 So, we will get started with our first lecture here. So, what I am going to do is get started with the introduction to probability, but in this statistics I am expecting you people already know little bit of probability. So, I will just quickly go through and those of you are especially MTech IOS students will be also doing IE 6 to 1 there you will see this more. But still I want to go through for the sake of other students in the class ok. Now, as you all can imagine there is a nothing deterministic in life right majority of the things are like random there is certain uncertainty associated with this ok like even they say that tomorrow sun will arise or not that is going to be not sure sure guaranteed there is certain maybe small very very small, but there is a possibility that it may not happen ok. So, there is randomness in many real world problems you believe and but we want to understand them right. So, that if you have to deal with those problems we need to understand and that is why we need to take into account their randomness ok. And now how we are going to do this if you have to deal with this real world problems which you know they are going to have certain uncertainty or randomness associated with this then what we are going to do is in that case if you want to analyze them they are going to allow or build models which themselves are probabilistic. As we go along in this course you will see that what is that probabilistic things that we are going to use in modeling real world problems. And also whatever the real world problems that we are going to see we actually do not know what is the probabilistic behavior or the probability model they are going to use, but what we get to know is their behavior property or nature through the data that they generate right. So, for example, we know the whether right whether we cannot predict whether it is difficult it is a random quantity like I do not know whether next year it is going to rain heavily in Mumbai or not or it is going to be the monsoon will going to be good bad or it is going to be normal. There is a certain kinds of underlying randomness there I do not know according to which what probability that is that is going to happen. But anyway next year I see whether it was a normal monsoon by the like by the end of when the monsoon season ends I know that it was normal or deficit or it was I mean over like it was more than normal. So, that is the data I observed and that data is like what is generated from that underlying real world problem ok. Now what we are trying to do is we get to observe this data and observing this data is what we are trying to go and build the underlying model may be like whether as a real world phenomenon is happening according to some probabilistic model I do not know that model behavior the only way I see it is through the data generated ok. So, I try to see the data and go back and try to see that a build model that can potentially better explain that real world problem. And why probability models and we really do not know like what is the model that is being used to decide all the weather conditions. But we know that there is some kind of randomness involved in that and that is why to at least understand that systematically we want to have to start using some model and because of this randomness we start thinking about some probability models here ok. And what we need to ensure is if we have able to suppose let us say there is an underlying model according to which the weather phenomena happens. And suppose you know that weather phenomena somehow you are able to model it well then how you are going to be sure about whether the weather model you are captured is going to be good or bad you are going to see whether it is good or bad based on the data that you are going to observe and see that that data being generated is consistent with your model ok. So, let us say that weather again weather is very classic example weather let us say our meteorological department build a model they build a model collecting our data what happened in the last 10 years 15 years 20 years 50 years whatever and now they build a model and then they make a prediction for the next year. They may say that ok maybe monsoon is going to be normal and then next year you actually see whether the monsoon is normal or not. If monsoon happens to be normal then maybe the underlying probability model you have built is actually is close to what the nature is trying to do and you are good. So, you have to try to understand that probability model if not then maybe something like you have your model is not at good your probability model is not good you maybe need to improve that and that is what happens like the weather prediction algorithm exactly does that like they make a predictions and if things are as per them fine if not they will go back take the data and try to improve the model ok. So, that is where it is the whole thing like you see that there is probability model and the data here we need to have probability models which tries to govern try to understand model the real world phenomena and data is what we observe and see that whether the data that is being generated is consistent with our probability model if it is so then we know that we have model the real world phenomena well ok and if we model the real world phenomena well that is good for us right like we can predict the things properly and accordingly we can take our actions ok. Now, let us start building the basics of probability maybe some of these things are already being parallely done in IE 6 to 1, but I will go through this quickly today ok. So, what we will start talk in this class is about simple sample space events so, probability conditional probability independent events and bias and formula like let us target to finish this today ok. These are all classics which everybody when the who knows already basic probability they already know this. So, if you are going to consider any experiment so, now and now we are we are talking about random experiments right like just to think of like let us think of whether itself like a weather is like an experiment like you want to see what is going to happen tomorrow that means whatever the underlying conditions that are going to make something to happen maybe like make the weather behave in certain pattern. So, these are all like we are just going to treat it as experiments random experiments ok the outcome what we are going to see they are coming through this random experience. Now, to understand that we will not go into anything complex like weather we do not know whether there are lot of things in weather right like where is a temperature humidity and what else maybe perception that density lot of things are involved that all are going to govern how the weather behaves, but to begin with we will not get into that complexity we will strip down the things to the bare minimum possible. So, what is the bare minimum possible we will go to the case of simple coin stores or a throw of a die to understand all these things ok. These are still random quantities, but they are pretty simple to understand and reason ok. So, we know about coin toss is coin toss is a random experiment or it is a deterministic experiment. It is going to be random experiment, because you cannot operate a predict what is the thing you are going to see whether it is a heads or tails. So, now the first thing we are going to look into is sample space. We are going to say possible outcome of an experiments as a sample space and we denote it by omega ok and the next thing of interest forest is something called event and we are going to treat any subset of the sample space is known as event ok. If you want to understand a set up a random experiment or understand or analyze that random experiment first thing you need to do is have some understanding about what are these possible outcomes and what are these possible events ok. Now, let us look into that. So, in case of coin flipping we know that outcome is going to be either head or tail right. So, that is why the sample space is simply going to be head or tail H represents head T represents tail and in terms of rolling of a dice the dice will have 6 faces right and any one of them come I do not know which one and that is why they are represented 1, 2, 3, 4, 5, 6 and other simple examples is like instead of one coin you may want to flip 2 coins and in that case there are 4 possibilities right either both of them may show head, both of them it show tail or each of them show different things the first one show head, second one tail or vice versa and similarly if you are going to take 2 dice there are now 36 possibilities right and that 36 possibilities is written in this matrix and that will constitute your omega and now your outcome or a sample space can be actually need not be as simple as that it could be actually continuous space or interval ok. For example, if you are interested in room temperature and that room temperature the possible range of that temperature could be anywhere between some value A and B. For example, if you are interested in room temperature in Mumbai it could be something anywhere between 20 to 45 degrees ok. So, then in that case you are going to say my outcome sample space is the interval A B and any value in the interval A B can happen ok. Now, events right we wanted to talk about events. Now, we said that event is nothing but subset of sample space. So, if you are if you take the coin toss subsets are H T are both H T and in fact null set can also be there ok. Now, what does event that H means? That means like you are interested in the event heads and T means you are interested in the tail and H T means you are interested in either of them like I mean whatever comes fine that basically and what is fee here nothing happening nothing happening, but when you are going to toss a coin can nothing happen no right either head tail happens or if you are interested in both either is fine, but nothing, but if you fee null set still trivial event we can consider ok. And in the rolling of a dice you may be interested in knowing an event in which the outcome is divisible by 2 or even number. In that case your event is 246 or you may be interested in an event where your outcome is odd number in which will be interested in 135 or you may be interested in an event where your outcome is divisible by 3 in which case your event is going to be 36 like that. And now if I have omega as my sample space how many events will be there 2 to the power and does this include your null set ok. So, there will be total 2 to the power cardinality of omega will be your thing, but this is make sense when your omega is finite ok. That is you are interested our possible outcomes are finite like if not this could be just infinity ok, but that is fine any event which is a subset of my outcome I am going to treat it as an event ok. In terms of flipping of coins like can somebody tell what does this event represents yeah here it is yeah here it is saying that I am interested in the case where the first outcome is head. So, in that case second one comes either head or tail I do not care. And similarly this event is representing that my first outcome should be tail ok. In the role of dice again you may be interested in an event where the sum of the 2 faces is going to be 5. So, in that case these are all the possible outcomes will be interested in. And similarly when it is sum is 6 you can consider all these possible outcomes as your events. Now in a case of a temperature we said that let us say a b was your interval your possible outcome this was our omega and event can be any subset of this. Let us say like let us say if my this is my a and this is my b room temperature and I will take some interval here between this c and d. And then in that case this is a subset and that could be an event. So, instead of asking if your possible room temperature is between 20 to 45 you want to see that will it be cold in my room like that in that case suppose let us say will the temperature will be between 20 to 24. And you will be interested in that particular subset now ok. Now given that we have this sample space and events you may want to do certain operations on events. For example, like the what is the what is the chance that like if you are throwing a dice that my outcome is even and also it is a divisible by 3. Now you are trying to play with you are interested in two events now like one event is it is an even and another is divisible by 3. Anyone of them happens you are fine and now you want to like now how to do now you are trying to look into multiple events and you are trying to do operations on that. So, what operations are possible. So, let us take two events which are in which are events in omega. So, obviously we are this and f is also subset of this. Now let us try to understand when we say event e occurs ok. So, we know that let us take I know e is a subset of omega. Now I am talking about event e happened what does that mean. So, what it means is let us say ok. So, before that let us say whenever any element in e is the outcome then I am going to say event e has happened. So, what does that mean? Suppose let us say you are rolling a dice and you are interested in the event 146 the face showing 146. Then we are going to say that event e has happened if either face shows 1 or 4 or 6. So, only if the face shows something like a 2, 3 or 5 then I will say that event e has not occurred. Otherwise we I will just say that event e has happened ok. Now if I have to I am asking if event e has happened then I may be also interested in some events not happening like I may want like my room temperature is not between 40 to 45 ok. So, then what is that complement? The complement is like of an event e is nothing, but you remove e from your omega and whatever remains is your complement. For example, if you have this let us say this is your omega and you have some set e here and now all the things that are outside they are going to form e complement. Now this naturally brings out couple of properties on an event and it is complement. So, if you are going to take union of e and e complement that has to be naturally omega right that is by definition and if you are going to take intersection of e and e complement that is a null set because there is nothing common between them ok. But let us this is a simple case when we looked into e and e complement their union and intersection that is clear because we know what happens. But now let us take two things two sets and try to define their union and intersection. Let us say I have one set e here and another set f here this is omega. Now what is their union? Now union is simply nothing, but all the elements in this. So, which is all of this and intersection similarly we are going to define to be simply all the elements which are common in that. And now let us say g is intersection of e and f two events. Now what like g is a new event I have defined using event e and f right. Now when I am going to say event g happened. So, it should that means both e and f happened that means the element which is common in both e and f if that has come as an outcome then I am going to say that event g has happened that means basically the outcome should be from the intersection if this has happened then I am going to say that event g has happened ok. Next mutually exclusive if there is nothing common in between these two elements ok let us take let us take two events e and f. And now what we are saying is if you take their intersection if it is null that means there is nothing common in between them like in this case right e and f there is nothing common in between them. And in this case we are going to say that mutually exclusive events ok. And now so this is a simple case of two events we have considered, but the same definition of union intersections applies even you have countably many events. For example, let us say e 1, e 2 all the way up to infinity then their union means that means this is if an element belongs to any of the event it is there in this union. And similarly an element belongs to the intersection often only if it is belongs to each of these events ok. And even though I have written it for this countably many events here this applies even if it is a finite I mean this is a standard definition. So, all of you understand difference between finite and countable here anybody who do not understand it ok fine. So, now let us get into probability. So, you people are already in the IE 621 you taught about sigma algebra ok. So, you will learn it, but I am not going to go into that. So, but in the any of you are aware of what is sigma algebra? Probability space what is probability space? Just tell me ok let me let me just will make it simpler for you. I just want to know do you know sigma algebra ok one let us just think like we talk about events right we said events which are nothing but subsets of your omega. Now, let us collect all subsets of omega ok we know that all subsets omega there are total 2 to the power cardinality of omega ok that size. And now so, there is a formal definition of sigma algebra I mean in this we will not go into that you will anyway learn it in the other course and that is also not of so importance for us. What we want to do is for every any subset we would like to know the likelihood of that subset happening ok. Let me ask this question like ok if you have a simple dice you are throwing it what is the likelihood that the outcome is divisible by 2? 1 by 2 ok what is the likelihood that that value is divisible by sorry it is divisible by 3? 1 by 3 right. Now, why you are saying that it is based on the likelihood right because I mean everybody right now you are also assuming that it is a fair dice. So, each one of them showing up has the same likelihood ok like and now I am interested in a particular like when I say divisible by 3 out of 6 possibilities were interested in 2 possibilities right 3 and 6. So, 2 by 6 is how you are computing it. Now, instead of going that see in probability we have to like define these things formally. Now, what we are now going to do is probability you are now going to assign probability to each of the possible outcome each of the events ok. So, ultimately when you are going to do an experiments you will be interested in different events happening and you would like to know their probabilities or their likelihood or at least you want to modulate you want to assign some probabilities or likelihood to them and that is where the probability comes into picture ok. So, probability is nothing, but like I mean the sigma algebra is denoted by f script here which is a subset of all set. So, this is now going to be a function from this subset of all samples place to 0 1 ok. And now this the way we want to assign probabilities we want to make sure that they satisfy certain basic properties. And what properties they should satisfy we are going to make some assumptions on that and that will make us make some axioms ok. The first property we are going to assume that probability of any event is going to be non-negative ok. So, we want to when we want to deal with probability which when we are also going to talk about likelihood right we want some non-negative numbers there like making saying that likelihood of happening something is minus 0.05 is not so intuitive right like you want to assign some non-zero numbers and that is why we want everything to be positive. And also when I consider the whole sample space itself I want that to be assigned some number and that number should be the largest right like for example like when I saying when you are saying let us say dies if I tell any number is fine to me the likelihood of that should be larger than any other possible event right. So, I am saying that let us say you are taking omega equals to 1, 2, 3, 4, 5 and 6. So, this is your dies which is this sample space. Now, let us take two events e 1 and e 2 e 1 is let us say 1, 3, 5 that is the odd number and e 2 is 2, 4, 6. And now omega is also unmoving event which is like entire thing whose probability whose likely should be more e 1 or omega e 2 and omega omega. So, omega will have the largest likelihood right when it is covering basically and that value I want to normalize. I do not want it to be take any value and that is why I will put a normalization. I say that that value should be equals to 1. And last point I will assume that finite additivity. We say that if you are going to take two mutually exclusive events, if I look into the union. So, union of two events is going to be another event. The probability of that event should nothing but should be equals to the sum of the two events. So, for example like let us take I have this omega and I have these two mutually exclusive event here I am denoted as e 1 and e 2. Then the likelihood of happening either of this that either of this is e 1 union e 2 right that is either of this should be nothing but then the probability that then the likelihood of this should add up because they are just separate and that is what we called mutually exclusive and if they are mutually exclusive then the probability of the union should be equals to sum of the probabilities.