 You're welcome everyone, second session. So today we will start with Rafael Cetri. Rafael is a professor at the University of Tecota, in Nice, France, in Serbia. He's an expert in private theory and also the graphic thermodynamics, and he will give a Blackboard lecture about the conditional problem. Thanks very much. First of all, I want to approach you because yesterday I was not here for welcome you. So I hope that all your eye-wild was good. Second thing that you will see, my English is very bad. I have a terribly French accent. So please, interrupt me when you are. It's not a problem. However, Tecota has not planned at the beginning. So don't worry. If I speak just five minutes and that you ask me 50 minutes of question, it's not a problem for me. So moreover, I think I have also a slot tomorrow for speak. So and I have not so much things to tell you. So don't worry. So finally, I decided to speak to you to something very elementary. So probably most of you listened this story before, but maybe it would be useful for you to have this example in head for the lecture of the second week, in particular the lecture of information that will arrive in the second week. These people can ask me questions also online. Okay. So I have also other problems that probably this is a very little charged. Nobody have an iPad here. Do you have an iPad? No. Okay. So I will start with this and maybe at the first time where it stopped, I will pass to Whiteboard. So the story that I want to start is that when you do base, the message will be very simple. The conditional probability from A to B, if I know B, is in general the dependence of B. Of course, it's different that A, if I know B prime. If I write this, this is for you evident. But no, I will tell you a story and I will see that in the real life, sometimes this totology for mathematics, sometimes you don't see really the difference between B and B prime. And sometimes this is the root of some paradox. So the story, the paradox that I want to tell you now, the first paradox, it's called, let's make a game. So it's a game, TV, TV game. So it's a real TV game. And it's called also Monty Hall paradox. So who is Monty Hall? He's a TV presenter, American TV presenter. So Monty Hall is a name which during no time presents this game at the TV, which was called, let's make a term. But what was the term? The deal was that there was in this game, three door, the door A, the door B, it must be the same. Sorry, it's not the same. And then in front of two doors, behind two doors, there was gold. So this was a losing door. And behind one door, there was Ferrari. So this was a winning door. Then some people, I don't know if in English, the player arrived and select one door. For example, what is your name? Can you pose it? Okay, perfect. So, by example, imagine that Jose arrived and then you're a bit in front with this. Okay, this is Jose. Then Jose to the door A, but he don't open the door. He put in front of the door and he said, okay, I think it will be the door with the Ferrari. He wants the Ferrari. This is a prerequisite. Then Monty Hall. This is important. Monty Hall, no, where is the Ferrari? Monty Hall, no, where is the Ferrari? He chose the place. So then after that Jose chose, Jose don't open the door. He is in front of the door. Then Monty Hall will open a losing door different that A, different that the door that Jose chose. So Monty Hall, open. This is the two notion that I know are important. A losing door. He can't open the door with the Ferrari. A losing door with a different that A, so different, different that Jose door. Okay, then from this door that Monty Hall open, imagine he opened B, then a gout go out. And then, and this is the main question which is interested for us, then Monty Hall asked Jose, no, that a gout go outside from B, it remains just two doors. Your door A or the door C. So what do you want to do? You can change if you want. You can go to a C or you can stay with a choice. Then do you want, do you want to change your choice? This is the main question of these patterns. So I must to tell that historically, this leads to a big debate in this TV show because the TV show runs a long time and after some times, some people, not scientific, try to understand if there is a good or not a good strategy. And then some reader asked to a woman, which was not a mathematician, which is called Marilyn Van Savant, which I think at this time a chronic on times kind of clinic where you can ask anything, any questions. Why? Because she has a very big, probably 210. And then in this sense, she was, she was a people think that she was good for understand to all type of question. Then they asked this question. And these girls give a good answer to the strategy. She gives a good answer. But then a lot of mathematicians, which was not probably probably, then try to tell, okay, this is very stupid, you are not a mathematician, your answer is totally stupid. Okay. And then there was a big conflict like long time between some people, mathematicians, physicists or not, or just the new fits, which arrives to the fact that at the end, she gives, she gives, she gives the good answer, the good answer. And then this lead to big debate, big debate. So at the end, there was so much debate, that Montiol himself, which is not at all scientific, decided to go in the countryside house, his countryside house with a lot of friends during our weekend, and to organize numerical simulation with the real life numerical simulation. So he tells to one friend, okay, you will change every time of the door, you will change, you will never change your choice. And then after playing many times, they see who is the winner. So you organize real life numerical simulation. I can do a break. Did you understand all the rules of the game? Okay. So now, since this is very popular, I am sure that you listened to the story before, it's arrived in movie, for example, Las Vegas 21. It's arrived in video game, strategic video game where you must choose between these three doors. So in this room, I can do things. Who thinks that we don't need to change the door? And who thinks that we must change the door? You must choose. Everybody must choose one choice. So who tells, okay, I don't care, I don't change the door. No, no, you can think that you take him around. No, no, no, but there is two choices. If you take him around, you must do one choice between the two. So who tells, I must change. Okay, so maybe here, I will tell 20%, who tells I don't change? No, but everybody must vote. Sorry. I come back. Who tells I don't change? I don't. I don't. One, two, three, four, five, six. Then who tells I change? Oh, you still don't vote. Sorry. What is your vote? No, no, no, the girl. Okay, okay. It's good. We pass. So there is two, there is two philosophy. There is no changest. And there is the changest. In this room, with the limit of the game, I will tell that, okay, maybe it was 70% of no changest, 30% of changest. Then I will try to to tell you what is more or less the argument of each family when I speak with them without math. The math will be very elementary after you see, but what is the argument of it? The non changest, the argument is based on equipability. If you speak with them every time at the end, the argument it is at the starting of the game, the three door, the three door was equiprobable. Equiprobable. What means this? It means that the probability was one third, one third, one third. Then after the effect that Bambtiol opens the B, it remains two door, and there is no reason why this two door is not equiprobable. I would. Lazar, do you think? Ah, okay, okay, no, no question. So the argument of them, it is after that the door B was open, it remains equiprobable. And then it is one half, one half. And then why it will take a risk to change if it is the same probability? In French, we tell the same or we can tell that the first intuition is always the better. Okay, then at the end, all this philosophy arrives to the fact that we don't need to change. Is the people in this room, which was non changest, agree with my explanation? Is their explanation is more or less the same? Okay, no, I will speak of the other philosophy, the other family, the changest. This is maybe a little more complicated, but not so much. So at the starting, there is two possibilities. Oh, José did a mistake. Oh José, as true. So if José did a mistake, what is the probability that he did a mistake? Two thirds, two on three. He has a probability of two thirds to do a mistake. So to be in front of the losing dough, to be in front of the losing dough, of a losing dough. And he has the probability one third to be in front of a winning dough. Okay, this I think everybody agreed. Okay, no, what will be a little surprise for you, but try to understand what I tell you. In all the case where he is in front of a losing dough, if he changes, he wins. Why? Because the Montreal opens the door B, a gout goes outside. If he was at the beginning in front of a gout, it means that C, the door C, was winning. No other choice. So if he changes, in this case, if he changes, he wins. Yeah, okay, but oh, you know that he was in front of a losing dough. Yeah, I don't know. For this, that I tell you that there is probability of two thirds, of two on three. There is another case where he is in front of the winning dough. But then in this case, if he changes, he loses. So conclusion, he must change. Two, if he changes in two, if probability two thirds, two on three, is a good strategy. If probability one or three is a bad strategy. So the conclusion here is terribly clear. He must change. So if there is in this room some non-changes people in the left column that know are convinced by this strategy of the right column or nobody is convinced by this. Is audio okay now? Okay, you can continue. Okay, so I came back. I hope it's good for everybody. And then, no, I will give for me ultimate argument. Is it to consider 1000 dough? So there is 1000 dough. And this 1000 dough, there is 999 goats. And one Ferrari. Then I will down the door quickly. 8, 9, 10. Okay, 500, 500, 1, 2, 3, 4, 5, 6, 7, 8, 9. Okay, 900, until 1000. This is 1000. This is 500. This is 1. So those arrive, they choose the same door, any door, but then by symmetry, we say that he choose the first door. Then Montreal arrive and you open the door too. A goat go out, a door three, a goat, goat, goat, goat, goat, goat, goat, goat, goat, goat, goat, goat, and suddenly it jumped the door 504. And then after goat, goat, goat, goat, goat, goat, until the end. And then it's remain on just two door, the first door that Jose is in front. And the door 504. And then he tell you, okay, no, what do you want to do? Because I jumped. If it's stupid, he tell you, I jumped the door 504. It's funny. In front of this 999 door, I just jumped this door. Why? Because she's winning probably. So do you change the point of view now? People which before was not inclined to change, no, do you change in this case? Yes. If there is somebody which still tell no, there is two door, the point is one half and then I stay with my door. Somebody? Okay, so in this case, if you change the door, the point is to win, will you become 999 on 1000? So in this case, it's very stupid to don't change. In the other case, I will prove no, it will become two-third by changing. But then you can tell me, okay, but two-third, I don't care. I mean, I prefer to have one-third and two, okay. But if you change this, in this case, it's 999. And also you can tell me, okay, but I don't care the theory. Okay, but then imagine the life, imagine the game where it is your life, which is in question. The good door is the fact that you stay in life. Then you see that it's better to have 999 on 1000 for the probability that you have one over 1000. So no, for still people which are not convinced, I will do the mathematical elementary proof. But you will see the mathematical elementary proof, in a sense, we don't give you the words of the problem. So I will call big F, it is a random variable. Where is, where is the Ferrari? Where is the Ferrari? F is for Ferrari. Now this belongs to A, B, C. I will call P the random variable, which is the door open by Monteo. And this belongs, a priori, also to A, B, C. And then one important thing, in all my conditioning probability, I will not put one important thing. It is, I put in red and this is true all the time. Jose is in front of the door A. He don't open, but he is in front. And so what I want to calculate for, for, good answer, mathematics to this conflict, it is the probability that the Ferrari is in A if I know that Monteo, the presentator, open the door B. Okay, so this, what I can find? For example, I can find maybe one half. And if I find one half, then people of no changing will be true. I don't need to change. It's equiprobable. It stays two doors. The two doors has one half, one half. But I can maybe, I can maybe find one third. And then this was the other column, the changes people, the changes people. And then I can find other things. I don't know. So I will do. So it's very elementary base probability. If the Ferrari is in A, is a presentator open B base, tell me, this is one of the most important formula, I will tell in science, but this is true just in classical physics, which is that it's a very easy A and the presentator open B divide by the fact that the presentator open B. Okay. No, once again, I will use for the numerator against base, but I use that it is the point is that the Ferrari is in A times the point is that the presentator open B. If the Ferrari is in A and in the denominator, I will use the total probability. I will decompose my event that I want for it is that the predator open B on the partitioning of the space, which is the place of the Ferrari. And then I will arrive to the Ferrari is in A times the presentator open B, if the Ferrari is in A plus the Ferrari is in Bay B times the probability that the probate predator open B. If the Ferrari is in B plus the probability that the Ferrari is in C times the probability that the presentator open still B, if the Ferrari is in C. Okay. Is everybody agree with this elementary question? Oh, so somebody have done so of what I did quickly. Jose, it's okay. Sorry. Okay, so the denominator is called the total root probability. In fact, for proof to pass from ear to ear, you must to put the information here. But now you want to add the decomposition until the partition of the space, which is the position of the Ferrari. So in fact, you will tell that P equal B, the event that you want, it is P equal B intersects with all the space. But all the space, it is Ferrari equal A, union Ferrari equal B, union Ferrari equal C. And then you do this intersection of union is union of intersection. And after you do the sum and you find this. In fact, we do this every time without giving a name. But okay, this is the name of total probability, but this, this is totally natural. But this is wrong. Okay, yeah, it can be strong for you. But I want to tell you that this is, for example, is this force is wrong in quantum probability. This is a very something trivial in a sum that we become where we would stop to be valid when you use this born projection and all this. Okay, so now you will help me to find the value numerical value of all the terms that they put in color. Okay. Okay, and last blue. Okay, so first, can you help me for tell me, tell me what is the value of the yellow. The yellow in front, it is the probability that initially without any conditioning, we need without any help from nobody. So it's at the starting of the experiment, what is the point is that the Ferrari is in C. One over three. Perfect. So all this go one over three. Okay, no, I will do, I will jump the green which is maybe the most hard. So I will focus on this, the blue. First, I will for you read in French, sorry, in English. So what is the probability that the presenter openly is if the Ferrari is in C and we don't forget the Ferrari is in C and Jose is in front of A. So remember that Montiol can't open a winning door and can't open the door for Jose is one. Exactly. Perfect. Okay, no red. The probability that the presenter open B is the Ferrari is in B. Zero. Perfect. And green. Green is more hard because we have three doors. The Ferrari is in A, is behind A, and Jose is in front of A. One half. Very good. He has two choices. He has two possible doors, B or C. So that you open B is one half. Okay. So is he already agree with this value? But then it's one third times one half on one third times one half plus one third times zero plus one third times one. So it is one over six. One over six plus zero plus one over three. It is one, one plus two. And surprise, we find one over three. The one over three of the challenges. Because no, what we find is the probability that the Ferrari is in A if the presenter Montiol open B. This is one third. And because the other, the point is that the Ferrari is in B, is the presenter open B, but this is zero. And then the last point is that the Ferrari is in C if the presenter open B, must be two third. And so you see that the conclusion is clear from a mathematical point of view. We must change. Is there any question? So in a sense, this gives you an easy answer to your problem. But my feeling is that this is what happened. So we will do a supplementary calculation. What is the probability that the Ferrari is in A if I know that the Ferrari is not in B? So once again, we will do very simple base. It will be the probability that the Ferrari is in A and that the Ferrari is not in B on the probability that the Ferrari is not in B. But on the numerator, if the Ferrari is in A, she is not in B. So one is included in the other. So the intersection is just the Ferrari is in A and in the denominator, it is just A or C. And then it is one third on two thirds. And then catastrophe, what I find here, it is one half, the very bad one half of the non-changing people. So this is very bad. So what I do here, I come back. I just I prove that if the Ferrari is in A before I prove that if the pre-emptator open B was one third and no, I prove that Ferrari is in A when Ferrari is not in B, equal one half. So the question which arrived, it is, what is the relation between these events, the pre-emptator open B and this event, the Ferrari is not in B. So did you see one event which is included in the other? Oh, do you think that they have to say? It's not the same, no, because the random variable which door the pre-emptator opens is not the same path, the same event that the Ferrari is. Okay, so can you tell me if there is one inclusion? I think it's more correct the second. Okay, first, forget the public story. Is this inclusion is true? If the pre-emptator open B, have you, the Ferrari which is not in B? This is true or not? Yeah, this is true. This is true, we are arguing. And we can't discuss this, this is true. If the pre-emptator open B, a gout go out, a Ferrari is not in B. The question is, do we have reciprocal? Is this true or wrong? This is wrong. The Ferrari can be outside B and if the Ferrari is in A, the pre-emptator can open C. So there is no reciprocal. So you see that we have an event, P equal B, which is more precise, which gives you a more precise information. So the word information will be important for the next week, that the events, big events, Ferrari is not in B. And then this is the root of the problem for the people which are non-changists. When they evaluate the story, they translate in their head, what is the information? The information is the Ferrari is not in B. Then they conclude one half an hour, if it's good. But they do a bad translation of the information because the most precise information of the experiment is not the Ferrari is not in B. The most precise information is the pre-emptator open B. And then this is a moral of my story. It's very dangerous and it's very hard. But when you take a decision for your life, for science, you must to condition with respect to the most precise information, to condition with respect. If not, this can totally change your decision as here. To the most precise, I will tell fine, fine, or most precise information. For example, if you want to bet on the next match of the niche football club, it's better if you look all the match, including the last match of last week. This is a very precise information. All the match since the 10th match, that if you arrive from the March planet, that you don't see the three last match and that you must bet with your information that you see of what the match that they did two months ago, then the information is less good, of course. So in this case, we will not do, but you'll see that in this paradox, very quickly, we translate with, because we are limited as human, we translate badly what is the information. And very quickly, we tell, okay, if you open B, okay, then then you're trying to draw. You're trying to draw? Good. Equal probability, one-half, one-half. This is bad. This is not the game. The game that's open B. It's more precise that a goat is in front of B. Okay, no. For finish, I think when I have to finish, soon, no? I am totally lost with the program. I have five minutes. Okay, I will take 10 minutes and I will stop. I will do the dual paradox. So the dual paradox is called... Ah, no, sorry. Before to do the dual paradox, I want to give you a little exercise for preparation of the next week. Can I ask the questions about... Hey, of course, for that, just I give the exercise after I go to the next week for preparation of the lecture of information of Dimitri Petritis. So, yeah, I will ask you to calculus the information of the random variable F, which was the position of the Ferrari. Okay, you must find log 3. And I will ask you to calculus the entropy of F if I know P. Conditional... Entropy of F, I know P. This, you must find log of 3 minus two-thirds log of 2. So, this is, finally, the entropy of the distribution that we find at the end, the position of the Ferrari if I know P. And then, I think next week, a notion of mutual entropy will be defined between the position of the Ferrari and the place of the Ferrari, which is, finally, the diminution of entropy between the initial and the final experiment. So, you must find something. And then, this, in a sense, when next week, people will speak of mutual entropy, then you must translate in your head that this is the information that Montiol put to you by opening the door. This is the information given by the presenter. Okay. So, for do these things, I think it's just the Shannon entropy that you know. This, you can look the definition on Wikipedia, but it will be, again, the Shannon entropy, but with respect to the conditional probability, F, if I know P, that we calculated before. Okay. Now, this is the definition. And this, I think, you will have next week probably another definition, then you must remember that in this experiment, this is this information which results from the fact that Montiol appeared in the story and give you an information by opening the door. And then, maybe this can help you to have some intuition for some abstract objects to the next week. So, before to, to pass two minutes on the next paradox, I will answer two questions of Elimir. Yes. Thank you. The first question is, at the beginning, the book is uniformly. Yes. And sorry if I missed. If it is not uniform, this information. Ah, it's changed totally the story. It's changed totally the story. Change. If it's not uniform, my complication is bad. Uniformity of the, of the theory was. I put this. Yes. If not, you change the story. But then, you know, you know, for, for that the paradox is apparent, you must take the simple. Okay. If you complicate the story, then the paradox will be less apparent. Okay. The same question. So, when you perceive the event, this is behind the first off and the computer. And then you open the door. This will be always reduced. When the presenter opened the door. No, but you, you don't open the door. The candidate don't open the door. It is Montreal, which opened a losing door. And then you are true. Exactly. Look, look at this formula. You are totally true. You will pass from this entropy, which is bigger, larger industry to this entropy, which is more little. Can you say something about your relationship with an entropy? For example, you open the second door, there is a book and there is no book. But this, yeah, but this is, yeah, yes, you are true, but this is another experiment. In this experiment, you don't open the door for the world. The presenter opened a losing door, who must be about to go out. He can't open a winning door. The presenter opened a losing door. Obligatory. But you are true. Your question is more or less this. Your question is more or less this. It means the fact that the presenter opened the door, change the information context of the experiment and then give you an information, which is, which will be the mutual property of the next week. And if it is not uniform, you only point that it will change in this year and three. Yeah, but probably it will also change the paradox. It's not that all will be changed. This will change, this will change also. So all must be changed. The paradox must remain here, but I don't know it. So I will not advise you. Thanks. Is there is other question? Yes. Can you repeat the part of the most precise information? Yes, of course. So are you agree that, are you agree that if the presenter opened B, then the Ferrari doesn't B. Are you agree that if the Ferrari doesn't B, this don't imply that the presenter opened B. So this is the most precise information of the experiment. If you translate the experiment by this, you forget the information. This is perfectly all the information from the opening to Monte Hall is here. This is a bad translation. This is the analogy in the football club where you look just the match three months ago. You forget the recent match. Okay. And so my conclusion was that in the life, it's very important when you take a decision with respect to what happened before, to translate exactly what happened before. In this experiment, if you translate by P different B, you take a bad decision, which is, I don't change the job. This was my, okay. In a sense, what I claim is very easy to claim, very complicated to do. If it was easy, everybody will be mired. So we are a failed machine for taking the decision because it's very complicated to translate all the information that we have in the good ensemble. The presenter opens the door, then you can use the event in different ways. You can use it by saying, okay, this is very useful information for people. Because also in the traveling in a different way, this is a non-change. We have to be very careful. There is a phenomenon happening, which is giving information. The way you process the information from this event can be very different. Okay. So now I will just speak of the dual during two minutes of the dual paradox, which is called, which is called, which is called prisoner paradox. Just in order to interest you, I will change the context of the prisoner in a maybe more modern context. Then there is three men in this room. I'll show you a change. What is your name? Yes. What is your name? Hi, it will be hard. Can you... Gilbert. Gilbert. Miguel. And Ravi. Sam, the player. Oh, are you with the X? No. Okay. Then the three men who... I don't know if you with a good name in English, the same girl. They want to go out with the same girl. The girl have no name. Josephine. But we don't know who Josephine chose. They don't know. Josephine knows. But Josephine has the best term, which is called, I don't know, Cunegonde. Cunegonde. And Josephine tells Cunegonde who she chose. Then they are all three in a room, and then Gilbert wants to know. So Gilbert wants to see Cunegonde, which is in this room, and tell, okay, Cunegonde, can you tell me who Josephine chose? And then Cunegonde tells me, no, no, I can't. I can't. Josephine asked me to don't tell to nobody. I can't tell you. It's the beginning of the story. So Gilbert comes back to see it and then things because he's a physicist and mathematician. So I have the solution. Then he comes back to see Cunegonde. Cunegonde is the best friend. And then he tells, okay, I understand that you can't tell me if you choose me or not. But then can you tell me between Miguel and Ravi? A man that she don't choose. Can you tell me between M and G? A man that Josephine don't choose. So Cunegonde, which is also a mathematician, thinks, thinks. Okay, I accept this. I accept you. I tell you that Josephine don't choose Miguel. Miguel is not chosen. So Gilbert is very happy and tell to Cunegonde, okay, you are very stupid because before my probability to be chosen was one-third. But know that Miguel disappeared from the story. It's remain just me and Ravi. And then, no, sorry, I have one-third. So you give me good information. And the question that I ask for you is, is this true or not? Yes, yes. So three man, Gilbert, Miguel, Ravi, which want to go out with the same girl, Josephine. They don't know who Josephine chose. But Josephine has a best friend, which is Cunegonde. And then Cunegonde, no. Who is the man from the three, which is chosen? So initially they have probability one-third. Okay, but then Gilbert want to know, then he go to see Cunegonde and ask if they are chosen and I am the chosen. She said, I can't tell you. I don't know. I can't tell you. I am not allowed to tell you. Okay, so then he came back to see Cunegonde and said, okay, I understand that you can't tell me, but then I have other question, which is between the two other, can you tell me one which is not chosen? And then Cunegonde said, okay, just like I can do for you. And I will tell you, Miguel is not chosen. So at the end he stayed Gilbert of the V. And then Gilbert said, okay, good. May I probably know his one-half. So who thinks that Gilbert is true? Nobody. Ah, one person. Okay. And who thinks that Gilbert is wrong? Okay, and all the other still thinks nothing. Oh, this is, ah, she's wrong? Wrong. Okay. Okay, so you know you can prove to Tom that he has wrong, means that this afternoon you will prove the probability that the, the, the lover is, no, the lower is Gilbert if Cunegonde tell Miguel is one third. Instead, the probability that Zollover is Ravi, if Cunegon tell Miguel is two thirds, you must prove this with exactly the same question that I did before. And so in a sense, the winner of this story is Ravi, which like nothing. And if you if you tell me okay, but what is the root of this? But it is the fact that when he asked his question, Gilbert, he is good himself. So he takes no risk. So the probability will not win. It will be not fair. The other take a risk. The other is not in the story, but the other can can be answering as people with which has not chosen. So finally, when he is not answering as not chosen, too much negation. Then this tell you telling something by the fact that maybe is more probable that he is chosen. And okay, the maths after very simple give you this. So innocent is the same paradox except that the story is a little different because you can't exchange your name. But if not, it is. Okay, I think I conclude what I want to tell you today. On this, my main message is warning conditioning probability can be not intuitive. And then conditional entropy will have the same kind of paradox of course, because it will be just the entropy related to this conditional probability. Okay, fine. Very much. No, okay. Good. Thank you.