Now, let's speak more about Finite Spaces. Let's start with what we're finished with is what is called the mutual exclusive event and the formula for some of the probabilities. So, the event is a set of outcomes and mutual exclusive event are just these joint set of outcomes. But there is a important thing. So imagine somebody asks you I toss a coin and get, I don't know, hand and another time I roll a dice and get three. So are these outcome events, are they mutually exclusive? Correct answer is not yes or no. The correct answer is that they are not defined on the same probability space, so the question has no meaning. It's often the case in probability theory when the question has no meaning. Anyway, if there are two mutual exclusive events, then the probability of their union A or B is the sum of probabilities. That's what we have seen for a specific example, and this will happen in the general case as well. A special case of this rule is with this mutual exclusive event are complimentary. So imagine we have some event A, then it defines another event which is called not A. This event happens when A doesn't happen. In terms of set theory, it's the complement of the set which form the event. And A and not A are mutual exclusive events and together they can cover the entire space, so the sum of probabilities should be one. So the probability of not A is one minus probability of A. So if you believe that probabilities, I don't know. To see a crocodile is one third for whatever reason, then you should agree that probability not to see is two-third at the same experiment. There is no reason of probability space here, but just it's a simple rule. This is the explanation why it happens, because they are mutual exclusive so we can apply the sum rule. The last thing we want to discuss about these two events, what if they are not mutually exclusive? So then this probability of A or B may be not equal to sum of probabilities. But now, the question is is it bigger or smaller, what do you think? Let's see an example and we will see the answer immediately. So imagine we have two events A and B and they are not mutual exclusive. There are some outcome three in both, and then probability of A is P1, plus P2, plus P3. And the probability of B is P3 plus P4. Now, there is common P3 here. So if we look at the union A or B, it just P1, P2, P3, and P4. And if we add this to probabilities look, we will count P3 twice. So to get the correct answer, we should compensate and subtract this intersection of these events which is now only one P3. And this rule probably you have seen this in combinatorics is inclusion exclusion formula. And it's true for general probability spaces, also not only for the equiprobable outcomes. Another story. When a final probability space with non equal probability of outcomes appears, it's what is called sequential choice. Let me explain. There are some prepared example. The question is how to select a reasonable probability model for some experiment and the experiment is like this. We have six balls, one, two, three, four, five, six and we have two boxes, Balls one and two are in one box, and three, four, five, six is in the other box. And then the probability experiment is described as we choose a random box and then choose the random ball in the box. So the question is what probability space reasonably corresponds to this process? And what we'll assume that the boxes have equal chances to be selected, and in each box all the ball have equal chances to be selected. We want to decide what are the probabilities of all six outcomes. Probably you can say the answer immediately, but still I prepared some slides to explain this. So just keep it if it's obvious. But it's useful to represent what is happening in a kind of a tree. So we have a tree, and this is the first choice whether we take box one or box two. And here we made the second choice, when we takes one ball either one or two if it's box one, or three, four, five, six if it's box two. So, in half of the cases we will go left, in half of the cases we'll go right, because the boxes are like probable. And here then we can go in half of this, half we go here and half of this half we go here, and here we have one of the eight. So, this is how we get the probabilities for six outcomes. And using the fact that at each point all choices, all five of choices are equiprobable. And then there's another explanation. So here is just the event we select box one, and here is the event we select box two and they should have probability one half. And also, this individual probability inside the boxes should also equal. So, there is no only one possibility for that, and that's written here is exactly the same numbers but explained a bit differently.