Now we consider other examples of probability spaces, we will have more later but just to illustrate the notions, let me show some. So first is a sequence of coin tosses. So imagine we toss a coin n times, where n is some probably a large number. So what is then the outcome? We just write down heads and tails of zeros and ones, so the outcome is just a sequence of n bits which show the outcome of first, second and so on coin tosses. And how many we can have of them. If you remember combinatorics, you know that it is 2 to the n. So each new bit increases the number of possibilities twice. So in total you have 2 to the n. And the assumption which usually is used in probability theory that all 2 to the n outcomes have the same probability. Appear with the same frequency in the long series of experiment. And it's not so obvious really, because imagine we toss, I didn't know, the coin ten times. And one of the possible outcome is just 000, ten zeros. And another one is something like I get it know. 0111011100, one, two, three, four, five, one, two, three, yeah. So this account, ten zeros and this, so what do you think, is which of them will appear more often. No, it's ten zeros, so you should. The dots are for zeros. So there are two specific outcomes, ten zeroes and this one. So which of one will appear more often in the real life? And maybe some people believe probably that this is more probable because the zeroes is something strange, you have y always zero, but. Actually, the current understanding of the real world, is that these two outcomes will appear equally often in the real world. But, this is about real world and mathematical assumption starts here. So we agree in advance before computing some probabilities, that they are all the outcomes are equally probable appear equally often. So for example, probability of all heads for n bits for it's just 1/2 to the n, when n is the number of tossed coins. Tossing and we have 2 to the n outcomes and interesting do one. You can consider more interesting events. Imagine this event is the "first bit = last bit". So for example, this event happens if we start with zero and end with zero and also happens when we start with zero and end, start with one, sorry, and end with one. So what is the probability of this event. What do you think? Actually, it's easy to think that it's one-half. Why? Because that actually four possible groups, you can have 001101 and 10, and here, in all four groups, you have arbitrary, n- 2 bits. So, this four, we divide all the outcomes into four groups. And this four groups, a, b, c, d, have the same number, the same number of outcomes. So, each of them is one-fourth of all possible outcomes, so our group here is just one-half, together it's two-fourth. So this event has probability one-half. If you remember combinatorics part probably you will remember computation like this. And then another question for this probability one-half, another question more difficult, and I will not tell the answer, so you can think about it. The event of the number of heads is even. So if it's sum of zeros and ones, we just have a number, or I don't know, maybe it's number of tails. Anyway, it doesn't matter. So, the number of heads is even. What is the probability of this event? Well, I will tell you the answer, one-half yeah, it's one-half, but what I won't tell you is why it's one-half. But maybe you remember something from combinatorics and you can see it immediately, if not think a bit, and I hope you will get the idea why it happens. Again we can also look at Galton board and here the outcomes are very similar. It's a sequences of not zeroes and ones just left and right. Of course, you can use any notation. And again, we have two to the n outcomes, and the probability space is made of all these outcomes. And if you're interested, they are assumed to be equiprobable, and if you're interested in the concentration event, if we want to compute the probability that the being will end in the middle, just consider the event of the number of Rs in the sequence is between 40% of n or 60% of a total number. And so this is an event, so we need to compute how many outcomes are in this event and divide by the total number of outcomes. So, according to general formula, favorable outcomes are just the outcomes where the number of Rs is between 40% and 60%. And so, this is a formula, it's the same formula that you have seen. This is the number of outcomes with exactly k, that is R and then we'll make a sum and divide the sum by 2 to the n and that's what we wanted. So now you probably will say that why should we learn probability theory, it's just combinatorics in disguise. You can ask about probability of something, or you can just ask about the number of outcomes. This essentially is the same question in different language. And it's true and it's mostly true in our course, but it's not completely true in general. So first we speak of only about mathematical part. So that is also a question of natural science, which models our good or bad but we decided it's not probability theory, so for us we can say this is a mathematical part. And mathematical, still there are important things. So, probability theory cares a lot about this independence thing. And in combinatorics, we don't usually speak about independence, we just count things. So, if we concentrate on whether some events are independent or not, this is more or not about probability theory. Also, if we're interested in finding model for real word for example, a coin maybe non-symmetrical or a dice can be not, could be one part could be here. So then of course they think our assumption that all the outcomes one, two, three, four, five, six are equal often is wrong. So we should somehow consider another models which are called non-uniform distributions, we will speak about this. And sometimes we don't know, we want to speak about probabilities of different event even if we don't know exactly the underlying distribution. So, the simplest example, imagine some event A has some probability, so then we can see with the negation of A, the event say that A didn't happen. And what is the probability of this negation? Of course, negation happens in cases when our event doesn't happen. So it's just the complement. So the probability's one minus probability of the event. And say, for saying this, we don't need to know anything about underlying distribution. And finally, there is a part of probability theory very important which we are completely ignore in our course. It's about continuous distributions. Let me just say one thing. So imagine, I don't know, there is rain outside, and you go with a piece of paper and you see where the first drop falls. And it can fall at any position but you are interested in cases when it falls into some region. You draw a picture on this paper and you see whether it goes inside the figure or outside. And of course that outcomes are not, they're infinitely many of them, the drop can go in at any point on the paper. So you cannot count them but what you do instead, you just look at the area of this thing. And the empirical effect is that the fraction of cases when it goes inside the zone is proportional to the area of the zone, so it's kind of, mathematicians say measure for areas and other similar things. So the probability theory is a part of measure theory and a part which deals a lot with independence. But it's a complicated things, you need to know how to, the notion of integrals, so we will not go into this. And you can see the only discrete distribution but the number of outcomes is finite.
