CBSE Class 12

The NCERT Solutions for Class 12 English Flamingo Chapter 5 Indigo Consists Of All The Answers From This pdf. You Can Download The Pdf For NCERT Solutions for Class 12 English Free And Refer To The Answers To Get A Better Understanding Of The Chapter. https://mcq-questions.com/ncert-solutions-for-class-12-english-flamingo-chapter-5-indigo/

Indigo Lesson NCERT Solutions for Class 12 English Flamingo Chapter 5

Indigo NCERT Text Book Questions and Answers

Indigo  Think as you read

Class 12 English Indigo Ncert Solutions Question 1.
Strike out what is not true in the following.
(a) Rajkumar Shukla was:
(i) a sharecropper
(ii) a politician
(iii) delegate
(iv) a landlord

(b) Rajkumar Shukla was:
(i) poor
(ii) physically strong
(iii) illiterate
Answer:
(a) a politician, a landlord
(b) physically strong

Indigo Class 12 Ncert Solutions Question 2.
Why is Rajkumar Shukla described as being‘resolute’?
Answer:
Rajkumar Shukla requested Gandhiji to go with him to his area called Champaran. Gandhiji was engaged at that time. However, Shukla did not let go of Gandhiji. He followed him wherever he went. Finally, Gandhiji had to arrange and fix time to go with him. This shows that Shukla was resolute.

Class 12 English Chapter 5 Question Answer NCERT Solutions Question 3.
Why do you think the servants thought Gandhi to be another peasant?
Answer:
Gandhiji was quite simple in his dress and manners. He never thought himself as a great leader. That is why, servants believed him to be another peasant.

Indigo Class 12 Questions And Answers NCERT Solutions Question 4.
List the places that Gandhi visited between his first meeting with Shukla and his arrival at Champaran.
Answer:
Gandhiji visited the following places: Patna, Rajendra Prasad’s house, Muzaffarpur, Professor Malkani’s home and lastly Champaran.

Question 5.
What did the peasants pay the British landlords as rent? What did the British now want instead and why? What would be the impact of synthetic indigo on the prices of natural indigo?
Answer:
The peasants had to grow indigo on 15 per cent of their land. This product was submitted as rent to the British landlords. Synthetic indigo was developed by Germany. The landlords thus, did not need to raise indigo on their land any longer. They demanded compensation from the peasants for freeing them from the indigo-raising agreement.

Question 6.
The events in this part of the text illustrate Gandhi’s method of working. Can you identify some instances of this method and link them to his ideas of satyagraha and non-violence?
Answer:
Gandhiji had always followed the voice of his conscience. He never supported anything immoral. He followed this principle all through his fight against the British injustice. He never paid evil for evil. He followed the principle of non-violence even as the authorities raised blows on him. His path was that of satyagraha non-violence for truth. Dandi March was a good example.

Question 7.
Why did Gandhi agree to a settlement of 25 per cent refund to the farmers?
Answer:
The whole situation in front of Gandhiji was a deadlock. He wanted to break this deadlock somehow. The British planters wanted some excuse for prolonging the dispute with the peasants. However, Gandhiji proved too wise for them. The deadlock was ended by accepting what the planters wanted.

Even so the British had to compromise with their pride. Gandhiji agreed to a settlement of 25 per cent refund to the farmers; in fact, the amount was less important than the fact that the landlords had to be forced to return part of the money and with it, part of their pride and prestige.

So far the planters had behaved as if they were above the law, they had to realise that Britishers were not above the law. The peasants now saw that they too had rights and defenders, and they learned courage. The peasants were also saved from the trouble of spending time and money on court cases. Within a few years, the British planters abandoned their estates and left. The land came back to the peasants and this was the end of indigo sharecropping.

Question 8.
How did the episode change the plight of the peasants?
Answer:
The peasants now had courage. They believed that they had rights which they could defend. Gradually, the British planters left their estates. These estates now came back to the peasants. Indigo sharecropping disappeared permanently. They were no longer indebted to the British planters.

Indigo Understanding the Text

Question 1.
Why do you think Gandhi considered the Champaran episode to be a turning point in his life?
Answer:
Gandhiji considered the Champaran episode a turning point in his life because this episode released the peasants from the mortal fear of British landlords and made them aware of their rights. Not only this, the farmers got back 25% of the compensation money. They also developed courage.

This episode proved to be the beginning of the cultural, social and economic transformation of the poor and badly exploited and terrified peasants. Thus, the British planters were forced to leave the land of the peasants and they became the owners of their lands. This was an attempt to fight injustice and remove sufferings of the peasants. It ignited the feelings of patriotism among simple farmers. It became the first success of Non-cooperation Movement for Gandhiji.

Question 2.
How was Gandhi able to influence lawyers? Give instances.
Answer:
The lawyers desired Andrews to stay in Champaran and help them. However, Gandhiji opposed them. He said that taking the help of an Englishman would be their weakness. They should learn to win the battle with their own strength. They should learn to be self-dependent.

Question 3.
What was the attitude of the average Indian in smaller localities towards advocates of ‘home rule’?
Answer:
The average Indian in smaller localities felt afraid to show sympathy for advocates of ‘home rule’. They probably feared the consequences.

Question 4.
How do we know that ordinary people too contributed to the freedom movement?
Answer:
The ordinary people stood with Gandhiji at every juncture. At Motihari, they flocked
in thousands as they learnt that Mahatma had some trouble with the authorities. The ordinary people supported in their own little way. Rajkumar Shukla and Professor Malkani defied all odds and contributed to the fight. Prof. J. B. Kriplani motivated a large number of students, and welcomed Gandhiji at Muzaffarpur railway station at midnight. The spontaneous demonstration outside the court was also quite significant. Civil disobedience could triumph in India only because of the unity of ordinary people.

Indigo Talking about the Text

Question 1.
“Freedom from fear is more important than legal justice for the poor.” Do you think that the poor of India are free from fear after Independence?
Answer:
In the chapter, Gandhi makes it possible for the sharecroppers of Champaran to shed their fear of the British landlords. According to him, the first step towards self-reliance is freedom from fear. Unfortunately, the poor people are not free from fear even after the Independence. The poor people live in a continual fear of the police, who instead of taking care, often end up maltreating them. Due to globalisation and the craze for the foreign products, the poor are becoming poorer.

Question 2.
The qualities of a good leader.
Answer:
A good leader is the person who leads the minds and convinces people to follow his set of ideas or beliefs. He thinks for the people and works for them. He should be sincere in his approach and should be a man of principles. A good leader inherits some qualities that set him apart from the rest. Truth, honesty, patriotism, morality, spirit of service and sacrifice are the qualities of a good leader. He should be courageous in the face of adversity and should never quit. He should encourage and motivate others to bring out the best in them, and should appreciate the efforts of others without being bias or partial.

Indigo Extra Questions and Answers

Indigo Short Answer Questions

Question 1.
How did Gandhiji react to the Commissioner’s advice? Where did he go?
Answer:
Gandhiji was asked to leave the Tirhut division at once by the commissioner. He did not leave, instead, he proceeded to Motihari, the capital of Champaran.

Question 2.
Why did the servants think Gandhiji to be another peasant?
Answer:
Gandhiji was a simple man and he used to dress in a dhoti, which was the dress that the farmers in India used to wear. Hence, the servants thought Gandhiji to be another peasant.

Question 3.
“The battle of Champaran is won!” What led Gandhiji to make this remark?
Answer:
Gandhiji said these words when he was able to win the lawyers’ trust. Earlier, these lawyers had certain misconceptions about Gandhiji, but as they saw his determination towards the peasants’ liberation, they came in his full support.

Question 4.
Why did Gandhiji go to Lucknow in December 1916? Who met him there and why?
Answer:
Gandhiji went to Lucknow to attend the annual convention of the Indian National Congress. A poor peasant named Rajkumar Shukla met him there. He was from Champaran. He wanted Gandhiji to come to Champaran to help the poor sharecroppers.

Question 5.
Why did the landlords compel the peasants to do as per the terms of a long-term contract?
Answer:
The landlords forced peasants to plant indigo on 15 per cent of their land. All the indigo produce had to be surrendered as rent. The peasants felt sour about it.

Question 6.
What did the British planters try to do when they came to know that synthetic indigo had been developed by Germany?
Answer:
The British planters realised that it was no longer profitable to produce natural indigo. The synthetic indigo was much cheaper. Thus, they compelled the peasants to give them compensation for not having to plant indigo on their land.

Question 7.
What happened when the British planters asked the peasants for compensation for releasing them from the 15 per cent agreement?
Answer:
The sharecropping agreement seemed irksome to the peasants. Therefore, many of them signed it willingly. However, others engaged lawyers to fight their cases. So the landlords hired thugs.

Question 8.
How was Gandhi treated at Rajendra Prasad’s house?
Answer:
Since Gandhiji was quite simple in his dress and manners, Rajendra Prasad’s servants mistook him to be a peasant. They did not allow him to draw water from the well lest it be polluted. They let him stay on the grounds.

Question 9.
What were the terms of the indigo contract between the British landlords and the Indian peasants?
Answer:
The fertile land was divided into large estates owned by Englishmen and worked by Indian tenants. The peasants had to grow indigo on 15 per cent of the land. This product was submitted as rent to the British landlords.

Question 10.
Why was Gandhiji opposed to C.F. Andrews helping him in Champaran?
Answer:
Gandhiji was opposed to C.F. Andrews helping him in Champaran because he was a foreigner. C.E Andrews was a social worker in Champaran. He was a close follower of Gandhiji. He felt that a foreigner’s help should not be sought to free India of foreigners. According to him, self-reliance was of utmost importance.

Question 11.
When Gandhi got the wholehearted support of the lawyers, he said, ‘The battle of Champaran is won’. What was the essence behind his statement?
Answer:
The essence behind this statement was that now he would be able to defeat Britishers who were exploiting poor peasants and would make the lawyers help poor sharecroppers to’ get back their lost respect and money as well. Further, Gandhiji was ready to tutor all the lawyers how to fight this struggle.

Question 12.
Though the sharecroppers of Champaran received only one-fourth of the compensation, how can the Champaran struggle still be termed a huge success and victory?
Answer:
The Champaran struggle was termed a huge success and victory because Gandhiji was able to make the landlords surrender part of the money and their prestige by making them agree to handover 25% of the money as compensation. More important was the fact that peasants understood that they also had rights and people to defend them if they had problems. They learnt to be courageous when they stood behind Gandhiji to break the deadlock between the farmers and the landlords.

Question 13.
The lesson, ‘Indigo’ highlights Gandhiji’s method of working. Can you identify them and link them to his ideas of Satyagraha and non-violence?
Answer:
Gandhiji opposed unjust laws; his politics addressed day-to-day problems of the common man. He showed a willingness to oppose laws and even go to jail. His disobedience was always peaceful, and for truth and justice. He led through embarrassing people who were hypocrites (lawyers).

Question 14.
How did Mahatma Gandhi uplift the peasants of Champaran?
Answer:
Gandhiji gave them economic relief, made them overcome fear and to be united, taught them courage, provided solutions for their cultural and social backwardness, and improved their health and sanitary conditions.

Question 15.
Why is Rajkumar Shukla described as being resolute?
Answer:
Rajkumar Shukla was a poor, illiterate peasant from Champaran. When he came to know that Gandhi was in Lucknow, he decided to meet him and ask him to help the poor sharecroppers of Champaran. He requested Gandhi to come to Champaran but Gandhi was not free. He had appointments in Cawnpore and in other parts of India. Shukla followed him everywhere and even to his Ashram at Ahmedabad and urged him to fix a date. Finally, Gandhi had to agree to visit Champaran. This clearly shows that Shukla was resolute.

Indigo Long Answer Questions

Question 1.
The Champaran episode was a turning point in Gandhiji’s life. Elucidate.
Answer:
Before the Champaran episode, Gandhiji was not aware of the reality of the peasants of his motherland. ,On the insistence of Rajkumar Shukla, a sharecropper, Gandhiji went to Champaran and saw the miserable condition of the poor illiterate farmers. It was an eye-opener for him. The Britishers exploited the farmers to grow indigo. When it was not needed, they had to render compensation in order to be freed from old agreement.

Gandhiji was shocked to see them going to the court. He gathered them. This was the first step to free them from their fear of the British. The officials felt powerless without Gandhiji’s co-operation. He made them realise that the power of the British could be challenged by Indians.

The peasants were made to realise that they too had rights. The British landlords left the estate to the peasants and returned to their land after some time, thus ending indigo sharecropping. Through the Champaran episode, he made it clear to the British that they could not order Indians in their own country and through his personal example taught masses to be self-reliant and motivated them into civil disobedience.

Question 2.
Why did Rajkumar Shukla invite Gandhiji to Champaran? How did Gandhiji solve the problem of the indigo farmers?
OR
Why did Gandhiji consider freedom from fear more important than legal justice for the poor peasants of Champaran?
Answer:
Rajkumar Shukla was a poor peasant from Champaran. Under an old agreement, the peasants were compelled by the British to grow indigo on 15% of their land and part with it as rent. For this, Rajkumar Shukla had been advised to speak to Gandhiji who he was told, would be able to do something about their problem.

The landlords had learned that Germany had developed synthetic indigo. They forced the sharecroppers to sign agreements to pay them compensation to be freed from the 15 per cent arrangement. The sharecroppers, who refused, engaged lawyers. The information about synthetic indigo reached the peasants who had signed the agreements. They wanted their money back.

Gandhiji organised a gathering of the peasants at Motihari around the court. This was the beginning of their liberation from fear of the British. Though Gandhiji co-operated with the British and regulated the crowd, but it was a clear proof that their might could be challenged. He inspired the lawyers to fight for justice for the sharecroppers.

After the inquiry committee’s report, the peasants expected the entire sum of money as refund, but Gandhiji asked for 50% only. He was offered a refund of 25%. Gandhiji accepted it.According to Gandhiji, at that stage, money was less important. The landlords had to surrender their prestige and the peasants realised that they too had rights. This was their first lesson in courage. This is how their problem was solved.

Question 3.
Which factors helped the fear-stricken peasants of Champaran to achieve freedom?
Answer:
There were several factors in which Gandhiji’s contribution was remarkable.
The peasants were sharecroppers with the British planters. According to an old agreement, the peasants had to produce indigo on 15 per cent of the land and give it as rent to the landlords. Around 1917, it was told that Germany had developed synthetic indigo. So the British planters now no longer desired the indigo crop. To release the peasants from the old agreement, they demanded compensation from them. Most of the illiterate peasants agreed to it. However, others refused. Lawyers were engaged to go to the court.

At that time, Gandhiji appeared in Champaran. He fought a long battle for the poor peasants for one year and managed to get justice for them. The peasants now became courageous and became aware about their rights. Along with the political and economic struggle, Gandhiji worked on the social level also. He made arrangements for the education, health and hygiene of the families of poor peasants by teaching the lesson of self-reliance. It was one of the ways to forward the struggle for Indian independence.

The peasants now had courage. They believed that they had rights which they could defend. Gradually, the British planters left their estates. These estates now came back to the peasants. Indigo sharecropping disappeared for all times to come.

Question 4.
Give an account of Gandhiji’s efforts to secure justice for the poor indigo sharecroppers of Champaran.
Answer:
Gandhiji went to Champaran on receiving reports of exploitation of the poor sharecropper peasants at the hands of British planters. He began by trying to get the facts. The British landlords as well as commissioner of Tirhut were non-cooperative. Lawyers from Muzaffarpur briefed him about the court cases of these peasants. Gandhiji and the lawyers collected depositions by about ten thousand peasants. Notes were made on other evidence. Documents were collected. The whole area throbbed with the activities of the investigators and forceful protests of landlords.

The lieutenant governor summoned Gandhiji. After four protracted interviews, an official commission of enquiry was appointed to look into the indigo sharecroppers’ situation. Gandhiji was the sole representative of the peasants. The official enquiry assembled huge quantity of evidence against the big planters.

They agreed in principle to make refunds to the peasants. After consolation, a settlement of 25 per cent refund to the farmers was agreed upon. This was a moral victory for the peasants. They recognised their rights and became courageous. Within a few years, the British planters gave up tVieir estates. These now went back to the peasants. They became the master of the land. Thus, indigo sharecropping disappeared.

Question 5.
How was the Champaran episode a big success? Elucidate.
Answer:
The fight and the success of Champaran was the success of Civil Disobedience Movement started by Gandhiji. It was the attempt of the poor peasants who were helpless to the fraud met out to them. One of them contacted Gandhiji. Gandhiji’s presence in Bihar raised a huge row in Champaran. Thousands of peasants held a demonstration to protest against the government. The government was baffled. The orders for Gandhiji to quit Champaran were disobeyed by him. Afterwards, an enquiry commission was set up which ordered the sharecroppers to get 25 per cent of their money. The cruel landlords were made to surrender the partial amount of the extorted money. The efforts of Gandhiji and the peasants made the government realise its mistake.

Question 6.
Exploitation is a universal phenomenon. The poor indigo farmers were exploited by the British landlords to which Gandhiji objected. Even after our independence, we find exploitation in unorganised labour sector.
What values do we learn from Gandhiji’s campaign to counter the present day problems of exploitation?
Answer:
The weak are exploited and the strong prey on them is a universal fact. In the case of the poor illiterate indigo farmers, they were exploited by the British landlords. Gandhiji objected to it and freed the farmers from the agreement and brought an end to indigo sharecropping. In his manner of tackling the issue, he went
stepwise:

• he gathered information
• fearlessly he stated’his points
• in the final negotiations, he did not bother about the money; it was the submission of the opponent’s pride and prestige.

Similarly, we can proceed with such issues as Gandhiji’s method of solving the problem has universality about it. Today, we can follow it this way: one must be fully aware of one’s weaknesses and must try to overcome them, find ways of getting justice, never give in to any kind of exploitation, if trapped, try to come out of it wisely, get united when in trouble and seek help. Do not compromise your self-respect, values or dignity at any cost. Try to come out of the darkness of ignorance as soon as possible. Mistakes once made, must not be repeated.

Question 7.
Though Rajkumar Shukla was an illiterate peasant; he was resolute and was able to bring a change in the lives of the people of Champaran. Taking hints from the text, write an article on the topic, “Grit and Determination can take you a long way”.
Answer:
Grit and determination plays a very important role in one’s life. A person who doesn’t give up too easily and has tendency to step ahead without thinking too much about the difficulties is able to accomplish anything. We can take the example of Rajkumar Shukla. He wanted Gandhiji to go with him to his area called Champaran. Gandhiji was engaged at that time.

However, Shukla did not leave Gandhiji. He followed him wherever he went. Finally, Gandhiji had to arrange and fix time to go with him. Shukla’s resolute nature led to a change in the lives of the people of Champaran. His persistence bore fruit. It is important to pursue our goals with grit and determination to be successful. The will to succeed, will one day result in triumph. It is possible that it might take a long time to succeed but success will definitely be achieved.

Question 8.
‘Dialogue and not violence can resolve situations of conflict and injustice’. Prove the statement with reference to the lesson, ‘Indigo’.
Answer:
Gandhiji met Rajkumar Shukla, a poor peasant from Champaran at Lucknow. Shukla wanted Gandhiji to come to Champaran to help the poor sharecroppers who were compelled by the British to grow indigo on 15 % of their land and part with it as rent. Since the development of synthetic indigo, cultivation of indigo had become a waste. The landlords wanted sharecroppers to sign agreements to be freed from the 15 per cent arrangement by paying compensation. After understanding the problem, Gandhiji wanted to meet the secretary of British Landlord’s Association, but he was refused.

Then he tried to meet Commissioner of Tirhut who bullied him and ordered to leave. However, he defied the order and organised a gathering of the peasants around the court. Gandhiji proved that British power was no longer unchangeable. The authorities got afraid and postponed the case.

Gandhiji was released on bail. He inspired the lawyers to fight for justice for the sharecroppers. The case was dropped and Gandhiji agreed for 25% refund as was agreed by landlords. Finally, indigo sharecropping was abandoned and land was given to peasants. This became the first success of Non-cooperation Movement for Gandhiji.

These NCERT Solutions for Class 12 Maths Chapter 13 Probability Miscellaneous Exercise Questions and Answers are prepared by our highly skilled subject experts. https://mcq-questions.com/ncert-solutions-for-class-12-maths-chapter-13-miscellaneous-exercise/

NCERT Solutions for Class 12 Maths Chapter 13 Probability Miscellaneous Exercise

Question 1.
A and B are two events such that P (A) ≠ 0. Find P(B|A), if
i. A is a subset of B
ii. A ∩ B = Φ
Solution:

Question 2.
A couple has two children.
Find the probability that both children are males, if it is known that atleast one of the children is male.
Solution:
The sample space, S = {MM, MF, FM, FF},
where M denote male and F denote female.
Let A: both children are males
B : atleast one child is a male
A = {MM},
B = {MM, MF, FM}
\(\mathrm{A} \cap \mathrm{B}=\{\mathrm{MM}\}, \mathrm{P}(\mathrm{A})=\frac{1}{4}\)
\(P(B)=\frac{3}{4}, P(A \cap B)=\frac{1}{4}\)

Question 3.
If a leap year is selected at random, what is the chance that it will contain 53 Tuesdays?
Solution:
A leap year contains 366 days = 52 weeks + 2 days
The last 2 days can be
i. Monday, Tuesday
ii. Tuesday, Wednesday
iii. Wednesday, Thursday
iv. Thursday, Friday
v. Friday, Saturday
vi. Saturday, Sunday
vii. Sunday, Monday
Of these seven possibilities, (i) & (ii) are favourable to 53 Tuesdays.
∴ P(53 Tuesday) = \(\frac { 2 }{ 7 }\)

Question 4.
An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be atleast 4 successes.
Solution:
Let p be the probability of a success and q the probability of failure.
Then p + q – 1 and p = 2q
Solving p = \(\frac { 2 }{ 3 }\) and q = \(\frac { 1 }{ 3 }\)
Let X be the number of success.
Then X is a binomial distribution with

Question 5.
How many times must a man toss a fair coin so that the probability of having atleast one head is more than 90%?
Solution:
Tossing a coin many times is a Bernoulli trial. Here success is obtaining a Head.
∴ P = \(\frac { 1 }{ 2 }\)
q = 1 – p = 1 – \(\frac { 1 }{ 2 }\) = \(\frac { 1 }{ 2 }\)
Let X be the number of heads obtained
Then X is a binomial distribution B(n, \(\frac { 1 }{ 2 }\))

We know 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32 and so on
Hence the minimum value of n is 4 i.e. n > 4
i.e. the man has to toss the coin atleast 4 times.

Question 6.
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.
Solution:
The game ends in the following ways.
i. The man gets 6 in 1^st throw. In this case, he earns ₹ 1.
P(getting 6 ¡n 1^st throw) = \(\frac { 1 }{ 6 }\)

ii. The man does not get 6 in 1^nd throw and 6 in 2^nd throw. In this case he earns ₹ 0
(In 1^st throw, he earns ₹ 1 and in 2^nd throw he loses ₹ 1)
P(not getting 6 on 1^st throw & 6 in 2^nd throw) = (\(\frac { 5 }{ 6 }\))(\(\frac { 1 }{ 6 }\)) = \(\frac { 5 }{ 6 }\)

Question 7.
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? box B?
Solution:
Let E₁ : selecting box A
E₂ : selecting box B
E₃ : selecting box C
E₄ : selecting box D
A : selecting a red ball
E₁, E₂, E₃ and E₄ are mutually exclusive and exhaustive events.
∴ \(\mathrm{P}\left(\mathrm{E}_{1}\right)=\mathrm{P}\left(\mathrm{E}_{2}\right)=\mathrm{P}\left(\mathrm{E}_{3}\right) \dot{\mathrm{P}}\left(\mathrm{E}_{4}\right)=\frac{1}{4}\)
\(\mathrm{P}\left(\mathrm{A} \mid \mathrm{E}_{1}\right)=\frac{1}{10}, \quad \mathrm{P}\left(\mathrm{A} \mid \mathrm{E}_{2}\right)=\frac{6}{10}\)

Question 8.
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.
Solution:
E₁ : a red ball is transferred from bag I to bag II
E₂ : a black ball is transferred from bag I to bag II
A : a red ball is taken from bag II after transferring a ball
E₁ and E₂ are mutually exclusive and exhaustive events

Question 9.
If A B are two events such that P(A) ≠ 0 and P(B|A) = 1, then
a. A ∩ B
b. B ∩ A
c. B = Φ
d. A = Φ
Solution:

Question 10.
If P(A|B) > P(A), then which of the following is correct:
a. P(B | A) < P(B)
b. P(A ∩ B) < P(A). P(B) c. P(B | A) > P(B)
d. P(B | A) = P(B)
Solution:

Question 11.
If A and B are any two events such that P(A) + P(B) – P(A and B) = P(A), then
a. P(B|A) = 1
b. P(A|B) = 1
c. P(B|A) = 0
d. P(A|B) = 0
Solution:

These NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.4 Questions and Answers are prepared by our highly skilled subject experts. https://mcq-questions.com/ncert-solutions-for-class-12-maths-chapter-13-ex-13-4/

NCERT Solutions for Class 12 Maths Chapter 13 Probability Exercise 13.4

Question 1.
State which of the following are not the probability distributions of a random variable. Give reasons for your answer.

Solution:
P (0) + P (1) + P (2) = 0.4 + 0.4 + 0.2 = 1
It is a probability distribution.

(ii) P (3) = – 0.1 which is not possible.
Thus it is not a probability distribution.

(iii) P(-1)+P(0)+P(1) = 0.6 + 0.1 + 0.2 = 0.9 ≠ 1
Thus it is not a probability distribution.

(iv) P (3) + P (2) + P (1) + P (0) + P (-1)
= 0.3 + 0.2 + 0.4 + 0.1 + 0.05 = 1.05 ≠ 1
Hence it is not a probability distribution.

Question 2.
An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X represent the number of black balls. What are the possible values of X ?. Is X a random variable?
Solution:
These two balls may be selected as RR, RB, BR, BB, where R represents red and B represents black ball, variable X has the value 0,1,2, i.e., there may be no black balls, may be one black ball, or both the balls are.black. Yes , X is a random variable.

Question 3.
Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. What are possible values of X?
Solution:
Let ‘w’ denote the number of heads and ‘n’ the number of tails when a coin is tossesd six times
X is the difference between m and n
∴ X = |m – n|

∴ The possible values of X are 0, 2, 4, 6.

Question 4.
Find the probability distribution of
(a) number of heads in two tosses of a coin.
(b) number of tails in the simultaneous tosses of three coins.
(c) number of heads in four tosses of a coin.
Solution:
i. When a coin is tossed twice, the sample
space S = {HH, HT, TH, TT}
Let X denote the number of heads.
Then X takes the values 0, 1, 2.
P(X = 0) = P(two tails) = P{TT} = \(\frac { 1 }{ 4 }\)
P(X = 1) = P(one head & one tail)
= P{HT,TH} = \(\frac { 2 }{ 4 }\) = \(\frac { 1 }{ 2 }\)
P(X = 2) = P(two heads) = P{HH} = \(\frac { 1 }{ 4 }\)
∴ The probability distribution of X is

ii. When a coin is tossed 3 times, the sample space S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Let X denote the number of tails.
Then X take values 0, 1, 2, 3.
P(X = 0) = P(no tail) = P{HHH} = \(\frac { 1 }{ 8 }\)
P(X = 1) = P(one tail & two heads)
= P{HHT, THH, HTH} = \(\frac { 3 }{ 8 }\)
P(X = 2) = P(two tails & one head)
= P{HTT, THT, TTH} = \(\frac { 3 }{ 8 }\)
P(X = 3) = P(three tails) = P(TTT) = \(\frac { 1 }{ 8 }\)
The probability distribution of X is

iii. When four coins are tossed, sample space, S = {HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT}
Let X denote the number of heads in the four tosses of a coin.
Then X can take the values 0, 1,2, 3 and 4
P(H) = \(\frac { 1 }{ 2 }\) ,P(T) = \(\frac { 1 }{ 2 }\)
P(X = 0) = P(TTTT) = \(\frac { 1 }{ 2 }\) x \(\frac { 1 }{ 2 }\) x \(\frac { 1 }{ 2 }\) x \(\frac { 1 }{ 2 }\) = \(\frac { 1 }{ 16 }\)
P(X = 1) = P{HTTT, THTT, TTHT, TTTH}

Question 5.
Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as
i. number greater than 4
ii. six appears on atleast one die
Solution:
i. When a die is tossed the sample space S = {1, 2, 3, 4, 5, 6}.
Let A denote the success
A: getting a number greater than 4
A = {5, 6}
P(A) = \(\frac { 2 }{ 6 }\) = \(\frac { 1 }{ 3 }\)
P(A’) = 1 – P(A)= 1 – \(\frac { 1 }{ 3 }\) = \(\frac { 2 }{ 3 }\)
Let X denote the number of successes in ‘two tosses of a die.
Then X takes the values 0, 1, 2
P(X = 0) = P(A’A’) = \(\frac { 2 }{ 3 }\) x \(\frac { 2 }{ 3 }\) = \(\frac { 4 }{ 9 }\)
P(X = 1) = P(A A’ or A’ A)
= \(\frac { 1 }{ 3 }\) x \(\frac { 2 }{ 3 }\) + \(\frac { 2 }{ 3 }\) x \(\frac { 1 }{ 3 }\) = \(\frac { 4 }{ 9 }\)
P(X = 2) = P(AA) = \(\frac { 1 }{ 3 }\) x \(\frac { 1 }{ 3 }\) = \(\frac { 1 }{ 9 }\)
The Probability distribution of X is

ii. Let B denote the event of getting 6 on atleast one die
∴ B = {(1, 6), (2, 6), (3, 6), (4, 6), 6), (6, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5)}
Let X denote the number of success
Then X takes the values 0 and 1
P(X = 0) = P(B’)= 1 – P(B)
= 1 – \(\frac { 11 }{ 36 }\) = \(\frac { 25 }{ 36 }\)
P(X = 1) = P(B) = \(\frac { 11 }{ 36 }\)
∴ The probability distribution of X is

Question 6.
From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.
Solution:
Let D denote a defective bulb
∴ P(D) = \(\frac { 6 }{ 30 }\)
∴ P(D’) = 1 – P(D) = 1 – \(\frac { 1 }{ 5 }\) = \(\frac { 4 }{ 5 }\)
Let X denote the number of defective bulbs in the sample of 4 bulbs.
Then X takes the values 0, 1, 2, 3, 4

Another Method:
There are 6 defective bulbs and 24 non defective bulbs
P(getting a defective bulb) = \(\frac { 6 }{ 30 }\) = \(\frac { 1 }{ 5 }\)
P(getting a non defective bulb) = \(\frac { 24 }{ 30 }\) = \(\frac { 4 }{ 5 }\)
Let X denote the number of defective bulbs.
Then X takes the values 0, 1, 2, 3, 4
P(X = 0) = P(no defective bulb)

Question 7.
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tads.
Solution:

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

Determine
(i) k
(ii) P(X < 3) (iii) P(X > 6)
(iv) P(0 < X < 3)
Solution:
(i) k

Question 9.
The random variable X has a probability distribution P (X) of the following form, where k is some number

(a) Determine the value of k
(b) Find P(X < 2), P (X ≤ 2), P(X ≥ 2)
Solution:

Question 10.
Find the mean number of heads in three tosses of a fair coin.
Solution:
When 3 coins are tossed, the sample space S = {HHH, HHT, FITH, HTT, THH, THT, TTH, TTT}
Let X denote the number of heads
Then X takes the values 0, 1, 2, 3
P(X = 0) = P(no head) = \(\frac { 1 }{ 8 }\)
P(X = 1) = P(one head) = \(\frac { 3 }{ 8 }\)
P(X = 2) = P(two heads) = \(\frac { 3 }{ 8 }\)
P(X = 3) = P(three heads) = \(\frac { 1 }{ 8 }\)
The probability distribution of X is

Question 11.
Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.
Solution:
Let X denote the number of 6’s when two dice are thrown.
Then X takes the values 0, 1, 2
P(X = 0) = P(no six on both dice)

Question 12.
Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find E(X).
Solution:
There are six numbers 1, 2, 3, 4, 5, 6 one of them is selected in 6 ways
When one of the numbers has been selected, 5 numbers are left, one number out of 5 may be select in 5 ways
∴ No. of ways of selecting two numbers without replacement out of 6 positive integers = 6 x 5 = 30

Question 13.
Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X.
Solution:
When two dice are rolled, then the sample space S has 36 simple events.
Let X denote the sum of numbers on the two dice Then X takes the values 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
P(X = 2) = P{(1,1)} = \(\frac { 1 }{ 36 }\)
P(X = 3) = P{(1,2), (2,1)} = \(\frac { 2 }{ 36 }\)
P(X = 4) = P{(1,3),(2,2),(3,1)} = \(\frac { 3 }{ 36 }\)
P(X = 5) = P{(1,4),(2,3),(3,2),(4,1)} = \(\frac { 4 }{ 36 }\)
P(X = 6) = P{(1,5), (2,4), (3,3), (4,2), (5,1)} = \(\frac { 5 }{ 36 }\)
P(X = 7) = P{(1,6), (2,5), (3,4), (4,3), (5,2), (6, 1)} = \(\frac { 6 }{ 36 }\)
P(X = 8) = P{(2,6), (3, 5), (4,4), (5, 3), (6,2)} = \(\frac { 5 }{ 36 }\)
P(X = 9) = P{(3,6), (4,5), (5,4), (6,3)} = \(\frac { 4 }{ 36 }\)
P(X = 10) = P{(4, 6), (5, 5), (6,4)} = \(\frac { 3 }{ 36 }\)
P(X=11) = P{(5,6),(6, 5)} = \(\frac { 52}{ 36 }\)
P(X =12) = P{(6, 6)} = \(\frac { 1 }{ 36 }\)
The probability distribution of X is

Question 14.
A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 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 X ? Find mean, variance and standard deviation of X?
Solution:
X denotes the age of the student selected.
∴ X takes the values 14, 15, 16, 17, 18, 19, 20, 21
The data can be summarised into the following table

Question 15.
In a meeting, 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take X = 0, if he opposed, and X = 1, if he is in favour Find E (X) and Var (X).
Solution:
X takes the values 0 and 1

Question 16.
The mean of the number obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face is
(a) 1
(b) 2
(c) 5
(d) \(\frac { 8 }{ 3 }\)
Solution:
Let X denote the number written on the face of the die.
Then X takes the values 1, 2, 5

Question 17.
Suppose that two cards are drawn at random from a deck of cards. Let X be the number of aces obtained. What is the value of E (X)?
(a) \(\frac { 37 }{ 221 }\)
(b) \(\frac { 5 }{ 13 }\)
(c) \(\frac { 1 }{ 13 }\)
(d) \(\frac { 2 }{ 13 }\)
Solution:
Let X denote the number of aces
Then X can take the values 0, 1, 2
P(X = 0) = P(no ace)

The NCERT Solutions for Class 12 English Flamingo Chapter 1 The Last Lesson Consists Of All The Answers From This pdf. You Can Download The Pdf For NCERT Solutions for Class 12 English Free And Refer To The Answers To Get A Better Understanding Of The Chapter. https://mcq-questions.com/ncert-solutions-for-class-12-english-flamingo-chapter-1-the-last-lesson/

The Last Lesson NCERT Solutions for Class 12 English Flamingo Chapter 1

The Last Lesson NCERT Text Book Questions and Answers

The Last Lesson Think as you read

The Last Lesson Question Answer Class 12 NCERT Solutions Question 1.
What was Franz expected to be prepared with for school that day?
Answer:
Franz was expected to be prepared with participles. Mr Hamel had told the class that he would be taking a test on the topic that day.

The Last Lesson Question Answers Class 12 NCERT Solutions Question 2.
What did Franz notice that was usual about the school that day?
Answer:
Usually when the school begins, there would be a lot of commotions. But that day, everything was quiet and it appeared to be like a Sunday, but the students were at their places and Mr Hamel was walking up and down with his terrible iron ruler under his arm.

The Last Lesson Class 12 Questions And Answers Question 3.
What had been put up on the bulletin board?
Answer:
The bulletin-board notified the general public about an order from Berlin. It stated that only German will be taught to the students in the schools of Alsace and Lorraine.

The Last Lesson Question And Answer NCERT Solutions Class 12 Question 4.
What changes did the order from Berlin cause in school that day?
Answer:
The order from Berlin brought all the routine hustle-bustle of the school life to a stand¬still. The teacher, M. Hamel became more sympathetic to his students and taught his lessons with more patience. The students became more attentive in their classes. The villagers were sitting at the usually empty back benches and had come to show their respect and gratitude to M. Hamel.

They regretted not going to school. The order also brought about a great change in people’s feelings towards their country and their native language. There was a general sadness about not being able to utilise the opportunities of learning French when it was possible to do so.

Class 12 English Chapter 1 Question Answers The Last Lesson Question 5.
How did Franz’s feelings about M. Hamel and the school change?
Answer:
Franz was shocked when M. Hamel told the students about the order from Berlin and that it was their last French lesson. He forgot about his teacher’s ruler and crankiness. He developed a sudden fondness for M. Hamel, and was disturbed by the idea of being separated from him forever. He understood the pain and agony his teacher was undergoing. And he became more sympathetic towards his teacher.

His school, too, now carried a different meaning. His books and lessons seemed old friends, whom he couldn’t give up. He realised with pain that how much French meant to him and regretted not being attentive in his classes earlier. Suddenly, he felt that the ‘difficult concepts’ had never actually been difficult.

The Last Lesson Understanding the Text

The Last Lesson Questions And Answers Class 12 NCERT Solutions Question 1.
The people in this story suddenly realise how precious their language is to them. What shows you this? Why does this happen?
Answer:
M. Hamel told the students and the villagers that henceforth only German would be taught in the schools of Alsace and Lorraine. Those who called themselves Frenchmen would neither be able to speak nor write it. He praised French as the most beautiful, the clearest and the most logical language in the world. He said that for the enslaved people that their language was the key out of prison. Only then the people realised the importance of their language. This shows people’s love for their own culture, traditions and country. Pride in one’s language reflects pride in the motherland.

Ncert Solutions For Class 12 English Flamingo The Last Lesson Question 2.
Franz thinks, “Will they make them sing in German, even the pigeons?” What could this mean?
Answer:
Alphonse Daudet’s ‘The Last Lesson’ very prominently raises the question of linguistic and cultural hegemony of the colonial and imperial powers and their lust for controlling the world and influencing their cultures and identities. Enforcement of German on the defeated nation was a way of realising this. The order to teach German rather than . French in schools was released.

Franz is flabbergasted on hearing this and understands that this order would deprive him of learning his mother tongue. He also wondered if the pigeons would have to coo in German. By compelling them to use a foreign language was like snatching away their language from them, which he felt would be unfair and unkind.

The language was as natural to them as cooing is to the pigeon. So compulsion to speak another language is like dominating the force of nature and enslaving it. As it is next to impossible to alter the way pigeons sing, the same way, it is difficult for people to accept a language which is forcibly imposed on them. Adopting a new language causes pain and discomfort.
Or
This sentence could possibly mean that however hard the authorities try to embed German language in the culture of Alsace and Lorraine, the natural status of French for them, will remain unchanged. French flows in the air and the entire place is full of its effect. Even though they train students in German, the basic mode of communication would remain unchanged like the cooing of the pigeons.

The Last Lesson Talking about the Text

Last Lesson Question Answer Class 12 NCERT Solutions Question 1.
“When people are enslaved, as long as they hold fast to their language it is as if they had the key to their prison.” Can you think of examples in history where a conquered people had their language taken away from them or had a language imposed on them?
Answer:
Some examples of the native language taken away from its people and/or imposition of the language of the conqueror are:

• Portuguese becoming the lingua franca of Angola.
• English imposed on the various Celtic people.
• Spanish imposed on the Basques and the Catalans.
• Turkish imposed on the Kurds.

The Last Lesson Class 12 NCERT Solutions Question 2.
What happens to a linguistic minority in a state? How do you think they can keep their language alive? For example:
Punjabis in Bangalore
Tamilians in Mumbai
Kannadigas in Delhi
Gujaratis in Kolkata
Answer:
A linguistic minority in a state does not have as much liberty to exercise linguistic skills as the natives of the state. They initially try to learn the jargons in order to cope with the day-to-day activities and finally begin to understand the native language with regular interaction. At the workplace and educational organisations, English or the link language helps a lot to cope with the work and learning process. But when it comes to understanding the basic norms of the society, in order to socialise, one does face a sort of linguistic barrier during communication.

To keep their language alive, the linguistic minorities can form small communities where . they can celebrate their festivals as per their traditions. Moreover, they can continue to speak their native language at their homes in order to make their children learn the language. People must even try to visit their native places at regular intervals in order to stay close to their roots.

The Last Lesson Ncert Solutions Class 12 Question 3.
Is it possible to carry pride in one’s language too far? Do you know what ‘linguistic chauvinism’ means?
Answer:
Yes, it is possible to carry pride in one’s language too far if one is fond of one’s own language at the cost of belittling of other languages. Indifference towards other languages is not healthy for any democracy like India.

When the sense of belonging to one’s own language crosses the thin line between ‘pride’ and ‘proud’, it becomes linguistic chauvinism. If people feel good about their language and traditions, they must have tolerance for other languages too. Everybody has the right to follow the religion as well as speak the language as per their choice.

The Last Lesson Working with Words

Notice the underlined words in these sentences and tick the option that best explains their meanings.
(a) “What a thunderclap these words were to me!”
The words were
(i) loud and clear.
(ii) startling and unexpected.
(iii) pleasant and welcome.

(b) “When people are enslaved, as long as they hold fast to their language it is as if they had the key to their prison.”
It is as if they have the key to the prison as long as they
(i) do not lose their language.
(ii) are attached to their language.
(iii) quickly learn the conqueror’s language

(c) Don’t go so fast, you will get to your school in plenty of time.
You will get to your school
(i) very late.
(ii) too early.
(iii) early enough.

(d) I never saw him look so tall.
M. Hamel
(i) had grown physically taller.
(ii) seemed very confident.
(iii) stood on the chair.
Answer:
(a) (ii) startling and unexpected.
(b) (ii) are attached to their language.
(c) (iii) early enough.
(d) (ii) seemed very confident.

The Last Lesson Extra Questions and Answers

The Last Lesson Short Answer Questions

Last Lesson Question Answers Class 12 NCERT Solutions Question 1.
How was the scene in the school, on the morning of the last lesson, different from thaton other days?
OR
How was M. Hamel’s class different the day Franz went late to school?
Answer:
Generally, there would be a great bustle, closing and shutting of desks, lessons repeated loudly in unison, rapping of the teachers’ ruler on the table, all of which could be heard out in the street. But that everything was quite different. There was no noise. All were in their seats, Franz walked in late and M. Hamel let him calmly. He then noticed that his sir was dressed in his best clothes and there were the elders of the village seated in the class. It was a bit later that Franz realised why the day was different. It was their last French lesson.

The Last Lesson Class 12 NCERT Solutions Question 2.
How does M. Hamel pay a tribute to the French language?
OR
What did M. Hamel tell them about the French language? What did he ask them to do and why?
Answer:
M. Hamel went on to talk about French language. He told that it was the most beautiful language of the world. It was the clearest and the most logical of all languages. He asked the people to guard it among themselves and never forget it. As long as people ‘hold fast to their language, they have the key to freedom’.

Question 3.
One order from Berlin changed the scenario of the school. Comment.
Answer:
The order from Berlin led to the announcement that French would not be taught anymore, and instead, German would be taught by a new master. This was to be their last French lesson. The class was quiet as it was a Sunday morning with no hustle and bustle. The teacher, M. Hamel was patient and calm but inwardly emotional. He was in his special dress. The sad villagers were sitting on the last benches like the other students and the teacher explained the lesson very patiently.

Question 4.
“What a thunderclap these words were to me!” Which were the words that shocked and surprised little Franz?
Answer:
M. Hamel said, “My children, this is the last French lesson I shall give you. The order has come from Berlin to teach only German in the schools of Alsace and Lorraine. The new master will come tomorrow. This is your last French lesson. I want you to be attentive”. These words of his teacher were a thunderclap for Franz.

Question 5.
Who did M. Hamel blame for the neglect of learning on the part of boys like Franz?
Answer:
M. Hamel blamed the parents for the neglect of learning of French language as they engaged the boys in farm work. He also blamed himself to some extent as he too assigned the work of gardening to boys like Franz. He also gave them a holiday whenever he wanted to go for fishing.

Question 6.
“This is your last French lesson.” How did Franz react to this declaration of M. Hamel?
OR
How did Franz react to the declaration that it was their last French lesson?
Answer:
The announcement made by M. Hamel left a great impact not only on Franz but all the other citizens. Franz was shocked to hear that M. Hamel was leaving and that it was his last lesson. He realised that he would not be able to read and speak his own mother tongue and regretted his lack of interest and carelessness.

Question 7.
How did M. Hamel say farewell to his students and the people of the town?
Answer:
M. Hamel looked very pale and tall when he stood up in his chair. All the students were quiet. The village people old Hauser, the former Mayor, the former postmaster and several others were present in the schoolroom. The teacher told the villagers that French was the most beautiful language in the world. He ended the lesson by writing Vive La France on the blackboard. He made a gesture with his hand to indicate that the school is dismissed and students could go home.

Question 8.
Why had the bulletin board become a centre of attention during the last two years?
Answer:
For the past two years, the news of lost battles, the draft and the orders of the commanding officer were displayed on the bulletin board. People thronged the bulletin board to read all this information. This was the reason why it had become a centre of attention.

Question 9.
What was tempting Franz to keep away from school ‘that morning’?
Answer:
Franz was supposed to learn participles as part of his schoolwork, which he had not done. Therefore, he was afraid of being scolded by M. Hamel. Also, he wanted to spend the day outdoors as it was warm and bright. The sight of the chirping birds and the Prussian soldiers drilling appealed to him more than the rules of participles.

Question 10.
What was unusual about M. Hamel’s dress and behaviour on the day of his last French lesson?
Answer:
Whenever Franz arrived late, he was met by an angry teacher. This time however, he was astounded when he was welcomed by a kind and polite M. Hamel. This was quite contrary to his nature. Moreover, he was dressed in his best clothes, a beautiful green coat, frilled shirt and an embroidered black silk cap, which he wore only on inspection and prize days.

Question 11.
Why had M. Hamel put on his fine Sunday clothes? Why were the old men of the village sitting there in the back of the classroom?
OR
Who occupied the back benches in the classroom on the day of the last lesson? Why?
Answer:
The back benches were occupied by the people of the village. Old Hansar, who had his three cornered hat, the former Mayor, the former post master and several other elders. They had come to express their respect and regard for M. Hamel and sorrow that he had to leave from their midst.

Question 12.
How did Franz perform when his turn came to recite? How did M. Hamel react?
Answer:
Franz’s name was called and he was asked to recite. Despite his best efforts, he got mixed up on the first words. He stood there holding on to his desk. His heart beat fast. And he did not dare look up. M. Hamel told him in a polite tone that he would not scold him as he was not the only one who neglected learning French. Many others in Alsace shared his fate because of procrastination. He said that every one had a great deal to reproach themselves with.

Question 13.
“We’ve all a great deal to reproach ourselves with.” Why did M. Hamel blame the parents and himself too for not showing due attention and care to the learning of French?
Answer:
M. Hamel did not hold Franz responsible for neglecting the learning of French. Most people of Alsace only pretended to be Frenchmen. But they could neither speak nor write their own language. The parents were not anxious to have them learn. They preferred to put children on a farm or at the mills to earn a little more money. He . even held himself responsible as he often sent his students to water his flowers instead of learning their lessons. He also used to give a holiday whenever he wanted to go fishing.

Question 14.
What does M. Hamel say about French language? What did he urge upon his students and villagers to do?
Answer:
M. Hamel talked at length about the French language. He considered French to be the most beautiful language in the world. It was the clearest and the most logical language too. He urged his students to guard it among themselves and reminded them never to forget it.

Question 15.
How does M. Hamel prove to be an ideal teacher?
Answer:
M. Hamel brings home the message of importance of love of mother tongue and patriotism. He explains things well and asks students to continue learning French even when he is gone. Hence, he proves to be an ideal teacher.

Question 16.
How was M. Hamel dressed differently that day? Why?
Answer:
M. Hamel wore a green coat, frilled shirt and black silk cap to the class. He announced that it was their last lesson in French and that German will be taught in the school in the future. He was proud of being French and was upset by occupation of Alsace by Germans. He was very attached to the town, the school and its people.

Question 17.
What had the narrator counted on to enter the school unnoticed?
Answer:
The teacher’s rap of the ruler, the banging of the desks, and the lessons repeated would be so loud that it could be heard in the street. The author thought this background would be a shield and he could enter the school unnoticed.

Question 18.
What changes did the order from Berlin cause in the school?
Answer:
The order from Berlin directed schools in the districts of Alsace and Lorraine in France to teach German instead of French.

Question 19.
Why were the elders of the village sitting in the classroom?
Answer:
The elders of the village came to the classroom to attend the last lesson of French in the school as a mark of respect to the French teacher, Mr Hamel who had been teaching there for the last forty years. These elders had not studied well, and could not read and write their mother tongue, French and so as it was the last opportunity for them, they came to attend the class.

Question 20.
How did Franz react to the declaration that it was their last French lesson?
Answer:
Franz was shocked and sad when he heard this news. Suddenly, he developed a liking for his language and was keen to learn French. He was remorseful for not learning well in the past and was sad that his teacher, Mr Hamel would go away.

Question 21.
What did Franz wonder about when he entered the class that day?
Answer:
He wondered why the classroom was still with no great bustle, the sound of desks opening and closing, lessons being repeated in unison, very loudly and M. Hamel’s great ruler rapping on the table.

Question 22.
Why was Franz not scolded for reaching the school late that day?
Answer:
Franz was not scolded that day as the situation was different than the other days. It was the last lesson in French by M. Hamel, who taught for forty years there. He regretted neglecting his classes earlier and wanted to compensate on the last day, before he left.

Question 23.
How were the parents and M. Hamel responsible for the children’s neglect of the French language?
Answer:
Parents were never keen or anxious to make their children learn French. They rather made them work in the fields or mills. Mr Hamel also lacked sincerity. He made the children water his garden during class hours or dismissed his class when he wanted to go for fishing.

Question 24.
“We’ve all a great deal to reproach ourselves with”, said M. Hamel. Refer to the context and explain what he wanted to convey to his students.
Answer:
M. Hamel wanted to convey to his students that still no loss has caused. If they desire, they can do a lot. Further, he advised them to move on and not to look back. He boosted the morale of his students by saying that though they have to blame themselves for not attending the school and he himself had to blame and disgrace himself for giving the holiday to students but hoped that they could mend their ways.

The Last Lesson Long Answer Questions

Question 1.
What is ‘linguistic chauvinism’? Analyse the order from Berlin in this light. How do you justify M. Hamel’s views about French and the new-found love of the people towards their language?
Answer:
Carrying pride in one’s language too far leads to ‘linguistic chauvinism’. We can analyse the order from Berlin in this light. It is nothing but a pure example of linguistic chauvinism. The imposition of German language over the French-speaking population can’t be justified at all. It is the worst kind of colonialism.

M. Hamel’s love for French is genuine. The shocking order from Berlin arouses patriotic feelings in him. He loves French and feels it to be the most beautiful language in the world. He calls it the clearest and the most logical language too. He regrets that the people of Alsace did not pay much heed to the learning of this great language. He asks the people to safeguard it among themselves.

It is the key to their unity and freedom. The people of Alsace, particularly the village elders, suddenly realise how precious their language is to them. Students like Franz too are not immune to patriotic feelings. Franz feels sorry for neglecting the learning of French. He hates the idea of German language being imposed on them. He remarks sarcastically, “Will they make them sing in German, even the pigeons?” The last lesson was so impactful that it helped to revive the love for the language among the people of Alsace.

Question 2.
How can you estimate M. Hamel as a man with a ruler and as a man with a gesture?
OR
How does M. Hamel prove to be an ideal teacher?
Answer:
In ‘The Last Lesson’, Alphonse Daudet presents M. Hamel’s character with due sympathy and respect. Initially, he is presented in the mould of a traditional schoolmaster. He used his terrible ruler under his arm. Everyone could hear the rapping of the ‘great ruler’ on the table even outside in the street. Franz reminds us ‘how cranky’ M. Hamel was. The students used to dread their schoolmaster. Franz was scared of being scolded as he had not prepared his lesson on participles. For a moment, he even thought of running away from school. Mr Hamel was a hard task master. He maintained discipline in the class.

The other side of Mr Hamel’s character is seen after the order from Berlin came. He had been transformed now. He became soft and gentle towards his students. He didn’t scold Franz for coming late. He did not even use his ruler when little Franz got mixed up and confused when his turn to recite came. He declared that it was his last lesson in French as from the next day German would be taught in the schools of Lorraine and Alsace. He would leave the next day. A new teacher would come in his place. He wore his best dress in honour of the last lesson.

M. Hamel was given respect not only by his students but even by the village elders. He was totally dedicated to the cause of teaching. He had been teaching for forty years in the same school. The village elders came to pay their respect to such a grand teacher. They sat on the back benches to listen to his last lesson.

M. Hamel loved France and French from the depth of his heart. He regarded French as the most beautiful language in the world. He told the people to guard it among themselves and never to forget it. On hearing the sound of trumpets of the Prussian soldiers under his window, patriotic feelings overpowered him. He mounted the chair and tried to speak, however something choked him. He wrote “Vive La France” with a piece of chalk on the blackboard and dismissed the class.

Question 3.
Write a character sketch of Franz.
Answer:
Franz was a student of a school in Alsace. His schoolmaster was M. Hamel. Franz was not brilliant. Franz enjoyed spending time out of doors. He liked the warm and bright day, and loved to listen to the chirping of the birds and watching the drilling of the Prussian soldiers. He preferred this instead of being in the classroom. He didn’t prepare his lesson on participles. When he was asked to recite, he got mixed up and confused. He was not excited to go to school and did not show any interest in M. Hamel’s teaching.But he was scared of M. Hamel’s scolding. He always dreaded the great ruler that M. Hamel kept under his arm. Franz knew how ‘cranky’ M. Hamel was.

However, Franz was forced to change his opinion about M. Hamel. An order came from Berlin pronouncing that German language would be taught in the schools of French districts of Alsace and Lorraine. On knowing that it was the last lesson that Mr Hamel was going to deliver, his views about him changed. He started respecting the man who had spent forty years in the same school. He felt sorry for not learning French.

He shared M. Hamel’s views about French. It was the most beautiful language in the world. Franz sarcastically remarked, “Will they make them sing in German, even the pigeons?” After the last lesson, his views about French took a patriotic turn. He listened to M. Hamel’s last lesson with rapt attention and dignity, and regretted having been careless and inattentive.

Question 4.
Our native language is part of our culture and we are proud of it. How does the presence of village elders in the classroom and M. Hamel’s last lesson show their love for French?
OR
Our language is part of our culture and we are proud of it. Describe how regretful M. Hamel and the village elders are for having neglected their native language, French.
Answer:
M. Hamel told the students and villagers that henceforth only German would be taught in the schools of Alsace and Lorraine. Those who called themselves Frenchmen would neither be able to speak nor write it. He praised French as the most beautiful, the clearest and the most logical language in the world. He said that for the enslaved people, their language was the key out of prison. Only then the people realised the importance of their language. This shows people’s love for their own culture, traditions and country. Pride in one’s language reflects pride in motherland.

When Franz jumbled while it was his turn to answer, M. Hamel expresses regret at the pathetic state of the language among the folks of Alsace. He regrets the fact that everyone chose to procrastinate. Also, he felt that the parents preferred their children worked in the farms for that extra income. He worried that the Germans would ridicule them for being incapable of speaking and writing their language. He blames everyone including himself for being careless, lazy and Lackadaisical (unenthusiastic and lack of determination).

Question 5.
Everybody during the last lesson is filled with regret. Comment.
Answer:
Everybody during the last lesson is filled with regret. There was a general sadness about not being able to utilise the opportunities of learning French when it was easily accessible. Franz wished that he had attended classes more often and regretted not being attentive in his classes earlier. He suddenly found his lessons more interesting and easy. The villagers, who were sitting at the usually empty back benches and had come to show their respect and gratitude to M. Hamel, regretted not going to school more than they did.

The order also brought about a great change in the feelings of the people towards their country and their native language. M. Hamel regretted sending his students to water his flowers instead of learning their lessons. He also regretted giving holiday to students whenever he wanted to go on fishing.

Question 6.
What changes did the narrator find in the school when the order from Berlin came?
Answer:
The order from Berlin prohibited teaching of French in the schools of Alsace and Lorraine. Instead, German was to be taught in the schools. Franz was late for school that day. He noticed that the hustle and bustle was missing. There was no opening and closing of desks, no repetition of lessons or rapping of the teacher’s ruler on the table could be heard. It was all very quiet and still.

Franz was further surprised because, instead of meeting an angry teacher, he was welcomed by a kind and polite teacher, who was dressed in his best clothes, a beautiful green coat, frilled shirt and an embroidered silk cap, which he wore only on inspection and prize days. The back benches were occupied by the village people who never came to school, as they were more concerned about their livelihood. He was further astounded to know that M Hamel was going to teach his last lesson that day.

Question 7.
Justify the title of the story, ‘The Last Lesson’.
Answer:
The title of the story, ‘The Last Lesson’ is self-relieving. The whole story revolves around the title. The beginning of the story serves as preparation for it. The title also conveys the central theme of the story—the fact that sometimes even the most precious things in our lives are taken for granted. The people of Alsace never gave much importance to the mother tongue, French.

They did not even insist their children to pay any attention to their language. They did not encourage regular attendance of their children in French classes. They preferred their children to work and earn, instead of studying. The order from Prussians made them realise the importance of their mother tongue. So they attend M. Hamel’s last lesson altogether. Thus, the title, ‘The Last Lesson’ is justified.

Question 8.
Write a character sketch of M. Hamel as a teacher.
Answer:
M. Hamel was a true French man who has been teaching French in the districts of Alsace and Lorraine for forty years. He loved his profession and was proud of his language, French. He had a deep sense of respect for his mother tongue. He considered French to be the most beautiful language of the world. As a teacher, he was very particular and strict in imparting knowledge to his students. When France was overtaken by Prussians, he was depressed because French was banned from being taught in the schools. While taking his last lesson, he tried his best to remain calm and composed.

His sorrow was evident in the way he was sitting in the class while his students were completing their writing assignment. He felt tormented at the fact that people had become indifferent to learning French and appealed to them to keep their language alive. He was a true patriot. He believed that mother tongue is a means of holding one’s identity and self¬respect. At the end of his last lesson, he writes ‘Vive La France!’ on the blackboard. This shows his love and concern for the people and the language of his country.

These NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.5 Questions and Answers are prepared by our highly skilled subject experts. https://mcq-questions.com/ncert-solutions-for-class-12-maths-chapter-13-ex-13-5/

NCERT Solutions for Class 12 Maths Chapter 13 Probability Exercise 13.5

Question 1.
A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of
(i) 5 successes?
(ii) at least 5 successes?
(iii) at most 5 successes?
Solution:

Question 2.
There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective item?
Solution:
This is a Bernoulli trial with n = 4
Here success is getting a doublet
p = probability of success
= \(\frac { 6 }{ 36 }\) = \(\frac { 1 }{ 6 }\)
q = 1 – p = 1 – \(\frac { 1 }{ 6 }\) = \(\frac { 5 }{ 6 }\)
Let X be the number of successes. Then X is a binomial distribution with

Question 3.
A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability, of two successes.
Solution:
This is a Bernoulli trial with n = 10
Here success is the inclusion of a defective item
p = probability of success = \(\frac { 5 }{ 100 }\) = \(\frac { 1 }{ 20 }\)
∴ q = 1 – p = 1 – \(\frac { 1 }{ 20 }\) = \(\frac { 19 }{ 20 }\)
Let X be the number of defective item in the sample.
Then X is a binomial distribution with

Question 4.
Five cards are drawn successively with replacement from a well- shuffled deck of 52 cards. What is the probability that
(i) all the five cards are spades?
(ii) only 3 cards are spades?
(iii) none is spade?
Solution:
This is a Bernoulli trial with n = 5
Here success is getting a spade
p = probability of success = \(\frac { 13 }{ 52 }\) = \(\frac { 1 }{ 4 }\)
∴ q = 1 – p = 1 – \(\frac { 1 }{ 4 }\) = \(\frac { 3 }{ 4 }\)
Let X denote the number of spade cards.
Then X is a binomial distribution with
n = 5, p = \(\frac { 1 }{ 4 }\) and q = \(\frac { 3 }{ 4 }\)
P(X = x) = \({ }^{n} \mathrm{C}_{x} p^{x} q^{n-x}\)
i. Probability of 5 spades = P(X = 5)

Question 5.
The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs.
(i) none
(ii) not more than one
(iii) more than one
(iv) at least one will fuse after 150 days of use
Solution:
This is a Bernoulli trial with n = 5
Here success is the bulb fuse after 150 days.
∴ p = probability of success = 0.05
∴ q = 1 – p = 1 – 0.05 = 0.95
Let X be the number of bulbs fuse after 150 days
Then X is a binomial distribution with n = 5, p = 0.05 and q = 0.95
∴ P(X = x) = \({ }^{n} \mathrm{C}_{x} p^{x} q^{n-x}\)
(i) When X = 0, P(X = 0)
\({ }^{5} \mathrm{C}_{0}(0.05)^{0}(0.95)^{5-0}=(0.95)^{5}\)

(ii) X is not more than 1 i.e. X ≤ 1
∴ P(X ≤ 1) = P(X = 0) + P(X = 1)
= \({ }^{5} \mathrm{C}_{0}(0.05)^{0}(0.95)^{5}+{ }^{5} \mathrm{C}_{1}(0.05)(0.95)^{4}\)
= (0.95)⁴ [0.95 + 5 x 0.05]
= (0.95)⁴ x 1.2

(iii) X is more than 1 i.e. X > 1
P(X > 1) = 1 – P(X ≤ 1)
= 1 – (0.95)⁴ x 1.2 from (ii)

(iv) X is atleast 1 i.e. X ≥ 1
P(X ≥ 1) = 1 – P(X = 0)
= 1 – (0.95)⁵ from (i)

Question 6.
A bag consists of 10 balls each marked with one of the digits 0 to 9. If four bails are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?
Solution:
This is a Bernoulli trial with n = 4.
Here success is the ball drawn is marked with number 0
p = probability of success = \(\frac { 1 }{ 10 }\)
∴ q = 1 – p = 1 – \(\frac { 1 }{ 10 }\) = \(\frac { 9 }{ 10 }\)
Let X be the number of balls marked with 0
Then X is a binomial distribution with

In an examination, 20 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 head, he answers ‘true’, if it falls tail, he answers ‘false’. Find

Question 7.
In an examination, 20 questions of true – false type are asked. Suppose a student tosses 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.
Solution:
Probability that student answers a question true = \(\frac { 1 }{ 2 }\)
i.e., when a coin is thrown, probability that a head is obtained = \(\frac { 1 }{ 2 }\)
Probability that his answer is false = \(1-\frac { 1 }{ 2 }\) = \(\frac { 1 }{ 2 }\)
Probability that his answer at least 12 questions correctly = P (12) + P (13) + P (14) +…….. P (20)

Question 8
Suppose X has a binomial distribution \(B\left( 6,\frac { 1 }{ 2 } \right) \). Show that X = 3 is the most likely outcome.
(Hint: P (X = 3) is the maximum among all P (Xi), xi. = 0,1,2,3,4,5,6)
Solution:
\({ \left( \frac { 1 }{ 2 } +\frac { 1 }{ 2 } \right) }^{ 6 } \)

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?
Solution:
P = \(\frac { 1 }{ 3 }\). q = 1 – P = \(1-\frac { 1 }{ 3 }\) = \(\frac { 2 }{ 3 }\)

Question 10.
A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is \(\frac { 1 }{ 100 }\) . What is the probability that he will win a prize?
(a) at least once,
(b) exactly once,
(c) at least twice?
Solution:
Probability that the person wins the prize = \(\frac { 1 }{ 100 }\)
Probability of losing = \(1-\frac { 1 }{ 100 }\) = \(\frac { 99 }{ 100 }\)
(a) Probability that he loses in all the loteries

Question 11.
Find the probability of getting 5 exactly twice in 7 throws of a die.
Solution:
S = {1,2,3,4,5,6},n(S) = 6
A = {5} ⇒ n(A) = 1

Question 12.
Find the probability of throwing at most 2 sixes in 6 throws of a single die.
Solution:
When a die is thrown,
Probabiltiy of getting a six = \(\frac { 1 }{ 6 }\)
Probabiltiy of not getting a six = \(1-\frac { 1 }{ 6 }\) = \(\frac { 5 }{ 6 }\)
Probabiltiy of getting at most 2 sixes in 6 throws of a single die = P (0) + P (1) + P (2)

Question 13.
It is known that 10% of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles 9 are defective?
Solution:
This is a Bernoulli trial with n = 12
Here success is selecting a defective item.
p = probability of success = \(\frac { 1 }{ 100 }\) = \(\frac { 1 }{ 10 }\)
∴ q = 1 – p = 1 – \(\frac { 1 }{ 10 }\) = \(\frac { 9 }{ 10 }\)
Let X is the number of defective items. Then X is a binomial distribution with

Question 14.
In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is
(a) \({ 10 }^{ -1 }\)
(b) \({ \left( \frac { 1 }{ 2 } \right) }^{ 5 }\)
(c) \({ \left( \frac { 9 }{ 10 } \right) }^{ 5 }\)
(d) \(\frac { 9 }{ 10 }\)
Solution:
p = \(\frac { 1 }{ 10 }\)
q = \(\frac { 9 }{ 10 }\)
n = 5,
r = 0,
P(X = 0) = \({ \left( \frac { 9 }{ 10 } \right) }^{ 5 }\)
Option (c) is correct

Question 15.
The probability that a student is not a swimmer is \(\frac { 1 }{ 5 }\). Then the probability that out of five students, four are swimmers is:
(a) \({ }^{5} \mathrm{C}_{4}\left(\frac{4}{5}\right)^{4} \frac{1}{5}\)
(b) \(\left(\frac{4}{5}\right)^{4} \frac{1}{5}\)
(c) \({ }^{5} C_{1} \frac{1}{5}\left(\frac{4}{5}\right)^{4}\)
(d) None of these
Solution:
This is Bernoulli trial with n = 5
Here success is that student is a swimmer.
p = probability of success
= 1 – P (not a swimmer) = 1 – \(\frac { 1 }{ 5 }\) = \(\frac { 4 }{ 5 }\)
∴ q = 1 – p = 1 – \(\frac { 4 }{ 5 }\) = \(\frac { 1 }{ 5 }\)
Let X denote the number of swimmers in the group of 5 students
Then X is a binomial distribution with

These NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.3 Questions and Answers are prepared by our highly skilled subject experts. https://mcq-questions.com/ncert-solutions-for-class-12-maths-chapter-13-ex-13-3/

NCERT Solutions for Class 12 Maths Chapter 13 Probability Exercise 13.3

Exercise 13.3 Class 12 NCERT Solutions Question 1.
An urn contain 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 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?
Solution:
Urn contain 5 red and 5 black balls.
(i) Let a red ball is drawn.
probability of drawing a red ball = \(\frac { 5 }{ 10 }\) = \(\frac { 1 }{ 2 }\)
Now two red balls are added to the urn.
⇒ The urn contains 7 red and 5 black balls.
Probability of drawing a red ball = \(\frac { 7 }{ 12 }\)

(ii) Let a black ball is drawn at first attempt
Probability of drawing a black ball = \(\frac { 5 }{ 10 }\) = \(\frac { 1 }{ 2 }\)
Next two black balls are added to the urn
Now urn contains 5 red and 7 black balls
Probability of getting a red ball = \(\frac { 5 }{ 12 }\)
⇒ Probability of drawing a second ball as red
= \(\frac { 1 }{ 2 } \times \frac { 7 }{ 12 } +\frac { 1 }{ 2 } \times \frac { 5 }{ 12 } =\frac { 7 }{ 24 } +\frac { 5 }{ 24 } =\frac { 12 }{ 24 } =\frac { 1 }{ 2 }\)

Ex 13.3 Class 12 Maths Ncert Solutions Question 2.
A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.
Solution:
E₁ : first bag is selected
E₂ : second
bag is selected A : ball drawn is red
E₁ and E₂ are mutually exclusive & exhaustive events
P(E₁) = P(E₂) =
P(A|E₁) = \(\frac { 4 }{ 8 }\) and P(A|E₂) = \(\frac { 2 }{ 8 }\)
By Bayes’ theorem

Class 12 Maths Chapter 13 Ex 13.3 NCERT Solutions Question 3.
Of the students In a college, it is known that 60% reside In hostel and 40% are day scholars (not residing In hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% of day scholars attain A grade in their annual examination. At the end of the year, one student Is chosen at random from the college and he has an A- grade what Is the probability that the student is a hostlier?
Solution:
E₁ : student is a hosteler
E₂ : student is a day scholar
A : student attains A grade.
E₁ and E₂ are mutually exclusive and exhaustive

Question 4.
In answering a question on a multiple choice test, a student either knows the answer or 3 guesses. Let \(\frac { 3 }{ 4 }\) be the probability that he knows the answer and \(\frac { 1 }{ 4 }\) be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability \(\frac { 1 }{ 4 }\) . What is the probability that the student knows the answer given that he answered it correctly?
Solution:
Let E₁ : student knows the answer
E₂ : student guesses the answer
A : student answers correctly
E₁ and E₂ are mutually exclusive and exhaustive events.

Question 5.
A laboratory blood test is 99% effective in detecting a certain disease when it is, in fact, present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e. if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?
Solution:
Let E₁ : the person has a disease
E₂ : the person is healthy
A : the test result is positive
E₁ and E₂ are mutually exclusive and exhaustive events.

Question 6.
There are three coins. One is a two headed coin, another is a biased coin that conies up heads 75% of the time and third is an unbiased coin. One of the three coins is choosen at random and tossed, it shows head, what is the probability that it was the two headed coin?
Solution:
Let
E₁ : coin tossed is two headed
E₂ : coin tossed is biased
E₃ : coin tossed is unbiased
A: getting head
E₁ and E₂ and E₃ are mutually exclusive and exhaustive events.

Question 7.
An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of mi accident are 0.01, 0.03, 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?
Solution:
Let
E₁ : insured person is a scooter driver
E₂ : insured person is a car driver
E₃ : insured person is a truck driver and
A : insured person meets with an accident.
E₁ E₂ and E₃ are mutually exclusive and exhaustive events.
Total number of insured person
= 2000 + 4000 + 6000 = 12000

Question 8
A factory has two machines A and B. Past record shows that machine A produced 60% of the items of output and machine B produced 40% of the items. Further, 2% of the items produced by machine A and 1% produced by machine B 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 B.?
Solution:
Let
E₁ : item produced by machine A
E₁ : item produced by machine B
A : item is defective
E₁ and E₂ are mutually exclusive and exhaustive events

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 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group.
Solution:
Let E₁ : first group wins
E₂ : second group wins
A : introducing a new product
E₁ and E₂ are mutually exclusive and exhaustive events

Question 10.
Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads.If she gets 1,2,3 or 4, 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 1,2,3, or 4 with the die?
Solution:
When she gets 5,6, she throws a coin three times.
The events are
E₁ : getting 1, 2, 3, 4
E₂ : getting 5, 6
A : getting exactly one head
E₁ and E₂ are mutually exclusive and exhaustive events
P(E₁) = \(\frac { 4 }{ 6 }\) P(E₂) = \(\frac { 2 }{ 6 }\)
When she gets 1,2,3,4, .she throws a coin once.
P(A|E₁) = \(\frac { 1 }{ 2 }\)
When a coin is tossed 3 times, the sample space is {HHH, HHT,- HTH, HTT, THH, THT, TTH, TTT}
One head is obtained as {HTT, THT, TTH} P(A|E₂) = \(\frac { 3 }{ 8 }\)

Question 11.
A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that it was produced by A?
Solution:
Let E₁ : item produced by operator A
E₂ : item produced by operator B
E₃ : item produced by operator C
A : event of getting a defective item
E₁, E₂ and E₃ are pairwise disjoint and exhaustive events

Question 12.
A card from a pack of 52 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?
Solution:
Let E₁: lost card is a diamond
E₂ : lost card is not a diamond
A : getting 2 diamond cards from 51 cards
E₁, E₂ are mutually exclusive and exhaustive events

Question 13.
Probability that A speaks truth is 4/5. A coin is tossed. A reports that a head appears. The probability that actually there was head is
(a) \(\frac { 4 }{ 5 }\)
(b) \(\frac { 1 }{ 2 }\)
(c) \(\frac { 1 }{ 5 }\)
(d) \(\frac { 2 }{ 5 }\)
Solution:
Let E₁ getting a head
E₂: getting a tail
F : A reports ‘head occurred’
E₁ and E₂ are mutually exclusive and ex-haustive events

Question 14.
If A and B are two events such that A⊂B and P (B) ≠ 0, then which of the following is correct:
(a) P(A | B) = \(\frac { P(B) }{ P(A) }\)
(b) P (A | B) < P (A)
(c) P(A | B) ≥ P(A)
(d) None of these
Solution: