 I am Siddhartha Ghosh, I belong to the civil engineering department and you have my contact phone number and the. So, for the next few days 3 to 4 days I will be your instructor and then professor Ashish thus will take over and then again I will come back for another 2 days or so. While I am here my primary instructions will be based on probability relation of probability statistics, application of probability etcetera. Well, I try to make my instructions interactive possible, but with such a large class like this I do not think I would be able to make it that way, but anyway if you have anything any question raise your hand interrupt me that would not be a problem. So, with that we will start. So, this is going to be the introduction to probability. So, far we have been discussing statistics various aspects of statistics and you have learnt to use mathematical tools like Sylab to get various parameters like mean, median and mode for a distribution and you have learnt about distributions of various kinds, how to deal with them, how to interpret the data in a statistical sense. Now, we are shifting a little bit towards something we call probability and I am sure all of you are familiar with the term probability to some extent may not be the technical way we always interpret probability, but to some extent you know what probability is and of course your mathematics courses at the 11th or 12th standard should have given you some introduction to what probability is, what are the different notions of probability. So, we will quickly go through those things and then discuss exactly where probability comes in the picture when we talk about our disciplines of engineering, technology, sciences like non-mathematic sciences, physics, chemistry, biology etcetera. So, here comes the introduction to probability, what we know a probability is that it is a branch of mathematics that is what we have learnt and we have so far considered it purely as a mathematics. For this course our focus is going to be in the application of knowledge about probability, I am trying to go slow here for today and I think I can pick a speed later on. So, what is probability? Probability the most cases what we do is to calculate the likelihood or possibility of occurrence of an event and I have emphasized those words like likelihood, possibility, event etcetera occurrence. I am sure you must have seen definition like this have you seen that yes right and you have some ideas about those typical terminologies what does it mean by likelihood, what does it mean by occurrence of an event ok. We will go into a little more detail about all those things specifically event what constitutes an event or so and so far what we did with probability is mostly calculating the probability or likelihood of occurrences such as when you toss a coin what is the likelihood that it would be heads or when you roll a dice what is the likelihood that it is going to be a five on the top face things like that or maybe the last example picking a red sock out of a box usually you say I said that there is a box which is like a black box you do not know what is there inside you only know that ok in total there are maybe 50 red boxes red socks and 20 black socks and you are supposed to pick at random for 3, 4 times and what is the possibility you get 2 red and 5 blacks or any other combination like this ok. So, this is what we have been doing so far in terms of probability. So, for this course we change it a little bit when you think about engine and technological application of probability ok. So, again rolling the dice is not our focus of course we are going to use a lot of example in terms of rolling a dice in terms of tossing a coin because those are really very useful tools to describe how we compute probability how we do counting to get a typical likelihood of the occurrence of some event ok. But again those are just examples when you go through our tutorials you will see that in a lot many cases we deal with again rolling the dice ok or picking out a colorful ball from black box those do not have any practical application, but fine, but they are good for explaining certain notions that as an instructor I would like to explain to you ok. So, our goal is to look at design of engineering systems like civil, electrical, mechanical, chemical etcetera you can add on I have just said engineering system it does not only relate to engineering also deals with modeling of processes of what kind as an example I tell you molecular processes it is very complicated or any natural phenomena. The natural phenomena are usually of very complicated type it is not very easy to know exactly what are going on inside a specific system take the case of cell motility ok. If you want to know about cell motility how human cells move how they change how they deform etcetera etcetera you need to know about them in detail and in many cases that detailed kind of knowledge is not available to us. So, what do we do we use the concept of probability and statistics to have some idea have some estimation of the behavior of those human cells and their motility. So, we use probability lot in modeling natural phenomena natural processes another example is this what is the average lifespan of a microchip let us say you built a semiconductor you built a microchip any customer any consumer of that product will ask you how much reliability do I have with that product. How many days how many years it is going to last and as a manufacturer you have to give the customer that can have information which will also determine the cost of the product and then you can have a proper marketability of your product ok. So, these things are very important and these things can be interpreted only from a probabilistic perspective we do not really have in deterministic system deterministic as opposed to probabilistic to know exactly what is the lifespan going to be for a typical product like a microchip or even take the case of human life we do not know, but what does an insurer do ok let us take any life insurer let us take the case of LIC or something else. So, you know you pay some premium and there is a typical amount that you are supposed to get if you die before these and that amount changes if you die after certain years and so on. Now all these calculations the amount the premium it all depends on what the insurer interpreted the probability of an event what is that event the death of a person based on some large statistics you do this kind of interpretation and then would arrive at some numbers to charge as a premium. So, probability has not only applications in science engineering technology, but also in finance management almost every field you can imagine. So, safety and maintenance of systems processes products well let us come to safety and maintenance. Whenever you are delivering a product or you are making a system let us take the case of an workshop when you build a workshop for any manufacturing process let us say. You need to plan for its maintenance you need to plan for its operation you need to know a little bit about what is the expected lifespan of that workshop. So, that overall you can have some idea about was the overall return from this kind of investment. See investment and return is one kind of measure of what the whole performance of this system that you have created is going to be. So, considering all that you can make a judgment if I am going to invest on building a workshop over here or not and the cost incurred on making or building the workshop is not only the initial cost of creating that, but also the running cost the operation cost the cost for repair and maintenance etcetera etcetera. You should consider all this to have a proper scientific judgment on if you should build that workshop or not am I getting my ideas clear. So, these are like practical applications of the theory of probability in our daily life as engineers as technologists as scientists. So, we are going to study basically this where is the probability in the scheme of data analysis and an interpretation as we are studying in IC 102. So, now I told you that I am a civil engineer. So, I will try to give example from the civil engineering perspective I know that most of you are not civil engineers and most of many of you are not engineers at all, but anyway it does not matter. Try to have a fill of real life applications real life notions of probability not just you know red stock picking red stock out of a box. So, what is civil engineering? Many of you may not be familiar with what it is. In civil engineering we basically deal with design and analysis also operation and maintenance etcetera etcetera of infrastructural systems. And by infrastructural systems we mean these like buildings, bridges like the drinking water network in a city, the waste water network in a city, analysis of a flood plain, designing canals, waterways, designing airports, highways etcetera. So, these are infrastructural systems that a civil engineer is supposed to know how to build how to design and for an existing one how to analyze, how to know what is the level of safety in that. So, as a civil engineer I come to these examples first well I should also tell you this that my specialization is in the area of earthquake engineering. Where what we do is to create systems construct buildings, bridges again all those infrastructure systems safe in case of earthquakes that is our basic purpose. So, to do that we need to know about two things. One is the buildings the bridges that you are building and on the other hand we should also know about the earthquakes. Once you have both sides of the coin then you can construct the system that you need. Now, the problem that an earthquake engineer faces all the time is that to know about the earthquake we do not have enough information on earthquake. So, whenever you are designing something against earthquake making something safe against earthquake the first question that you should ask is when is the next earthquake going to be and how big it is going to be. Now, there are some typical measurements you know we use terminologies like magnitude of an earthquake and I am sure you have heard things like well take the case of the recent Haiti earthquake have you read about that? Yeah of course, do you remember what was the magnitude of that earthquake 7 point something. So, that is one measure we call it the Richter magnitude moment magnitude and so on. There is also other measures like intensity have you heard of anything like that well it is not that important, but anyway. So, these are measures that one needs to know before somebody does a design safe against earthquakes and as you can very easily see that these are things that you do not know these are things that are uncertain these are things that are probabilistic. We have no information when the next earthquake is going to be and how big that earthquake is going to be. I give you another example of design of storm sewers are you familiar with the term storm sewers these are those big drains that carry rain water to some you know water sink water basin. For example, there is a big storm sewer this open channel that comes from the swimming pool alongside the road then crosses the main road next to the tennis court and goes to power lake have you seen that you all must have seen that right. So, that is a big open channel what is it supposed to do when it when there is a storm and by storm I mean a huge rainfall when there is a storm it should carry the storm water safely into a water sink. So, that the areas which we need for our daily functions are not flooded that is the purpose of a storm sewer. So, to design a storm sewer what do we need to do we need to know when and how big is the next storm again storm does not mean only wind storm here means rainfall. So, that is the primary information that we need to design a storm sewer also need to know the intensity and duration. So, that is like the experience that a civil engineer has while building or constructing different infrastructure systems. So, the primary thing that we get here is the notion of uncertainty or randomness whenever you are designing something let us say you are designing a satellite and let us say you are an electrical engineer looking at the moving of arms of the satellite you are a control engineer. Again you will deal with this kinds of uncertainties and randomnesses. So, that is going to be the primary focus for this part of the course that we treat probability primarily as uncertainties randomness that we have to deal with that we have to face in designing, analyzing, maintaining real systems where real includes both nature and man made. What we got to know from the previous examples is that design and analysis parameters are uncertain you also call them probabilistic and the fact is that we cannot avoid these uncertainties. Can we really say that I know that the next earthquake is going to be on this state and it will be of magnitude 6.7 I cannot say that. So, uncertainties in real systems are unavoidable. So, better we deal with it in a scientific way that is the purpose of learning probability in this course. As an example I cite here something that you are very familiar with the Heisenberg uncertainty principle which you use a lot in quantum mechanics do you remember that well as you know it is a it was not first proposed by Heisenberg. It is a more general law where you say that a pair of parameters cannot be identified both at the same time for a given system. So, see the whole subject of quantum mechanics is based from this uncertain principle without that you cannot go any further with this subject. So, you can see here that if we accept that there are uncertainties and we develop our models our theories to deal with that uncertainty then we can deal with the systems along with those uncertainties that we have in real life and that is the kind of purpose for this. Now, we come to two concepts two notions of probability the first one is known to be the subjective notion we call it subjective is notion here it is also sometimes known as Bayesian you might be familiar with the term Bayesian name after the theologist Thomas Bayes I will discuss about that later and it goes something like the example that is given over here that there is a 40 percent chance that a student will get an AA in this course a student says that. So, that is the kind of statement that you would see in case of subjective is notion of probability it is a probabilistic statement we always make a statement like that we say that it looks like this 50 percent chance of having rain today again that is a subjective is notion of probability it describes someone's belief or that belief comes out of experience or some other knowledge but, not on concrete data someone's belief that this event is going to occur with a likelihood of so and so percentage that is the subjective is notion of probability the other notion is known as the frequentist which was first formulated by the French mathematician and astronomer and politician Laplace have you heard of Laplace you are going through Laplace transformation and stuff like that in mathematics course some of you might be going through that as he was a very famous one I think if I remember correctly he lived from the middle of the 18th century to early part of the 19th century. So, he went through the French revolution and so on anyway. So, this famous scientist from France sometimes known as the Newton of Isaac Newton of France he first defined the concept of this frequent is notion of probability which is kind of based on counting which is based on data where we say that when an event can be treated as an experiment then what we can do to know the likelihood of that event is to repeat that experience several times and look at the output look at how many times event a is occurring how many time event b is occurring and then compute the probability. So, this is based on the frequency of occurrence of an event that is why it is called the frequentist notion of probability and it is based on counting and we will know more about counting later on. So, here is an example that there is a 1 6 chance that the dice will show 3 at the top that is true for a fair dice right. So, we can say that very easily that is based on counting if you keep on rolling the dice you will see that 1 6 of the times it should be 3 at the top or 1 6 time it would be any number at the top for this course we will mostly adopt a frequentist approach, but I do not if you can read that it says the mathematics is same for both approaches. So, it does not really matter if you have a subjective is notion of probability or the frequent is notion of probability how we deal with it in a mathematical perspective remain same well we say that probability is 60 percent. So, you deal with that 0.6 that number 0.6 in the same way be defined in the frequentist sense or be defined in the subjective sense. So, our theories or a method of calculations won't change and we will see when we deal a little bit about Bayesian notion Bayesian definition we will see how we can incorporate this subjective is this belief kind of notion of probability along with the frequentist more objective kind of notion of probability. Now, I come to another example this is an example from well you can say it is mechanical engineering example this is the second law of thermodynamics have you gone through thermodynamics anybody yes are you familiar with the second law of thermodynamics very good in very simplistic terms the second law of thermodynamics says that the entropy of a closed or isotropic system keeps on increasing this is not a rigid definition. So, you should not quote this what is entropy it is a kind of measure of the disorder in a system if a system is totally mixed up we say that it has high entropy if it is more order less mixed up then you say it has low entropy. So, that is the kind of idea of entropy. So, the second law says that for a closed and isotropic system closed means it is not affected by any external input it is separated from the rest. So, for a closed system the entropy keeps on increasing now we are trying to see here in a probabilistic sense how is that possible ok. If you use the notions of probability you can very easily see that this is going to be true in most cases and I will try to use an example to describe that and here we cite the case of a library without a librarian rearranging the books. So, the librarian rearranging the books is external to the system we make the system isolated now the system the library is without a librarian rearranging the books. So, what is going to happen that the book shelves or how the books are arranged is going to be more and more disordered as time passes do you see that users will be using that they will keep the books I think you do that I do that too in all kinds of wrong places and slowly you will see that the disorderedness will keep on increasing in that library that is kind of the second law of thermodynamics the example is not from thermodynamics of course ok. Why does that happen a frequentist probabilistic notion can explain that very clearly you know we go by counting for the frequentist notion of probability. So, we are looking at the possibility that the books can be in an ordered way and the possibility that the books can be in a disordered way. Now, the possibility of having the books in an ordered way is very small because there is only a single way of having them in a ordered fashion as opposed to numerous ways of having them in a disordered fashion. So, what you get really is that the probability of disorder that is how we put it here this is the probability of disorder this is the kind of common notation of putting probability is much greater than the probability of having the books in order. So, just by counting just by the frequentist notion of probability you can say that the second law of thermodynamics is true and it can be applied to systems even beyond the general thermodynamic systems. Now, so far I have been telling you about all these problems that we have with uncertainty that we do not know how this is going to be we do not know how much that is going to be we do not know when this is going to be and things keep on being disorderly things keep on being random. So, is randomness all that bad and I at least can cite one example that it is not all that bad from this example of Darwin's theory one of the greatest theories of modern science. So, the theory of natural selection which you have gone through well I am not an expert. So, I try to explain in layman's terms it is based on certain notions which are listed here the first one is the random genetic mutations. So, if these genes do not have any random mutations they do not come up with variations. So, it is only the random genetic mutations that will produce different traits in a species and in a cumulative sense as these differences keep on adding up and the differences between two species or two genes keep on increasing you have different species finally, appearing in the natural system. So, the first requirement for the Darwinian theory of natural selection is the random genetic mutation which results in variation in traits then along with things like heredity and finally, the selection of a preferred trait or preferred traits by nature will lead to what we call evolution of living beings. Primarily thing here is the random genetic mutation take any example very simple example I will tell you there was a single type of betel and due to some random mutations there came two types of betels one of this color one of that color that is one is brown one is green. Now, browns are more camouflaged in the woods. So, the greens are eaten by the predators. So, finally what happens that the brown betels did I say betels or worms whatever brown betels passes on their heredity and they are favored by the nature for having this within codes positive kind of attributes this is the notion of evolution in a very simplistic sense you have learned about you have studied about evolution. So, you know what it really is I do not have to go into the detail the basic thing that I want to emphasize is that it all comes from the random genetic mutation. Why do we call it random? Because it does not have any specific direction when the genetic mutations occur and it creates these two traits of betels green and brown you do not know which one is going to have a better success in keeping its species on. So, it is not directional it does not have any specific sense towards the forward direction of evolution. So, that is why it is a random there is a typical analogy that people give for this process of natural selection which is of the passing of a hurricane through a junkyard junkyard of metals scraps etcetera. So, the hurricane passes through the junkyard and the end result is the creation of a big airplane that is how random the system of evolution is that is what many people say what do you think is it a right kind of an analogy? Can you really say that this notion is similar to the notion of evolution that is the Darwinian evolution do you think so? No why no and the evolution is very probable. Well let us see this way I will give a simpler example they say the numbers may not be right exactly, but I will try to give an example. Let us say enough typical family in certain family they have only sons getting born over last let us say 40 generations. What is the likelihood of that event? Once in billions trillions I do not know it is very very less right so very improbable. Now think of your family you go backward you can go backward in various ways you can track back your mother your mother's fathers fathers mothers in various ways you can go back, but now if you are considering that and try to find out what is the likelihood or possibility of having that trait what do you think for the last 40 generations if you want to see the probability of that event occurring it would be of the same order right. Again having boys boys girls mixed up, but any specific train it would be of the same order very very less. Yes evolution if you consider any specific trait is has that kind of likelihood very very low, but when we see the cumulative nature of it that it happens in a very slow way then it makes sense that is where the primary difference from this analogy comes into picture. It is not like that this airplane gets built in seconds or in seconds 30 minutes. And if you consider that there were several other part airplanes different looking airplanes etcetera etcetera by several I mean millions then you would understand that ok it could be an analogy to the system of evolution. And here I quote one to this is from Dawkins ancestor stale where he says that natural selection is the only explanation we know for the functional beauty and apparently design complexity of living beings. Which essentially says that if we did not have that kind of randomness in the process of mutation then evolution or nature would not come up with the kind of variety in living beings the kind of complexity in living beings the kind of grandeur that you see in the world around you. Now, we will shift again from various sciences to evolution mechanical systems to going back to the theory of probability. Since you are going to deal with uncertainties mostly there are two types of uncertainty there are two different categories broad categories that you usually deal with. One is the type one I said here is the one that is inherent in the system. Take the case of earthquakes we do not know how big and when it is going to be when you design storm stairs you have to know about monthly total rainfall in a city or a simpler example is this the age of a student in this class that is also uncertain. You take any student you do not know exactly you cannot in a deterministic sense say that this is going to be the age of a student age of this student. So, the outcome of this event is an uncertain thing and this kind of uncertainty which is inherent in the system is known as the aleatory uncertainty. The other kind of uncertainty is known as the epistemic which is associated with the imperfect knowledge that we have in explaining a system. Take any system take the universe or take simple motion of rigid bodies take Newton's laws of motion then you can understand whatever laws whatever rules whatever theories we have for describing any system tracking down any system are at best approximate. You think that if you do some calculations based on whatever equations like f is equal to m times a and if you measure the force would you really get the exactly same number no you never get that why because our laws the knowledge that we have which describe a system is approximate one. We do a lot of idealization any real system whenever we try to theorize that we go through some simplifications some approximations to develop our theory. So, these things also finally result into a kind of uncertainty which is called the epistemic uncertainty. So, any system or any model or law which tries to describe a natural phenomenon for example is giving you some kind of epistemic uncertainty. Can you cite some other example anybody give me an example of epistemic uncertainty any law may be the second law of thermodynamics what do you think would it give rise to epistemic uncertainty think about it coming to more concrete stuff these are the tools that you have to deal with the notions of probability in our domain that means the domain of science engineering and technology. We define things like sample space and events and you guys may be familiar with these definitions. Sample space usually donated by S is the set of all possible outcomes of an event all possible outcomes is the universal one. So, the examples you can see here outcome of rolling a regular cubic dice. So, this set the sample space includes 1, 2, 3, 4, 5, 6 there is no other outcome possible. So, that is the sample set for the experiment for the process of rolling a regular dice. If you take that four sided dice then the sample space would not have 5 and 6, but it would rather have 1, 2, 3 and 4. The last example is grade of a student in this one it has to be one value out of all these listed over here. So, the sample space would have everything every possibilities of outcome from an experiment. Then we define sample event which is defined as any subset of the sample space. For an example the outcome of rolling a dice is 2 that is one possible outcome. So, that we will consider to be a sample event e and we said that the sample event can be any subset. So, the subset can also be a combination of 2 individual events like the second example that the outcome of rolling a dice is either 2 or 5. You can consider this also to be an event not 2 events. You can consider the rolling of dice and getting 2 as one event you can consider rolling of dice and getting 5 as one event you can also consider rolling a dice and getting either 2 or 5 as another event. So, any subset combination of individual is also possible. The third example is again another subset that the grade of a student in this course would be more than above BC and you can see that for these there are several outcomes like A, A, B, B, B, etc. But the whole thing can also be considered to be a sample event. These concepts can be best described considering the notions of set theory and using Venn diagrams and I hope all of you are familiar with these things yes right. So, we will go through these things quickly can you can you read this. So, the notions of union of sets intersection of sets complementary set subsets and the null and void set. So, this is just a simple review for you guys you have gone through this we know what these things are and we will use Venn diagrams to explain these things. For an example how do you represent sample space a rectangle you can draw as bigger rectangle as you want a smaller rectangle as you want as long as all the possible outcomes are within that rectangle it is a sample space. And here we have an event E here we have an event F. So, the intersection E F or this is this shaded portion right you remember this what about concepts like subsets it does not have to be a circle it can be an ellipse whatever a geometrical form as long as it is a closed one. So, here F F is a subset of E right or E is a subset of and what is this shaded area the complementary set. So, you are all familiar with these things then there are certain rules of basic set theory which we need to go through and I guess you are familiar with these things as well the commutative law the associative law distributive law and de Morgan's rules. These are the prime rules preliminary rules you need to know when you deal with set theory and Venn diagrams. We are going to use this tool the set theory and that will have Venn diagram to explain notions of probability they are very good at explaining things that we want to explain through this. So, all of you are familiar with these rules take the first one the commutative does it make sense E union F is same as F union E. So, this is the union set of E and F which are way you write it similarly this is the intersection which are way you write it well what you can do instead of me explaining all these things whoever has forgotten about these basic rules try to use Venn diagrams to get through these basic laws very simple ones and these are the kinds of things that we are going to do in the next tutorial also whatever we get out of these rules like. So, here we comes to the axioms of probability and these axioms again can be explained very easily using Venn diagrams using the notions of set theory. The first axiom says that the probability of an event that P E or denotes probability of the event E that number has to be within 0 and 1 does that make sense to you. So, probability can at most be 100 percent at least can be 0 it cannot have a negative value can it no with that we proceed to the second axiom which says the probability that S is the sample space probability of having any of the all outcomes what is that 1 the outcomes have to be from the sample space. So, that is why we say that P of S is equal to 1 we go to the next axiom where we say that for a set of disjointed sets E i and E j all E i and E j are disjointed that means the intersection of E i E j for any i and j are equal to 0 the null set. So, for that the union of all these sets is nothing but the summation of all these sets I interpret that in a probabilistic way again try Venn diagrams to get through this basic law. These are the three axioms of probability which we use to build an axiomatic system of probability theories. We will not get into the details of probability theory we will just use these things for our daily life applications in engineering and sciences. Here are two outcomes from those axioms the first one says that the probability of the complementary of E is 1 minus probability of E is that does it make sense to you one can be replaced as probability of S. Since E and E c are disjointed the intersection of E and E c is 0 you can write that E plus E c is S and S is the samples set having a probability of 1. So, we can get the first one similarly you can get the second one. I do not think I have to explain that if you have gone through set theory you know these things in detail. So, try using Venn diagrams for things like this you will do this similar examples in the next tutorial. So, this is the review of what we have gone through today. We discussed the notions and the occurrence of probability in engineering technology etcetera I gave some examples from civil engineering perspective. Then we dealt with uncertainty of different kinds you remember the names I gave you epistemic and aleatory due to different reasons we have different names for these uncertainties. Then the notions of probability the frequentist and subjectivist notion of probability then we have gone through set theory very quickly because you know that all and then finally axioms of probability and the next state is going to be we are building upon this axioms of probability and to get to some basic theories that we can apply. I am going to leave with you with two things two simple problems. So, this one example in a city the success state for a particular disease for every doctor let us say is 90 percent and one out of the last nine died under doctor one of the doctor's treatment the other doctor that you have nearby nobody died out of the last nine. So, the question is who do you go to how do you decide what is your statistical inference from the kind of known data that you think about it the first one who let us say killed one percent of the last nine is it a zero probability for you if you go there to die under him think about that a similar example is there in the next slide I think you can read this let us say it is known that one in 50,000 air passengers carries a bomb in the flight to blow the airplane up and you know what is the maximum number of passengers that one carrier can have. So, what you do to make sure that there is no explosion you carry the bomb yourself and assure a safe journey does that make sense no right you say no, but why no well you can think about that with that we conclude for today and tomorrow we will start again with Venn diagram statistics and some problems. Thank you.