Now we consider a famous quite well known paradox. It's called Monty Hall paradox and/or Monty Hall problem. And the name is because of some show of this name when this paradox was appeared. Okay, [INAUDIBLE] and I start with a short description just extracted from Wikipedia. There is a picture, you see this picture. And you see the short description here. And the story is like this. So there is a show, and the player, which comes to the show, sees three doors. He wants to find a car which is behind one of them, and he picks the door. And the game host opens one of the other doors, and, for example, the third door. And there is no car, there is no prize. Just a goat is just to make fun of the participant. So then the host asks the player whether the player wants to keep the guess or change it. So the player is now allowed to rethink and choose not the door 1, but door 2. And the question is whether the player should do this, or what is the best strategy for the player? And to make the game more clear, we should specify the protocol more exactly, and this is very important, as we will see later. Let, I prepare the slide with things in detail. So first, there is a TV show and there is a host and a guest. And there are three identical doors 1, 2, 3. And there is a prize behind one of these doors, so forget about goats. And there is a prize, a car, behind one of the doors. And this door is randomly chosen. And the guest makes a guess where the prize is, also randomly. And then the host looks open the other door with no prize, this. So there is one door with a prize and one door chosen by a guest, maybe the same one, but in all cases, there is another door which is not chosen by the guest, and also, there is no prize there. So the host opens this door and if there are two possibilities, the host chooses the random one. And then the guest is allowed to keep the guess or to change the guess. And then the door is opened, and the prize is given or not given, depending on the final guess. If it's the final guess is correct, the player gets the prize. So this is the game setting in full detail, I hope. And now you can think what were the question? So what should you do as a guest on the show? Should you keep your guess after the door has opened, or should you change it, or should you make a new random choice over? And now I'll try to present several arguments in favor of different theories. And I want to present it in a very convincing voice so you would be convinced, but then the claim is different, so you can't be really convinced at all, but let me try. So why you should keep the guess. What is argument why you should keep the guess, and that one is like this. So opening the other door, it's always, the host opens the door, but there's no prize. So opening this door doesn't really prove anything about the original door. So there is no reason to change the guess because this new events do not prove anything to you. So you should keep the old guess because there is no significant new information. Or why to make to make a new random guess among two doors. Enough of your recommendation, and this is supporting evidence. So now, after one door is opened, you can have a prize behind this or behind that door. So the best way, it's a random event, so the best way you can behave is just to also to make a random choice and choose the random door between two, and then you get the car. With the probability you want to have, so it's new random guess, it makes things better. So here is the reasoning. So there are two doors, don't know what is the correct one, but you can make a random choice. And finally, the argument why to change the door, and the argument is like this. The first door, if there was no second part, the first door has a prize probability one-third, and so the second door is just the complement, event which is not the first door does not have the prize. So the compliment has the probability two-thirds because there is a rule saying that the complement of event has a probability one minus the probability of the event, so as better probability, you should do this. So which argument is more convincing for you? And let me now try to move you in other directions, explaining why the arguments are not convincing. So in the first argument, here is it. The opening of the door by the host doesn't prove anything. Indeed, it doesn't prove, but it still provides some new information, even though this information is not decisive, it can change the probabilities, so why should you ignore this? So this is the refutation of this argument. It's valuable information, but even if it's not decisive. For the second argument, the second argument says that since you don't know where the car is, the only thing you can do is just to take a random door. But actually, there are several things which can be contested here. So for example, the idea that if there is some random process, to get the best chances, you must simulate the process. It's just the wrong assumption. Imagine we have a coin which more often falls head, so it gives head in 70% of cases, and you want to bet against this coin. And the wrong idea is to imitate the coin, is to try to say head in 70% of cases and tail in 30%. And you will win more if you play again this coin if you always say head. Of course, because there are more chances if you just say tail in some case, in some trial, and then you decrease your chances in this trial, so why do this? So this is why the second argument is bad. And the third argument, the probability is one-third. Then the complement has probability is two-thirds. And it's an error which we discussed, because this probability one-third, was for the one experiment, for the first experiment without this change, opening the door part. And then we changed the experiment, we have a new experiment now. So new probability is just the probability in the new experiment and we cannot apply the formula for the sum of probabilities. Because they were for the events in the same space. Okay, so now I tried my best to say convincingly what are the argument for, and the argument against. So where are you now? What do you think about this question and what is the correct answer? Stop here if you want to think and we will give the answer later.