If ‘getting an odd number’ is a success, what is

the probability of

at least 5 successes?

Answer:

X be a number of success of getting an odd number.

X has a binomial distribution.

Question:1(iii) A die is thrown 6 times. If ‘getting an odd number’ is a success, what is

the probability of

at most 5 successes?

Answer:

X be a number of success of getting an odd number.

X has a binomial distribution.

Question:2 A pair of dice is thrown times. If getting a doublet is considered a

success, find the probability of two successes

Answer:

A pair of dice is thrown times.X be getting a doublet.

Probability of getting doublet in a throw of pair of dice :

X has a binomial distribution,n=4

Put x = 2

Question:3 There are defective items in a large bulk of items. What is the

probability that a sample of items will include not more than one defective item?

Answer:

There are defective items in a large bulk of items.

X denotes the number of defective items in a sample of 10.

X has a binomial distribution, n=10.

Question:4(i) Five cards are drawn successively with replacement from a well-shuffled

deck of cards. What is the probability that

all the five cards are spades?

Answer:

Let X represent a number of spade cards among five drawn cards. Five cards are drawn

successively with replacement from a well-shuffled deck of cards.

We have 13 spades.

X has a binomial distribution,n=5.

Put X=5 ,

Question:4(ii) Five cards are drawn successively with replacement from a well-shuffled

deck of 52 cards. What is the probability that

only 3 cards are spades?

Answer:

Let X represent a number of spade cards among five drawn cards. Five cards are drawn

successively with replacement from a well-shuffled deck of cards.

We have 13 spades.

X has a binomial distribution,n=5.

Put X=3 ,

Question:4(iii) Five cards are drawn successively with replacement from a well-

shuffled deck of 52 cards. What is the probability that

none is a spade?

Answer:

Let X represent number of spade cards among five drawn cards. Five cards are drawn

successively with replacement from a well-shuffled deck of cards.

We have 13 spades .

X has a binomial distribution,n=5.

Put X=0 ,

Question:5(i) The probability that a bulb produced by a factory will fuse after days

of use is . Find the probability that out of such bulbs

none will fuse after days of use.

Answer:

Let X represent number of bulb that will fuse after days of use .Trials =5

X has a binomial distribution,n=5.

Put X=0 ,

Question:5(ii) The probability that a bulb produced by a factory will fuse after days

of use is Find the probability that out of such bulbs

not more than one will fuse after days of use.

Answer:

Let X represent a number of the bulb that will fuse after days of use. Trials =5

X has a binomial distribution,n=5.

Put ,

Question:5(iii) The probability that a bulb produced by a factory will fuse after days

of use is Find the probability that out of such bulbs

more than one will fuse after days of use.

Answer:

Let X represent number of bulb that will fuse after days of use .Trials =5

X has a binomial distribution,n=5.

Put ,

Question:5(iv) The probability that a bulb produced by a factory will fuse after days

of use is . Find the probability that out of such bulbs

at least one will fuse after days of use.

Answer:

Let X represent number of bulb that will fuse after days of use .Trials =5

X has a binomial distribution,n=5.

Put ,

Question:6 A bag consists of balls each marked with one of the digits to If four

balls are drawn successively with replacement from the bag, what is the probability that

none is marked with the digit ?

Answer:

Let X denote a number of balls marked with digit 0 among 4 balls drawn.

Balls are drawn with replacement.

X has a binomial distribution,n=4.

Put X = 0,

Question:7 In an examination, questions of true-false type are asked. Suppose a

student tosses a fair coin to determine his answer to each question. If the coin falls

heads, he answers 'true'; if it falls tails, he answers 'false'. Find the probability that he

answers at least 12 questions correctly.

Answer:

Let X represent the number of correctly answered questions out of 20 questions.

The coin falls heads, he answers 'true'; if it falls tails, he answers 'false'.

X has a binomial distribution,n=20

Question:8 Suppose X has a binomial distribution Show that is the

most likely outcome.

(Hint : is the maximum among all of , )

Answer:

X is a random variable whose binomial distribution is

Here , n=6 and .

is maximum if is maximum.

is maximum so for x=3 , is maximum.

Question:9 On a multiple choice examination with three possible answers for each of

the five questions, what is the probability that a candidate would get four or more correct

answers just by guessing ?

Answer:

Let X represent number of correct answers by guessing in set of 5 multiple choice

questions.

Probability of getting a correct answer :

X has a binomial distribution,n=5.

Question:10(a) A person buys a lottery ticket in lotteries, in each of which his

chance of winning a prize is What is the probability that he will win a prize

at least once

Answer:

Let X represent number of winning prizes in 50 lotteries .

X has a binomial distribution,n=50.

Question:10(b) A person buys a lottery ticket in lotteries, in each of which his

chance of winning a prize is . What is the probability that he will win a prize

exactly once

Answer:

Let X represent number of winning prizes in 50 lotteries .

X has a binomial distribution,n=50.

Question:10(c) A person buys a lottery ticket in 50 lotteries, in each of which his

chance of winning a prize is . What is the probability that he will win a prize

at least twice?

Answer:

Let X represent number of winning prizes in 50 lotteries.

X has a binomial distribution,n=50.

Question:11 Find the probability of getting exactly twice in throws of a die.

Answer:

Let X represent number of times getting 5 in 7 throws of a die.

Probability of getting 5 in single throw of die=P

X has a binomial distribution,n=7

Question:12 Find the probability of throwing at most sixes in throws of a single die.

Answer:

Let X represent number of times getting 2 six in 6 throws of a die.

Probability of getting 6 in single throw of die=P

X has a binomial distribution,n=6

Question:13 It is known that of certain articles manufactured are defective. What

is the probability that in a random sample of such articles, are defective?

Answer:

Let X represent a number of times selecting defective items out of 12 articles.

Probability of getting a defective item =P

X has a binomial distribution,n=12

Question:14 In a box containing bulbs, are defective. The probability that out of

a sample of bulbs, none is defective is

(A)

(B)

(C)

(D)

Answer:

Let X represent a number of defective bulbs out of 5 bulbs.

Probability of getting a defective bulb =P

X has a binomial distribution,n=5

The correct answer is C.

Question:15 The probability that a student is not a swimmer is Then the probability

that out of five students, four are swimmers is

In the following, choose the correct answer:

(A)

(B)

(C)

(D) None of these

Answer:

Let X represent number students out of 5 who are swimmers.

Probability of student who are not swimmers =q

X has a binomial distribution,n=5

Option A is correct.

NCERT solutions for class 12 maths chapter 13 probability-Miscellaneous Exercise

Question:1(i) A and B are two events such that Find if

is a subset of

Answer:

A and B are two events such that

Question:1(ii) and are two events such that Find if

Answer:

A and B are two events such that

Question:2(i) A couple has two children,

Find the probability that both children are males, if it is known that at least one of the

children is male.

Answer:

A couple has two children,

sample space

Let A be both children are males and B is at least one of the children is male.

Question:2(ii) A couple has two children,

Find the probability that both children are females, if it is known that the elder child is a

female.

Answer:

A couple has two children,

sample space

Let A be both children are females and B be the elder child is a female.

Question:3 Suppose that of men and of women have grey hair. A grey

haired person is selected at random. What is the probability of this person being male?

Assume that there are equal number of males and females.

Answer:

We have of men and of women have grey hair.

Percentage of people with grey hairs

The probability that the selected haired person is male :

Question:4 Suppose that of people are right-handed. What is the probability that

at most of a random sample of people are right-handed?

Answer:

of people are right-handed.

at most of a random sample of people are right-handed.

the probability that more than of a random sample of people are right-handed is

given by,

the probability that at most of a random sample of people are right-handed is given

by

.

Question:5(i) An urn contains balls of which balls bear a mark and the

remaining bear a mark A ball is drawn at random from the urn, its mark is noted

down and it is replaced. If balls are drawn in this way, find the probability that

all will bear mark.

Answer:

Total balls in urn = 25

Balls bearing mark 'X' =10

Balls bearing mark 'Y' =15

balls are drawn with replacement.

Let Z be a random variable that represents a number of balls with Y mark on them in the

trial.

Z has a binomial distribution with n=6.

Question:5(ii) An urn contains balls of which balls bear a mark and the

remaining bear a mark A ball is drawn at random from the urn, its mark is noted

down and it is replaced. If balls are drawn in this way, find the probability that

not more than will bear mark.

Answer:

Total balls in urn = 25

Balls bearing mark 'X' =10

Balls bearing mark 'Y' =15

balls are drawn with replacementt.

Let Z be random variable that represents number of balls with Y mark on them in trial.

Z has binomail distribution with n=6.

Question:5(iii) An urn contains 25 balls of which 10 balls bear a mark and the

remaining 15 bear a mark A ball is drawn at random from the urn, its mark is noted

down and it is replaced. If 6 balls are drawn in this way, find the probability that

at least one ball will bear mark.

Answer:

Question:5(iv) An urn contains 25 balls of which 10 balls bear a mark and the

remaining 15 bear a mark A ball is drawn at random from the urn, its mark is noted

down and it is replaced. If 6 balls are drawn in this way, find the probability that

the number of balls with mark and mark will be equal.

Answer:

Question:6 In a hurdle race, a player has to cross hurdles. The probability that he

will clear each hurdle is . What is the probability that he will knock down fewer

than hurdles?

Answer:

Let p and q respectively be probability that the player will clear and knock down the

hurdle.

Let X represent random variable that represent number of times the player will knock

down the hurdle.

Question:7 A die is thrown again and again until three sixes are obtained. Find the

probability of obtaining the third six in the sixth throw of the die.

Answer:

Probability of 6 in a throw of die =P

Probability that 2 sixes come in first five throw of die :

Probability that third six comes in sixth throw :

Question:8 If a leap year is selected at random, what is the chance that it will contain

53 tuesdays?

Answer:

In a leap year, there are 366 days.

In 52 weeks, there are 52 Tuesdays.

The probability that a leap year will have 53 Tuesday is equal to the probability that the

remaining 2 days are Tuesday.

The remaining 2 days can be :

1. Monday and Tuesday

2. Tuesday and Wednesday

3. Wednesday and Thursday

4. Thursday and Friday

5.friday and Saturday

6.saturday and Sunday

7.sunday and Monday

Total cases = 7.

Favorable cases = 2

Probability of having 53 Tuesday in a leap year = P.

Question:9 An experiment succeeds twice as often as it fails. Find the probability that

in the next six trials, there will be atleast 4 successes.

Answer:

Probability of success is twice the probability of failure.

Let probability of failure be X

then Probability of success = 2X

Sum of probabilities is 1.

Let and

Let X be random variable that represent the number of success in six trials.

Question:10 How many times must a man toss a fair coin so that the probability of

having at least one head is more than ?

Answer:

Let the man toss coin n times.

Probability of getting head in first toss = P

The minimum value to satisfy the equation is 4.

The man should toss a coin 4 or more times.

Question:11 In a game, a man wins a rupee for a six and loses a rupee for any other

number when a fair die is thrown. The man decided to throw a die thrice but to quit as

and when he gets a six. Find the expected value of the amount he wins / loses.

Answer:

In a throw of die,

probability of getting six = P

probability of not getting six = q

There are three cases :

1. Gets six in the first throw, required probability is

The amount he will receive is Re. 1

2.. Does not gets six in the first throw and gets six in the second throw, then the

probability

The amount he will receive is - Re.1+ Re.1=0

3. Does not gets six in first 2 throws and gets six in the third throw, then the probability

Amount he will receive is -Re.1 - Re.1+ Re.1= -1

Expected value he can win :

Question:12(i) Suppose we have four boxes A,B,C and D containing coloured marbles

as given below:

One of the boxes has been selected at random and a single marble is drawn from it. If

the marble is red, what is the probability that it was drawn from box A ?

Answer:

'

Let R be the event of drawing red marble.

Let respectively denote the event of selecting box A, B, C.

Total marbles = 40

Red marbles =15

Probability of drawing red marble from box A is

Question:12(ii) Suppose we have four boxes A,B,C and D containing coloured marbles

as given below:

One of the boxes has been selected at random and a single marble is drawn from it. If

the marble is red, what is the probability that it was drawn from box B?

Answer:

Let R be event of drawing red marble.

Let respectivly denote event of selecting box A,B,C.

Total marbles = 40

Red marbles =15

Probability of drawing red marble from box B is

Question:12(iii) Suppose we have four boxes A,B,C and D containing coloured

marbles as given below:

One of the boxes has been selected at random and a single marble is drawn from it. If

the marble is red, what is the probability that it was drawn from box C?

Answer:

Let R be event of drawing red marble.

Let respectivly denote event of selecting box A,B,C.

Total marbles = 40

Red marbles =15

Probability of drawing red marble from box C is

Question:13 Assume that the chances of a patient having a heart attack is It is

also assumed that a meditation and yoga course reduce the risk of heart attack

by and prescription of certain drug reduces its chances by At a time a

patient can choose any one of the two options with equal probabilities. It is given that

after going through one of the two options the patient selected at random suffers a heart

attack. Find the probability that the patient followed a course of meditation and yoga?

Answer:

Let A,E1, E2 respectively denote the event that a person has a heart break, selected

person followed the course of yoga and meditation , and the person adopted

the drug prescription.

the probability that the patient followed a course of meditation and yoga is

Question:14 If each element of a second order determinant is either zero or one, what

is the probability that the value of the determinant is positive? (Assume that the

individual entries of the determinant are chosen independently, each value being

assumed with probability ).

Answer:

Total number of determinant of second order with each element being 0 or 1 is

The values of determinant is positive in the following cases

Probability is

Question:15(i) An electronic assembly consists of two subsystems, say, A and B. From

previous testing procedures, the following probabilities are assumed to be known:

P(A fails) =

P(B fails alone) =

P(A and B fail) =

Evaluate the following probabilities

Answer:

Let event in which A fails and B fails be

Question:15(ii) An electronic assembly consists of two subsystems, say, A and B.

From previous testing procedures, the following probabilities are assumed to be known:

P(A fails) =

P(B fails alone) =

P(A and B fail) =

Evaluate the following probabilities

Answer:

Let event in which A fails and B fails be

Question:16 Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5

black balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from

Bag II. The ball so drawn is found to be red in colour. Find the probability that the

transferred ball is black.

Answer:

Let E1 and E2 respectively denote the event that red ball is transfered from bag 1 to

bag 2 and a black ball is transfered from bag 1 to bag2.

and

Let A be the event that ball drawn is red.

When a red ball is transfered from bag 1 to bag 2.

When a black ball is transfered from bag 1 to bag 2.

Question:17 If A and B are two events such that and then

Choose the correct answer of the following:

(A)

(B)

(C)

(D)

Answer:

A and B are two events such that and

Option A is correct.

Question:18 If , then which of the following is correct :

(A)

(B)

(C)

(D)

Answer:

Option C is correct.

Question:19 If A and B are any two events such

that then

(A)

(B)

(C)

(D)

Answer:

Option B is correct.