Now, we want to consider a paradox. A paradox happens when there are two different arguments which give different translations you should- that is a kind of a conflict between reasonings, two different way of reasoning. So paradox is known in the setting with children but. So imagine some woman has two children and you know that at least one of them is a girl. So what is the probability that there are two girls? The other child. She has two daughters? Okay, This is a biological question and whether they dependent, the sex of the two siblings is dependent or not. But just to ignore this problem, let's consider the cleaned formulation without any biological information. So Mary just make two times she tossed a coin. And it's known that at least one of outcomes is a tail. And so, what are the probability to have two tails? And I will show you two arguments, which give different answer. The first argument. First solution. Of course, if there is one of the outcomes is a tail, then there are two cases. Either the first bit is a tail or a second bit is a tail. And of course this case are completely symmetric, so we can consider one of them. We may as they say, we may assume without loss of generality that it was the first case, for example. First bit a tail. And then there are two possible outcomes, tail head and tail tail. And this is a favorable outcome when we have two tails. So there are two outcomes equiprobable and one of them is favorable, so the probabilities one have. So are you convinced? What do you think? Let me show you another solution. It's more formal without this loss of generality stories. Here is again the question. And now we're just consider four equiprobable outcomes: head head, head tail, tail head, tail tail. And just don't do the boring thing we just look at all the other events. So event here this event, C, at least one tail. Where is it? It's here, here and here. So it's probability three over four. And this event probability of two tails is just this one. So among the three events we have only one. Event E is two tails it's only one outcome in probability one over four. And actually it's now easier to compute the conditional probability here, is it? So we divide this one fourth over three fourth, is one over three, or we can say that there's a condition is this, and the event is this. Now event is a part of the condition. So out of three possible outcomes, our event selects one, so it's one over three. So what do you think? Which argument is correct? Actually the second one is correct. And to understand what is the error in the first argument we should look at a picture. And first we should read the problem statement and parse the problem statements to say. So here as shown in colors, I prepared the slide. So it's just the same statement, but there are three parts. The blue one, this one, yellow one, and the red one or magenta one. Okay. What is the role of the three parts? The first part is just sets the probability space. Mary tossed a fair coin twice means that there are four outcomes and they are equiprobable. This is not said explicitly but if you are in probability theory textbook, you always assume this, If it's not said even if it's not said explicitly. And then here's the condition, at least one of the outcome is a tail. So it's a condition which is event made of three outcomes. And the probability of this event is three over four, here it is, the outcome of probability. And so there is certain space, the condition and now the event we are interested in. So event E is two tails and this is one, is part of the C. And this had probability one fourth, and the condition of the probability is one over three. So I just repeated the second argument but there are only difference is I clearly separated here, the setting of the probability space and the condition. So it's, when you read this, it's not clear just Mary toss the coin at least of the one of the outcomes like kind of story. But if you want to ask a probability question, you should clearly see, clearly distinguish the probability space and the condition. And actually it was the source of many errors. I think that just in 19th century people do not understand this clearly. And this was a source of many paradoxes and so on. But hopefully now people are better at this. And we can look on the fourth, first solution wrong solution. Why, what we really did there and there is a picture. This is a picture for the correct solution. We have this space, this blue one, things, condition, this olive one, and event. This is event. So what we did in the first solution, we consider two other events. C prime and C double prime. And this is the event in the second tier, this is the first tier. Maybe they should be denoted. But anyway there are two events and together they make the condition C. But and for each of them the conditional probability is different. It's one over two. And for the entire condition it's one over three. So when we, when we say without loss of generality, we do a very bad thing. We replace this by this, only because this event C is a union of these two symmetric events, C prime and C double prime. But for each of them, the conditional probability is different so you cannot replace. And kind of my take home message. That information is not I can not always, you should distinguish between information (which explain the space) and the condition (which is part of the problem). So your first, you should explain what the probability space is, and then you say the condition in the event, and then the problem is set. And if you just make some kind of a story, it's very easy way to get into trouble. You should should avoid this and be clear.