 Dear students, let us commence the discussion of one of the most well-known probability distributions, that is the binomial distribution. Let me discuss with you what is the equation when we encounter or deal with the binomial distribution. The first thing is that the random experiment we are conducting, if it is a binomial experiment, then the probabilities are computed through the formula of the binomial distribution. If it is not a binomial experiment, then the binomial distribution will not be applied. So the crucial question is what kind of a random experiment can be called a binomial experiment? The answer to this is that four conditions should be fulfilled. If four conditions are fulfilled, then we say that our random experiment is a binomial experiment. First and foremost, the first condition is that the outcome of each repetition of the experiment, which is technically called a trial, so that I can say that the outcome of each trial may be classified into one out of two categories, which are called success and failure. Now whatever I have said, I will explain it to you. First, let us toss a coin. There are only two possibilities. Either we will get a head or we will get a tail. So if we regard head as success and tail as failure, then this first condition is obviously being fulfilled. Whenever I repeat or toss the coin, either I will be getting a head or a tail. So I repeat the first condition. The outcome of each trial may be classified into one out of two categories, success and failure. After that, let me tell you that if we apply this to real-life problems, the things we are usually interested in are called success and the other things are called failure. But sometimes we do the opposite. By and large, success is what we are interested in. For example, if we are interested in whether this person is a smoker or not a smoker, then at this time, if we want to do an analysis on smoking, we will regard smoking as a success. So every person we will examine, either he or she will be a smoker or not a smoker. So either success or failure. Students, what is the second condition? Second condition or third or fourth are actually intelligent. Second condition is that the probability of success for every trial, that probability of success which is denoted by small p remains the same. For example, if I toss the coin, if it is a fair coin, then every time the probability of getting ahead, if that is success for me, it will be equal to 1 by 2. Every time it will be 1 by 2. It is not that now it is 1 by 2 or next time it is 1 by 7. Third property is that the successive trials are independent. Now see, in coin tossing, it is very easy to see. Every time you have tossed it, it has nothing to do with the old toss, with the new toss. So those trials are independent. But if that second example which I have given you, the smoking one, how will we regard that one person and the other person are independent according to the phenomenon of smoking? Look, for example, if they are real brothers, suppose, then if there is a tendency in the family, then one is doing it and the other is doing it, then we cannot say that they are independent. However, if it is a large population out of which you are drawing a few people at random, then it is obvious that we can assume that the randomly selected people are not related to it. And so, with regard to smoking, their curiosity is that whether they are or are they not smokers, according to them those people are independent. And as I said earlier, in reality, these two conditions are interlinked. If this phenomenon of randomness is happening and they are independent, then we can say that the probability of success remains the same. After this, let me talk about the fourth property. The fourth property is, I mean the fourth condition rather, my dear students, is that the experiment is repeated a fixed number of times. And that number is denoted by N. Now, what is this? See, this means that if we have to toss a coin as an example, then we have already decided how many times we have to do this work. The best example is that if we think of cricket or hockey as a tournament, if a tournament has been decided on three games, then if we have already decided that if our team won the first two games, then we still have to play the third game. That number three is fixed in advance. So, what I said here is that the experiment is repeated a fixed number of times, whether we win one game, whether we win one, whether we win two, whether we win three. It is already decided that we have to play three games. There is another distribution, my dear students, in which this number is not fixed in advance. For example, there is a geometric distribution, that you keep doing that experiment until you get your first success. That is, if you get a tail in the first time and you do not get a head, then you toss again and if you get a head, then the game is over. If you do not get a head in the second time, then you toss again. If you get a head in the second time, then the game is over. So, you saw that it is not fixed in advance, number of trials or number of repetitions of your experiment. When these four conditions are complete, then we have the binomial distribution coming up, that is, a formula, which is called a binomial formula, that applies. It has been mathematically derived. I will not do the derivation right now, but I would like to present to you that formula. Of course, formula to me, I would like to say the PMF, the probability mass function of the binomial distribution. So, I will now say it in a formal manner. Let the random variable x denote the number of successes in n trials when a binomial experiment is being performed. Then the PMF of x is given by p of x is equal to n c x, p raised to x into 1 minus p raised to n minus x. After this, we will put a comma and write x is equal to 0, 1, 2, so on, so on, up to n. And along with this, we will also write that p of x is equal to 0 elsewhere. Whatever I said, please understand this. The thing is, what x denotes in the case of a binomial experiment and a binomial distribution. It does not represent variables like height, weight, or anything else. In a binomial distribution scenario, x will always, always represent the number of successes in n trials. For example, if you are tossing n times, then what are the possible values of your x? Exactly what you have written after the comma. x can be 0, i.e. 1 times b, you did not get a success. x can be 1, 1 times success, or n minus 1 times you got a failure. The last possibility is that x is equal to n. i.e. n times n times n times b, you got your success. So this is the scenario, my dear students, of the binomial distribution.