Now let us discuss the very important binomial distribution. Binomial distribution is a distribution of random variable that counts the number of heads in n tossings of a coin that is not necessarily fail. So our random experiment is to toss n times a coin that not necessarily fail. We will denote the probability of head as p, and so probability of tail is 1 minus p. Now X is number of heads. We can think about X in the following way. Let us assume that we have a sequence of independent trials. Each trial can result either in success or in failure. If we identify success with heads and we say that the probability of success is p, then X is number of successful trials. This is a rather general model for different natural phenomenons. So now let us find the distribution of X. It is easier to begin with some simple example. Let us assume that n equals to 5. Let us find probability that X equals to 0. If we have zero head, then it means that we have five tails in a row. So the probability of this event is probability of the following outcome: tail, tail, tail, tail, tail. As the probability of each tail is 1 minus p and all these tails are independent of each other, we can say that the probability of this outcome is 1 minus p to the power 5. In the same way, we can find, for example, probability that x equals to 1, but it becomes a little bit more complicated. Because in this case, this one head can be placed into one of these possible five places. In this case, we have not one outcome, but five different outcomes. However, as we discussed before, the probability of each outcome of this kind does not depend on the actual place of this head. It can be found in the following way. Probability of, for example, head, tail, tail, tail, tail equals to p times 1 minus p to the power 4. Here, we can write p_1. Here, one is the number of heads and four is the number of tails, and we have the same number, the same probability for each of these outcomes because this probability does not depend on the place of this H. So the overall probability here equals to 5 the number of outcomes times this thing. For other values of X, things become more complicated and we have to use binomial coefficient to find the number of appropriate outcomes. Indeed, for example, what is the probability that X equals to 2? Again, we have to find a probability of one outcome that satisfies this condition. For example, of outcome like head, tail, head, tail, tail. This probability equals to p_2 times 1 minus p to the number of 3. Here, this two is the number of heads and this three is number of these tails. Now we have to find the number of all outcomes for which number of heads equal to 2. From combinatorics, you can remember that this number is denoted by the following thing. This is binomial coefficient five choose two, and we have to multiply this binomial coefficient by this probability. By the way, five is binomial coefficient five choose one because it is exactly the number of ways to choose one element out from five. Now let us consider the general case. Probability that X equals to some number k equals to the following thing. This is binomial coefficient n choose k times p_k times 1 minus p to the power n minus k. Here, this value is number of outcomes with exactly k heads. Indeed, this number is the number of ways to choose k elements out of n. If we have n tossings, then it means that we have n places, and in k of these places, we have to put letter H and the rest will be T. So we have to choose k places out of n places. So this can be done with this number of ways, and each of this outcome corresponds to this probability. These outcomes are mutually disjoint. So the probability that any of these outcome take place is this product. Here in this formula, k is arbitrary integer number from zero to n. Now, if random variable has this distribution, we will say that this random variable is binomially distributed with parameters n and p. We'll write it in the following way. X is distributed as binomial distribution with parameters n number of trials and p probability of success. In this way, we described a family of important distributions. We'll use them in the future.