This week, we begin studying probability theory. Probability theory provides a foundation for Data Science and especially for mathematical statistics. But we begin with a very simple branch of probability theorem, which is called classical probability or discrete probability. But it is important to study these simple examples because they give us some intuition about the notion of probability. We begin with the notion of random experiment. It is not very easy to give a precise definition for random experiment. So let me begin with several examples instead. First example of random experiment is called tossing. If I toss a coin, the result of this experiment can be either head or tail. I don't know in advance before I toss this coin what will be the outcome of this experiment. So I have an experiment and I have two outcomes, head and tail. Of course theoretically, it is possible that I have some other outcomes of this experiment if we think in physical terms. For example, it is probable that the coin stops on the edge or something unusual happens. For example, the coin will throw out of the window. But we will consider only these two outcomes and assume that nothing else can happen. The set of all outcomes is called sample space, and it is usually denoted by Greek letter Omega. I will use just letters H and T to mention this head and tail. So Omega is the set of two elements, H and T. This is called the sample space. Now, let us consider something a little bit more complex. Assume that I toss a coin two times, and after each tossing, I record the result. What is the sample space for this experiment? What kind of outcomes can we have? It is easy to see that we have four different outcomes. We can have a head, then again head. We can have head then tail, tail then head, and tail then tail. This is a space of two tuples and each element of the tuple is either head or tail. So in this case, our Omega, our sample space, consists of four elements. We can abbreviate them by using letters HH, HT, TH, TT. Instead of considering a coin, we can use some other device that will give us randomness. For example, we can roll a dice. In this case, we have six outcomes because we have a dice with six edges. So in this case, the space of outcomes consists of six edges. To simplify the notation, I can just enumerate these outcomes and say that it is the same as just the set of 1, 2, 3, 4, 5, 6. Consider another example. Let us assume that we have a box or a room, that is full of some balls. We can shuffle these balls and select randomly one of them or several of them. This is a very important and popular example. Assume that our balls are enumerated and we select some of them. For example, two. Let us assume that when I choose one ball, l I will not return it to the box at the second time. So I choose one ball, and then I shuffle this ball cartons and choose the second ball. So this two balls have to be different. In this case, if we are not interested in the order of these balls, then what is the sample space for this random experiment if we have five balls and we select two of them randomly? In this case, all outcomes of this experiment are two element subsets of the set of all balls. So we can find the number of such outcomes. So Omega is two combinations of five elements. What is the number of elements in Omega? As we discussed previously in our course, we know that the number of two combinations of five elements is given by the binomial. This is 5 factorial over 2 factorial, 3 factorial, which equals to 10. Now, we discussed random experiments, and discussed their outcomes. Let us define what event is. Informally, event is a description of some situation that can happen when we perform our random experiments. For example, if we roll a dice, we can ask, will we have more than three points or not? Or if we toss a coin, we can ask, is it true that the result of the first tossing is the same as the result on the second tossing? These kind of conditions form what is known as event in probability theory. For each event, we have outcomes that satisfy this event and outcomes that do not satisfy this event. For example, let us consider this dice rolling. We roll dice, and we have event A. We get more than three points. This is event in this random experiment and we can find which outcomes satisfy this event, satisfy this condition. We see that out of these six outcomes, only three satisfy this condition. So we can say that this event A is the following subset of the set of all outcomes. This is 4,5,6. Another example. We toss a coin two times, and we have event B, which is both times has the same result. It means that out of all these four outcomes, we have two outcomes that satisfy our condition; head head or tail tail. We can consider another event for the same random experiment. For example, event C. We can ask, is it true that we get at least one head? How many outcomes satisfy this event? You see that there are exactly three outcomes: head head, head tail, and tail head. So events are just subsets of the sample space or the set of all outcomes. This gives us ability to investigate them and to consider some variations for events and to define a probability of events. We will do it in the next videos.