Probability Related

Arun, Hema, Lakshmi and John are students of Hindu College, Pattabiram. It was their first day in Class with Prof.A.R. as he is known in Hindu College.

Professor A.R with the Gang of four..

Prof.A.R., who  is the Professor and Head of the Department of Mathematics, teaches the following subjects to different classes.

  1. Mathematical Statistics

  2. Numerical Methods

  3. Operations Research

  4. Business Statistics.

These students had their first class on Mathematical Statistics and at the end of the day, before going home, they are sitting in the ground and discuss about Prof A.R. and his approach to teaching Statistics.

Let’s now listen in as they discuss and they are not aware that others are listening in on their discussion.

Hema

Lakshmi! How do you feel about today’s class by A.R?

Lakshmi

Some of our seniors had already warned me that A.R sir will be asking more questions than teaching. I think that’s what happened today.

Hema

Yes. I also felt that he was asking us a lot of questions> Moreover he repeated the same question several times knowing fully well that we could not answer that question.

Lakshmi

Yes. I really don’t know why he kept on repeating the question “What is Probability?”

Hema

Not only that. When we could not answer the question, he was asking “What is the probability of —– “. Only thing is he changed the problem each time and most of us could answer that question easily because those were the problems that we had solved in our 12th Standard Mathematics.

Lakshmi

But then he would repeat the question “What is probability?”. I wonder if there was any reason for him to repeat that question so many times or he was just finding a way to waste the time.

Hema

I think the two boys Arun and John are walking towards us. Perhaps they may join the discussion. Let’s ask them how they felt?

Arun

Hello, Hema! I think you did not like the way A.R started his class today.

Hema

I really don’t know what to say Arun. I think almost every one of us felt that way only. What about you?

Arun

Not everyone. John was really surprised to see how A.R sir was so patient in trying to make us understand the difference between the two questions he kept on asking us.

Lakshmi

What difference. He was only asking us questions on Probability again and again. Even when we answered, he never said whether the answer was correct or not. When we didn’t answer also, he was repeating the same question in another form.

John

I know that it was kind of irritating when he was repeating the same question. But then, after the third or fourth time, I understood that there was a big difference between the two questions.

Hema

So, you feel that we are all fools or we don’t understand English that well? What was the difference that you found out?

John

I also felt the same way to start with. But when I started writing down the questions, I could notice the difference. I am sure that none of you wrote down the questions.

Arun

Where was the need to write. He was only asking us two questions “What is probability” and “what is the probability of “

John

Yes. You want to tell me that there is no difference between the two questions?

Arun & Lakshmi

Yes. There is no difference.

Hema

Let me think. John, as you say, there is a difference. In the first question “the’ is absent. “What is “and ‘What is the —” . So definitely ‘the’ is missing. But what’s the big deal about it?

John

That’s what I am trying to find out. I think I may not be able to sleep today until find out the answer to that. Why not you also think about it and let’s discuss again tomorrow.

As it was getting late, everyone agreed to the suggestion and left for their home.

 

Day 2.

Arun Hema, Lakshmi and John reached the college earlier than usual so that they could meet and discuss before the classes began. Arun and Lakshmi had not bothered to think further as they had other things to do. Hema said that she thought hard but could not understand the difference between the two questions. John said after thinking about it for some time, He could make out that in the first question, the focus was more on the word probability. In the second question, the focus was more on the happening of an event such as getting a spade, getting a head, getting an even number on throwing a dice etc.

Before he could elaborate further, the bell rang and all of them went inside the class.

 In the evening, they met again to continue the discussion further.

 

John I don’t know whether I should cry or laugh after what happened in the class today.
Hema I also feel the same. I never expected that the answer would turn out to be so simple in the end.
Arun Yesterday, it was Ramesh who was naughty in the class. He was always making some odd comments and never looked serious enough to listen.
Lakshmi But today, as soon as the class began, he gave the answer which was right and made us all look silly in the end.
John Once Ramesh told the answer, it became obvious to everyone. Now I understand why A.R was asking us repeatedly the same two questions1.     What is probability?  2.     What is the probability of —- ?
Hema Every time, we answered the question “What is the probability of –?” correctly. But we never realized that the answer was always a number. So, we failed to connect. Now it’s clear that the answer to the question “what is probability?”,   the answer is probability is a number.
Arun Now, we can never be in doubt while answering the question ‘What is probability? ‘
John Before I could ask the question ‘does it mean any number can represent probability? ‘, Ramesh himself asked that question and also answered that only a number from 0 to 1 can represent probability.
Hema That was when I realised even A.R would smile in a class.
Lakshmi In school, we have been taught how to calculate the probability but never got an answer for the question ‘what is probability?’
Arun We have also never asked questions such as “What is Probability?’ or Why Probability?’ or ‘When & where probability?’. In fact, I realised that such questions exist only after being asked by A.R
John The way he connected all these questions and managed to get answers in bits and pieces from us, was totally different from other classes we have attended so far.
Lakshmi I wonder how he manages to take a class without going to the board even once. He was always facing us and was in direct eye contact.
Arun Finally, when he said ‘Probability is a measure of certainty in an uncertain world’ I could not understand the significance of the word ‘measure’. Can anyone help me?
John We measure height, weight, income etc and the measure is always expressed in terms of a number. Similarly, a measure of our understanding a subject is expressed in terms of the mark that we get. The mark is again a number. So, a number is always a measure. What it measures differs from case to case.
Hema That part of it is okay for me. But how do we say that ‘Probability is a measure of certainty?’. As we all know that we use probability only when we are not sure, i.e. uncertain.
Lakshmi I also agree with Hema. Suppose if you ask me whether I would come to college tomorrow, if I am certain then I would answer yes. If I am certain that I won’t come then I would answer ‘no’. Only when I am uncertain I would answer ‘probably’
John That is why he left us with the final question
Arun Can you repeat that question to me?

Hema

He asked us to explain the meaning of the answer ‘The Probability of getting a 5 when a pair of dice is thrown?’. I know the answer is 4/36 or 1/9. Tomorrow, we must explain its meaning. Then we may be able to understand how probability is a measure of certainty in an uncertain world.
At this point they decided that it was time to leave and continue the discussion the next day.

 

Day 3.

After a break of two days for the week-end, the Gang of four (Arun, Hema, Lakshmi and John) met again on Monday morning. Since they did have A.R’s class that day, they just proceeded to the class straightaway without stopping for a brief discussion. During the day, due to the absence of a lecturer, they had a lecture off and so they were let off free. Hence, they decided that it would be better to spend the time in the library.

As they came out of the library, John started the discussion. He said he thought hard and spent time reviewing the simple problems done on probability but could not find any reason to justify that probability measured certainty in an uncertain world.

To their surprise, Ramesh also joined the discussion. He chipped in saying that all of them had been mechanically finding answers to question on ‘What is the probability of —- ‘and that a majority of them (including himself) did not know what the answer really meant.

Lakshmi was stung by the criticism and was quick to retort saying that it was not the case with all.

Then Ramesh said he had talked to a few seniors about A.R’s classes and they had told him that in the coming week, there would be more penetrative questions.

Asked about the kind of questions to expect, Ramesh informed the Gang that they could expect the following question.

Question: When a student comes out after completing an exam, he says that there is a 75% chance (i.e. ¾) probability of his getting an A grade. Explain the meaning of ¾ here.

 Lakshmi was quick to respond that she had never thought such a question would be asked. She further added that all she knew was how to calculate probability of an event. She even said that the probability of a student getting a first class could never be ¾. Her logic was that you either get a first class or do not get a first class. Since there were only two possible outcomes, the correct answer could be only ½ and not ¾.

Hema agreed with what Lakshmi said, She said that the Mathematical definition of probability was very simple and clear. Probability is always the ratio of favourable cases to total number of cases. In an examination you either get a first class or not a first class. So, it had to be ½.

John was thinking. He said that in an examination, the possible outcomes were

  1. Failure

  2. Third class

  3. Second class

  4. First class

If that was considered then probability of a student getting first class could be ¼ but not ¾.

Ramesh was inclined to agree with what John said. But he knew that A.R had put such a question to the seniors.

At this confused state of mind, they decided to go home and meet again, the next day. They were hoping that they might get some divine help to clear the mystery.

Day 4.

As they alighted from the train, it started drizzling. Hence the gang of 4, as they were called, hurried to the class. To their utter surprise, Prof. A.R walked into the class during the first hour itself. As was his habit, he just took a quick survey of the class. After marking attendance, he became his usual self and started the class with the question whether any student had any doubt on what was explained in the previous class. There was absolute silence more out of fear or contempt as the students were aware that any response would only invite more questions and not clarification.

With a wry smile, he then asked if there were any questions from any student. The entire class was surprised when Balachandran (known as Bala) put up his hand. Bala had shown himself to be naughty and was more noisy than attentive in any class. He was closer to getting identified as the bad boy in the class already. Hence it was a matter of great surprise to all when he put up his hand saying that he had a question to be asked, Everyone was thinking why he was digging his own trap by daring to ask a question in A.R’s class let alone trying to answer a question.

As is his wont, A.R just smiled and let him ask the question. However, even he would never have dreamt that Bala would come up with the question he eventually asked.

When A.R told Bala that he was free to ask his question, Bala suddenly felt nervous. He looked around and took a few seconds to steady himself. Meanwhile A.R prompted him saying that he was free to ask whatever was in his mind and that it would be answered. Finally, Bala had the courage and energy to come out with his question. The moment the question was completed, there was a moment of stunned silence in the class. Even A.R was slightly taken aback but recovered quickly enough to smile back at Bala. The other students could hardly believe what they heard.

The question from Bala was this

“Sir, everything else is fine. But I am unable to understand this one thing. For me it appears that you are looking through the window and not at us while asking your questions. Why can’t you look at us and ask your questions?”

Bala went on to add that it was perhaps a reason that students were not interacting more with him.

Everyone in the class was looking at A.R anxiously. Ramesh was telling himself that he had never noticed this personally but had felt that something was strange. John was waiting for A.R to burst out in anger and ask Bala how he dared to ask such a question. Hema and Lakshmi could hardly believe what they heard? Many others were silently cursing Bala as they felt A.R would react strongly and everyone would feel the brunt.

A,R was equally surprised. But then the question brought out a genuine smile and seeing that everyone felt relieved. Now they were eagerly waiting for his response.

He had recovered quickly. With a smile on his face he asked Bala to sit down. He thanked Bala for pointing out this mannerism as he was not aware of it. He said that he himself had wondered many a time why he failed to strike a chord with the students in the class. He genuinely thanked Bala for pointing this out. He also suggested that if he happened to look through the window, Bala or anyone else could point it out so he could correct himself.

None of the students had expected this response. They were waiting for a reaction from A.R but got a genuine response. Suddenly, the atmosphere in the class changed. Now, they could look at each other and exchange a smile. Everyone was totally relaxed. Each one wanted to thank Bala for the change in the class environment.

For A.R, it was an important lesson learnt that day. He was always under the impression that he had been looking at the students straight in the eye. Had Bala not opened up, he would never have realized this flaw in his approach. That day brought about a change in his approach to the class also. As he started looking directly in the eyes of the students, they also started smiling more and naturally there was less tension. The result was even more astonishing. Students started enjoying his questions and attempted to answer the questions without fear.

It was perhaps the first time, A.R left the class with a genuine smile on his face and the students were also equally relaxed and responded with a genuine smile themselves.

Day 5.

By now, the gang had made it a practice to walk together. They discussed a new film that was released the previous day and went to the class. They were waiting for the 3rd hour as that was A.R’s class.

Strangely enough, A.R began his class without asking a question. While many students were happy, there were a few who were disappointed. As usual, A.R had finished a quick survey of the class and went to the board and wrote down ‘Different approaches to Probability”.

Then he turned towards the class and said there were basically three approaches to probability

  1. The frequency approach.

  2. The Classical or the Mathematical approach.

  3. The axiomatic approach.

He went on to explain how the human brain is the best statistician ever. While the students were wondering how could one say that the human brain was the best statistician, he asked if there was anyone in the class who could say ‘I have a favourite pen’ or ‘I have a favourite dress’ or similar statements. It was one question that every student in the class could answer immediately and responded positively.

Each student did have a favourite pen. Each student did have a favourite dress. Each student also had a dress that he or she considered unlucky.

Smiling at the enthusiastic responses, A.R then asked why and how did they identify a lucky pen or an unlucky dress. The students were back to their silent mode. After some minutes, it was Harini who broke the silence. She said she could make the identification based on past experiences. She added that for her green saree was a lucky dress because whenever she wore a green saree some good thing happened. To illustrate it further she went on to say that what was happening in the class was another example. As everyone looked at her curiously, she went on to say that she was wearing a green saree and that for the first time, she had felt confident enough to stand up and answer a question in the class. Immediately everyone started laughing. A.R also joined the fun and even encouraged the students to clap in appreciation.

Allowing the moment to die down, A.R waited. Then he asked Harini whether she decided it was her lucky saree the first time she wore it and as expected got the response ‘no’. He then asked her every time she wore a saree did she note down the colour and recorded the event that followed to decide whether it was her luck saree or not. Everyone laughed and agreed with Harini when she said hardly anyone did it.  At that, A.R said that was the exact point he wanted to convey when he had said that the human brain was the best statistician ever. He said the human brain had a knack of registering each and every happening in one’s life and then add it up to bring out at the right time. This is what we call experience.

This is known as the frequency approach, he explained. The first time, no one notices it. The second time it happens, it makes you think and the third time it happens, you start noticing it and start analysing. The more it repeats, the more you are convinced. Thus, the frequency approach measures the certainty in an uncertain world. It is because of this, one starts saying I know for sure that if I wear the green saree 10 times to attend an interview, I will succeed 7 times or you turn around say that the probability of my success in an interview when I wear a green saree is (7/10) or 0.7.

Thus, while success in any given interview was uncertain, the individual was certain that if she wore a green saree her probability of success would be 0.7.

AS the bell rang, A.R left the class and the students were silently mulling their thoughts over the way he had explained the whole concept.

The gang had a very short meeting in the evening. Hema was not in the mood to join and stay for a long discussion. Lakshmi also said that she would go with Hema as they both came from the same place. They told each other that they could understand the concept of a favourite pen or dress but still felt they would not be able to explain the statement ‘There’s a 75% chance of my getting grade A’

Everyone agreed that it needed further explanation and left.

Day 6.

Arun had missed the train. By the time the next train arrived, it would be late for the class. So, the other three decided to go to the class straight. There were one or two students who were trying to complete the problems given by the other professors. They were busy exchanging notes and copying from each other’s notebook. Soon, it was time for the regular classes and the bell rang.

  1. A.R’s class was slated for the afternoon only. It was the first hour after the break. Many students were feeling sleepy and tired. It was A.R’s first class in the afternoon.

Since he had known that it was always a difficult time to discuss anything serious, he asked if any student can sing a song. All the students were taken by surprise and one or two hands went up slowly. A.R asked Nalini to sing a song of her choice. Once the song was completed and the students applauded, A.R began his lecture.

He started by asking if anyone in the class could explain the statement that the probability of his or her attending class the next day is (3/5). Arun looked at the other members of the gang with a sly smile.

Bala answered by saying that using the mathematical approach, out of a total of 5 possible outcomes, 3 were in favour of his attending the class and hence the probability would be (3/5). When A.R asked him to list the 5 possible outcomes, he could not justify his answer. So, A.R chipped in saying that you would either attend the class or you would not. Thus, there would be only two possible outcomes. So, he said the probability could be only ½ and many students nodded.

He then came back to his question again. This time he asked if the statement was wrong and when the students nodded, he asked how many people made statements like

  1. Probability of my going for shopping tomorrow is 2/3.

  2. Probability of Pataudi winning the toss next time is 4/7.

  3. The probability that it would rain next Wednesday is 1/45.

He gave time for the students to write down these questions and waited for someone to answer any of the questions.

After some 15 minutes of silence, Nalini stood up hesitatingly and said

“Sir can I explain it this way”. As A.R prompted her to proceed, she said

      My three options are

  1. I want to go for shopping

  2. I don’t want to go for shopping

  3. I want to go for shopping.

Everyone laughed at this and said in a chorus she was in fact saying only two options. To their surprise, A.R told her that she was in the right direction but the answer in the present form was not correct. He gave her time to think more and correct her answer if she could.

The class went silent. Nalini said she could not offer a different explanation and sat down. It was Bala’s turn again to surprise the class.

Bala was an ardent follower of the game of cricket. So, he stood up to explain the statement that the probability of Pataudi winning the toss the next time being 4/7.

He said in the 28 tests that Pataudi went out to toss so far, he had won the toss 16 times and based on that statistics, it could be safely said that the probability of Pataudi winning the toss the next time would be 4/7. As A.R was also a vivid follower of cricket and went through all the newspaper articles, he just nodded his head.  He then commented that the frequency approach to probability helped in obtaining a measure of probability in situations like this.

That made Nalini and a few others think hard. After a while Nalini stood up again. She said the next day was a Thursday. She was mentally going through the pattern and found that she had gone for shopping or taken for shopping more on Thursdays than any other day of the week. So, she could certainly say that the probability of going for shopping would be more than ½ but she was yet to figure out the number 2/3.

That prompted Hema to ask a question. She said in the cases dealt so far, there was a history (either written or mentally recorded) and so one could say a value for probability. In the case of the examination, there was no previous History as she would be writing that examination for the first time. So how could she explain the statement that the probability of her getting grade A is 0.75?

As it was time for the class to be over, A.R said he would take up that question again in the next class.

The gang met in the evening as usual. Arun started saying he could not understand whatever A.R and others had said about pattern in finding an answer to a question on Probability. He said he had felt confident on working problems related to probability earlier. It was just applying some rules of Mathematics and he was good at that. So he had felt that it would always be easy for him to get answers quicker than others. But today’s discussion had led him to rethink.

John then asked him a question. He asked Arun if he had ever missed a class by being absent for whatever reason. Arun replied saying that he had never achieved 100% attendance so far and invariably he had missed classes either due to illness or some other reason. After a pause he added that most of the time he had missed the classes on Mondays compared to other days of the week. Continuing his loud thinking further, he said that the last 7 occasions he missed the class happened to be 3 Mondays, 1 Tuesday, 1 Thursday, 1 Friday and 1 Saturday. Based on that he could now tabulate the probability of his missing a class on different days as

          

Day of the week

Prob. Of missing the class

Monday

3/7

Tuesday

1/7

Wednesday

0

Thursday

1/7

Friday

1/7

Saturday

1/7

Lakshmi thanked Arun because she could now understand how pattern helped in answering questions on Probability. She said she could work out her patter because she had noted in her diary all particulars and she has her diary for the last 5 years. That could give her a better insight and she could come up her own table.

As it was getting late, they left to catch the train back home.

Day 7.

The next morning each member of the gang had come with a table as prepared by Arun, the previous evening. Everyone agreed that it was fun going through the past events and looking for a pattern. They further agreed that when they sat down to write things, each one could come across a pattern in several activities. For example, Hema Lakshmi said that she had missed the classes for 18 days in the last two years and that it was 3 days on each day of the week. Thus her probability of missing a class any day of the week turned out to be 1/6.  Hema came up with the information, she had gone for shopping 10 days in a year in each of the last three years and 20 of them were only on Saturdays and 10 on Sundays. So, the probability of her going for shopping on a Saturday would be 2/3 and on a Sunday, would be 1/3 and the other days it would be zero. It was John who reminded them that they were yet to explain the question relating to getting grade A in the examination.

When A.R started the class, he went on to discuss the Mathematical Approach to Probability and the next 30 minutes were spent on working some direct problems relating to calculating probability using the mathematical approach. Then Hema brought up the question of Grade A related problem.

A, R waited for someone in the class to respond. Arun stood up and told the class about the table he had worked out the previous day. Lakshmi also shared her table but eventually agreed that they could not explain or connect it to the question on grade.

A.R. smiled and acknowledged the work done by the gang. He said one explanation could be this.

The student who said the probability of his getting grade A in the exam is ¾, could have said because he felt that if 4 different teachers valued his paper, 3 of them would give him grade A. He was not sure that any teacher who valued his paper would give him grade A (that was the uncertainty) but if 4 different people valued his paper then 3 of them would certainly give him grade A (the certainty).  Then he asked the students to think if there could be any other explanation and left the class at that.

When the gang met in the evening they greeted each other with a sheepish smile. They agreed that the final explanation turned out to be so simple. Lakshmi reminded the others that A.R. had asked them if there could be any other plausible explanation. John said another possible explanation could be that the paper was so easy that on average 3 out of 4 students who had taken the examination would get grade A. Ramesh said that there could be another explanation also. He said that if he was asked to answer the same question paper then 3 out of 4 times he would get grade A.

Slowly but surely, they had begun to appreciate A. R’s way of teaching though most of them did not like his putting a lot of questions to the students. They were wondering if A.R would be able to finish the syllabus at this rate.

Just as they were about to leave, one of the senior students came to the and asked what they would say on the way A.R. taught the subject. They told him that the initial fear and hesitation is now gone but were afraid that the syllabus may not be completed in time. Hearing this, he laughed and said that they would be surprised to see that A.R. was the first one to complete the syllabus and would even give some of his classes to other staff and help them complete the syllabus. While others would conduct special classes and take extra classes to complete the syllabus, he assured them that A.R would never take an extra hour for completing the syllabus.

The gang could hardly believe what they heard. All they could do was to wait for the term to be over so that they could know.

With that in mind, they left for the day.

Day 8.

As there was nothing new to discuss, the gang went straight to the class and shared jokes with others who were already there in the class.  In response to q question from two or three of them, Ramesh gave his explanation to the question of grade and probability.

As the bell rang, they stood up for the morning prayer and then waited for the classes to start.

During the second hour, A.R. walked in. As he was surveying the class, Ramesh and Hema put up their hands to explain the answer. When they completed, A.R just nodded and without any further question asked them to sit and said he would proceed to the next topic. Though they were pleased, they were also wondering why they were left without any further questions,

A.R. began by saying that the question ‘What is probability’ had been discussed in detail but there were other questions to be answered. He listed the questions

  1. What is probability

  2. Why probability.

  3. When probability.

  4. How probability.

He asked the question ‘Why probability’ and then without waiting for anyone to respond, started giving the reasons himself.

He said that the human life was so unpredictable that no one could confidently say that he or she would be alive after 5 minutes but everyone had a plan as to what they would do after 10 years, 20 years and so on.  As the students nodded their heads, he continued with the statement, no one wanted to deal with uncertainties and only cared about certainty. Thus, each person was living in a world of uncertainties but still believed what would happen in his or her personal life was a certainty and hence planned for the next 30 or 40 years. Otherwise how people could think of a marriage, building a house, career development etc, he asked.

He concluded saying that is where probability as a measure of certainty in an uncertain world was required.

From the overall discussions that took place in the last several classes, most of the students were aware of when. It was obvious to everyone that probability came into play only when there was some kind of an uncertainty – whether it was choosing a dress to wear, or winning a certain game or choosing an option when there were several options available.

When it came to the question of how, A.R said that by and large, the mathematical approach helped. However, there were times when the mathematical approach failed to give an answer. He illustrated the issue with a simple example. The question “What is the probability that a person selected at random is a male?” could be answered in several ways depending on the context.  The mathematical approach could give the answer as ½ but in a class of 50 of which 40 were boys and 10 girls, the answer could be 4/5. If the whole institution was considered, the answer could be different and if a country was considered, the answer could be still different. If both the numerator and the denominator turned out to be infinite the mathematical approach would fail to give an answer. The other alternative was the frequency approach. But the frequency approach was too impractical because without a past history, the frequency never existed and not everyone could be expected to have a complete knowledge of the history and hence the answer could never be relied upon. On the other hand, if the history was known then the frequency approach was the best, he said.

This brought in to focus the third approach – ‘The axiomatic approach’. As it was time for the bell, he left the class by asking the question ‘What is an axiom?

They had an hour off and so the gang went to the library. They were surprised to see another 8 to 10 of their class-mates. John commented that slowly but surely things were changing. Earlier as soon as the class was over students rushed to catch the train. Now the number of students marching towards the library was on the rise. The grumbling against A.R’s habit of asking questions still persisted but the level was showing a slight downward trend. There seemed to be a divide where a section hated A.R while another section was trending towards acceptance and admiration.

While they were perusing the text-books, they could not get much details on ‘what is an axiom’. All text books just recorded the subheading ‘Axiomatic Approach’ and listed the three axioms. But there was no tangible explanation or an answer to the question ‘What is an axiom’. Ramesh and Lakshmi felt frustrated but Hema suggested that they should look in a dictionary. John agreed with the suggestion and straight away got the answer

Axiom:

                  1.An axiom is a statement that everyone believes is true.

  1. A formal statement or principle in mathematics, science, etc., from which other

                      statements can be obtained:

  1. A self-evident truth that requires no proof.

                  4 A universally accepted principle or rule.

                 5.Logic, Mathematics. a proposition that is assumed without proof for the sake

                    of studying the consequences that follow from it.

 

They felt the more they read about it, the more confused they were. So, they decided to wait to hear the explanation in the class and left.

 

 

Day 9.

 

While the Gang was walking towards the class, Ramesh said that a senior told him that an axiom is a statement or a concept which cannot be proved but is used in proving other statements or concepts. He further informed them that he was also given an example that the point is an axiom in Geometry.

John was thinking. No one has ever been asked to define a point. Its existence has always been taken for granted. If a point does not exist then the whole Geometry fails to exist. Yet, there is no formal definition of a point and it is difficult to answer the question ‘What is a Point?’

We understand when someone says a straight line is the shortest distance between two points. A triangle gets its name since it is a connection of three distinct points. But the point is undefined.

When he explained this, they said that they could now understand that an axiom is a universal truth requiring no proof.  Now, they were waiting for A.R to explain the axiomatic approach.

When A.R entered the class, he took a brisk survey of the class. He could see that quite a few students had come prepared to explain or answer the question relating to axiom. He just smiled to himself and began the lecture straightaway.

He said the same thing as what Ramesh had explained earlier. After giving the example of a point as an axiom in Geometry, he suggested that the students find more examples of axioms in different areas, He said that the probability theory was based on three axioms.

Before talking about axioms and the axiomatic approach to probability, he said that certain terms or vocabulary needed to be introduced.

  1. Sample Space.

  2. Mutually exclusive events.

  3. The sure event or the certain event.

  4. The impossible event or the null event.

An event is something that has happened or whose happening is being discussed. For example, when a coin is tossed, there two possible events ‘Head’ or ‘Tail’.

When a child is born, it can be a ‘boy’ or a ‘girl’. When a card is drawn from a pack of cards, it can be a ‘Spade’ or ‘Hearts’ or ‘Diamond’ or ‘Clover’.

A sample space is the set of all possible events that can be listed.

For a coin tossing experiment, the set of possible outcomes would be {Head, Tail}.

When a dice is thrown, the sample space would be {1,2,3,4,5,6}.

Mutually exclusive events are those events that when they occur, they prevent the occurrence of other events. For examp0le, when a coin is tossed, if a head occurs then occurrence of a tail is prevented and vice versa. So, Head and tail are mutually exclusive events.

For understanding the axiomatic approach to Probability, basic concepts of sets and relation between sets is a prerequisite.

The Sample space is denoted by the Master set S and every possible subset of S can be an event. Since events are subsets of S, they are denoted by the letters A, B, C, D ,.. etc.

The occurrence of at least one of the two events A or B is denoted by A∪B.

The simultaneous occurrence of two events A and B is denoted by A∩B.

The empty set or the null set is denoted by Ø.

He went on to give examples.

  1. The experiment is to throw a dice.

A dice is a six faced shape

When a dice is thrown, the set of all possible outcomes ‘S’ is listed as

S = {1,2,3,4,5,6}

The event that the result of throwing a dice is an odd number can be listed as

A = {1,3,5} and the event that the result is an even number is listed as

B = {2,4,6}

If we define A as the event of getting an odd number and B as the event of getting a number less than 3, then

A∪B = {1,2,3,5} since A = {1.3.5} and B = {1,2}.

A∩B = {1}.

The event that the result of throwing a dice is > 6 is impossible and an impossible event is denoted by the empty set Ø.

Two events A and B are said to be mutually exclusive if

A∩B = Ø.

If the event is the entire sample space ‘S’ it is known as the sure event.

The three axioms of probability are

  1. For any event A, p(A) ≥ 0.

  2. For the sure event S, p(S) = 1.

  3. For any two mutually exclusive events A and B,

                        P(A∪B) = p(A) + p(B).

The first two axioms together imply that probability is always a number in the range 0 to 1. The third axiom ensures that it does not violate the classical or mathematical approach to probability.

By this time, the class got over and the gang met as usual at the end of the day.

As there was nothing specific to discuss on the axioms and the axiomatic approach, they just discussed issues relating to other subjects and then dispersed.

Day 10.

It was another lacklustre day as the classes began. Those who were always capable of finding their own ways to entertain themselves during the class were busy doing just that. There were a few passive onlookers while a few felt disturbed as they were unable to concentrate.

A.R entered the class and again started explaining some more terms relating to probability. The following terms were introduced to the class that day.

  1. Independent events

  2. Simultaneous events.

  3. Conditional probability.

  4. Random Experiments.

  5. Joint Probability.

He began with the term ‘Random Experiments’. He said that experiments could be classified either as ‘Definite Experiments’ or ‘Random Experiments’.

Experiments which resulted in the same outcome irrespective of the place or time or the person conducting the experiment fell under the category ‘Definite Experiments’. On the other hand, experiments whose results could not be predicted in advance or whose results varied from person to person or from place to place or from time to time fell under the category of ‘Random Experiments’.

He then cited some examples illustrating the difference between the two types of experiments. The kind of experiments that a student performs in the Physics or Chemistry Lab results in the same outcome irrespective of who does the experiment. The results of such experiments could always be predicted in advance and could be verified by anyone anywhere anytime. The theory says that the boiling point of water is 1000C. This result does not change from place to place or person to person or time to time.

On the other hand, in a simple experiment of tossing a coin, the result cannot be predicted in advance. The result can vary from person to person or from time to time or from place to place even if the same coin is used.

As another example, he said that many children are born each second all over the world but how long the child born at a particular moment would live cannot be predicted in advance. Thus, human life is an example of a random experiment.

He then looked at the class and said a batch of students taking an examination is also a random experiment as the outcome (the grade or mark) cannot be predicted in advance and varies from person to person.

From the different examples cited in the class it was crystal clear that human life is full of random experiments and hence the concept of probability and its applications play a major role in shaping the human life.

  • Independent Events: A set of two or more events are known as independent events if the occurrence or non-occurrence of one event does not in any way affect or influence the occurrence or non-occurrence of any other event.

For example, when a pair of dice is thrown, the outcome of one dice is independent of the outcome of the other dice.

Another example of independent events – two students are taking the same examination seated in two different places. The mark scored by one student does not in any way influence the mark scored by the other student.

Similarly, if a coin is tossed and a dice is thrown, the outcome of the toss is independent of the result of throwing the dice.

  • Simultaneous events: A set of two or more events is called simultaneous events if they occur simultaneously. i.e. they take place at the same time.

For example, if a pair of dice is thrown and A is the event that the outcome is an odd number and B is the event that the outcome is < 10, then if the outcome is 3 or 5 or 7 or 8, then both A and B have occurred simultaneously. The simultaneous occurrence of two events A and B is denoted by (AB) or A∩B   

 

          P(A∩B) is known as the joint probability of the occurrence of the simultaneous event A  

         and   B.

  • Conditional probability: If two events are not independent events, then obviously they are dependent events – i.e. the occurrence or non-occurrence of one event depends on the occurrence or non-occurrence of another event.

As an example, when we choose two cards one by one from a pack of cards, the probability of the second card is a spade depends on the card chosen first. If the first card is a spade, the probability that the second card is a spade will be 12/51 and if the first card is not a spade then the probability will be 13/51. Whenever we use the word ‘if’ while explaining an event, the corresponding probability is known as the conditional probability.

Having explained the basic terms, A.R paused a while. He waited and gave time for the students to note down the points mentioned and think over the same. Then he suggested that interested students could discuss among themselves and come up with different examples to explain the terms. Finally he wanted the students to justify the need for the axiomatic approach to probability and also to give examples of situations where probability can be used in day to day life.

AS the bell rang, he left the class saying that the discussion would continue in the next class.

The gang met in the evening as a routine. Hema asked what A.R meant by situations in day to day life where probability could be applied.  Arun responded saying that many text-books explained probability using the coin tossing experiment. Though it helped in understanding the concept of probability, hardly anyone used it in the day to day life.  John chipped in saying that he read in a book that probability had its applications in gambling. Perhaps that was one reason why there were so many problems using the pack of cards or throwing a dice etc.

Lakshmi then suggested that they could do better by discussing day to day life situations where there were uncertainties. As an example, she said that as she left the house in the morning she could never be sure that the train would be on time or late. So how could she calculate the probability that the train would be on time on any given day?

This made others pause and think a bit more. John said that even if the train was on time at a particular station there was no guarantee that the train would be on time when it reached the Hindu College Station.

Arun said that they had now seen that A.R did not ask even a single question on some days. So on a given day whether A.R would ask questions or not would also be an example of a situation for applying probability. That made everyone laugh but they agreed with Arun. They dispersed with a smile on their face.

Day 11.

 

Next morning, as they walked their way to the class, they had more to share. Arun said there was no power supply in his are as he woke up in the morning. Because of this, the daily routine was upset, and his brother had to leave for office without taking breakfast. So, here was an example of joint probability if he could describe having a power cut as event A and missing breakfast as event B, then missing breakfast due to power cut would be event A∩B.

Hema then said her brother also leaves for office early every day and on occasions, he had to skip the breakfast. One day he had to skip breakfast because he got up late, another day my mother was unwell, another day his friend had invited him for breakfast and so on. So , missing breakfast because of power cut could be explained as conditional event. She wanted to know how it could be represented?

John came up with the answer. He said he had looked up for an answer to this question and found that (A/B) represents the conditional event of A because of B and (B/A) represents the conditional event of B because of A.  As we say missed breakfast because of power cut, we write (B/A) as B represents missing breakfast and A represents having a power cut. He added that conditional events are read as B given A or A given B as the case may be.

By then, they had reached the class and waited for the day’s routine to begin.

A soon as A.R entered the class, it was obvious that he would be asking a lot of questions that day. Before he could begin, Nalini said that she had performed a random experiment in the house that morning. As she was speaking, she was also laughing, and all could see that she was really enjoying what she was describing. So, A.R asked her to complete what she wanted to say.

Nalini said that her mother was unwell and asked her to cook. As she was cooking, she realised that cooking was a random experiment. She used the same ingredients, but the result varied on each occasion. In other words, the taste or the final quality of the cooked food could not be predicted in advance. Everyone started laughing at her explanation, but it also set them thinking.

Taking the cue from Nalini, the gang also shared their examples with the class.

Bhaskar came up with an example of telecommunication. He said that his sister was a student of Engineering and she gave an example of signal processing. The sender types in a message at one end and the receiver gets the message at the other end. What’s the probability that the message gets delivered correctly without any errors?

Bala gave another simple example. He said that when he switched on the tube light in his room today morning, it failed to burn. The life of the tube-light was over. He said that as in the case of human life, the life of bulbs also cannot be predicted in advance as they can fail any time. It would be the same with different electrical and electronic appliances used by many people. So, he wondered how the manufacturer could give warranty / guarantee of such products and why there was no guarantee for human lives.

A.R was certainly pleased with the way the class went. More people had participated in the discussion and different examples were given. More importantly, many students had started looking at things happening around them with a better insight than earlier.

When the gang met in the evening, they were naturally excited. They could now understand that Statistics was visible in every walk of life. Almost all the examples cited in the class were such a question of probability could not be answered using the mathematical approach or the frequency approach. Such examples clearly argued for the need of the axiomatic approach to probability.

It was Lakshmi who commented that in these 15 days, A.R had used the blackboard just on two or three occasions only. Hema added to that saying that students were now giving examples by just looking around them instead of giving bookish examples.  John said that he now understood the value of A.R’s lectures. Arun also joined by saying that he did not find it necessary to spend hours in reading the textbook and could explain the basic concepts just by looking around and relate things. At the same time, he was also wondering whether all these could result in scoring marks when it came to the examinations.

The last question from Arun only helped in bringing down the enthusiasm level and the gang dispersed in a rather pensive mood.

Day 12.

It was the first day of the new week. During the weekend, the gang members had tried solving some problems from the text book. They had looked at the text book examples and tried some problems on their own. Arum was quite happy with the results since he could solve all the problems. He said that understanding the concepts had made the procedure look simple and as he was good at calculations, he also enjoyed solving problems on his own. For Hema and Lakshmi it was a first-time experience at solving a problem without looking at the board. John said some problems he could do but he also had some difficulty with a few problems. Lakshmi said she could not understand why in some cases, the different probabilities were added and in some cases, the different probabilities were multiplied to get the final result. Hema confessed that she found it difficult to do problems on her own by looking at the examples. She had always looked at the board to solve a problem and once it was understood, she had no difficulty in answering that question in any kind of test or examination.

Since it was time for the class, Arun said he would help others in the evening so they also could start solving problems on their own.

A.R walked in briskly at the allotted hour. He said that he would give hints on solving the problems and that the students should do it on their own.. This would give him time to go and check individual students who had difficulty in solving. He added that if in a bench, one student was able to solve the problem then he or she could help others in solving the problem. In addition, he said that the students were free to move around and seek the help of any student of their choice. He could then concentrate on students who depended only on him for help.

He wanted the students to remember the basic rules while calculating the probabili8ty.

  1. When you use the word ‘or’ in explaining a situation, you must add the probabilities.

  2. When you use the word ‘and’ in explaining a situation, you must multiply the probabilities.

At this, Arun turned to look at Lakshmi and she nodded her head. Hema was tense as she had just heard that A.R would not solve any problem on the board but would only help them to solve problems on their own.

A.R started with simple examples. His first question was to find the probability of getting a number 2 when a dice was thrown. Many students shouted that the answer is 1/6.  He then asked the probability of getting a head when a coin is tossed and all replied ½. He now modified the question. A coin is tossed, and a dice is thrown. What is the probability of getting an even number and  a tail?

All of them knew that the probability of getting tail is ½ and that of getting an even number is ½.  As some students were hesitating, A.R asked them to read the question aloud. The question was about getting a tail and an even number. Then they remembered the rule. After that it was easy to multiply the two answers so that the end result would be ¼.

He then gave some problems relating to a pack of cards, throwing a pair of dice, etc. Every time, he wanted the students to pause and think which word would explain the situation better. ‘Or’ meant addition, ‘and’ meant multiplication.

Finally, he gave this problem for the students to work out and come back with the solution in the next class.

Question: Four students are tossing a coin each at the same time. What si the probability that at least two of them get a head?

When the gang met in the evening, Hem said that the answer should be just ¼. When asked to explain, she said there must be two heads and since probability for getting a head is ½ using the rule, the answer must be ½ x ½ or ¼.

Lakshmi said that the answer could not be so simple, but she could not find anything wrong with the answer.

John said that they had not; looked at the question properly. The question was getting at least two heads whereas Hema had answered for getting two heads. Arun nodded and said what A.R wanted must be different. As he was repeating to himself at least two three times, John said at least should mean two or more heads. Arun explained it further. He said when four coins are tossed, the number of heads can be 0 or 1 0r 2 0r 3 0r 4. At least 2 would mean that the number of heads should be 2 or 3 or 4.

Hema now said that 2heads means probability is ¼, three heads mean probability is 1/8 and four heads mean probability should be 1/16. So, applying the rule, the answer should be ¼ + 1/8 + 1/16 which would simplify to 7/16.

Everybody agreed that it could be the answer and congratulated Hema as she had answered on her own. It was time and they dispersed.

Day 13.

Next morning when the gang met, John said he was not sure of the answer they had agreed on the previous day. He said that he approached the solution in a different manner and got the answer as 11/16 and not 7/16. When asked to elaborate, he said he wrote down all possible outcomes and counted how many outcomes had at least 2 heads. He showed the he had worked to others.

 

Possible Outcome

Coin I

Coin II

Coin III

Coin IV

1

T

T

T

T

2

T

T

T

H

3

T

T

H

T

4

T

H

T

T

5

H

T

T

T

6

T

T

H

H

7

T

H

T

H

8

H

T

T

H

9

T

H

H

T

10

H

T

H

T

11

H

H

T

T

12

T

H

H

H

13

H

H

T

H

14

H

T

H

H

15

H

H

H

T

16

H

H

H

H

As could be seen from the table, except the first 5 possible outcomes, there were 11 outcomes favourable for at least 2 heads and so the probability must be 11/16 and not 7/16.

Hema was naturally upset on being told that her answer was incorrect. But she was happy that she had attempted a problem on her own, the first time.

In the class, A.R asked if any student would come forward and solve the problem on the board so that he others could understand.  John and went to the board and explained the procedure as he had explained it to the gang. While A.R acknowledged that John was correct but he wanted to find out if there was any other way of getting the same answer.

Bala said he could try. He said since 4 persons were tossing, the two persons who got ‘H” could be selected in 6 ways using algebra (permutation and combination). Similarly, 3 out of 4 would be 4 and all the 4 would be 1. Since it is 2 or 3 or 4, adding all, one would get 11. But he said he could not explain the total no of cases as 16.

A.R said if anyone could help Bala. John said he had already worked out the total cases as 16. Bala wanted the algebraic explanation and not the tabular form. Tabulating such cases would be very cumbersome if the number of coins are to be 10.

A.R agreed with Bala on that point. Arun said in each coin, getting a head or tail, probability is ½. Whatever the combination, it could be explained by ‘and’ (Head, Tail, Head, Tail). So by multiplying ½ x ½ x ½ x ½ we get 1/16 for each case, he said.

Thus, they could arrive at the answer 11/16. AS Arun glanced towards Hema, she nodded her head.

Nalini then stood up to say that the answer could be obtained in another way also. She said from the 16 possible outcomes, remove the outcomes that do not satisfy the condition at least 2 heads.

A.R acknowledged what she said with a smile and said that she had just proved an important rule of probability with an example.

If A is any event and AC  is its complement then from algebra of sets, we know that    

       is the Sample Space.

Further by the third axiom of probability,  

This leads to the result

What Nalini did was to use this result, he concluded.

AS the time was up, A.R left the class saying that the discussion would continue in the next class.

AS it was cloudy, the gang did not meet in the evening. They left as soon as the college was over.