False Monte Conclusion - Carlo or Player Error. Logical paradoxes

The players are undoubtedly aware of the Monte Carlo false conclusion. Some, however, will be surprised to learn that this is a false conclusion - they then consider it a "Monte Carlo strategy." Well, this is exactly what the croupiers are counting on.

We all know that there are half black and half red sections on the roulette wheel, which means that we have a 50% chance that when the wheel is turned, red will fall out. If we spin the wheel many times in a row - say, a thousand - and at the same time it will be in good order and there will be no cunning devices on it, then red will fall out about 500 times. Accordingly, if we spin the wheel six times, and all six times it comes up black, we will have a reason to think that by betting on red we will increase our chances of winning. After all, red should fall, right? No it is not true. On the seventh time, the probability that red will fall out will be the same 50%, as well as every Next time. This is true regardless of how many times black is dropped in a row. So here's a very reasonable piece of advice based on the Monte Carlo error.

If you are going to fly on an airplane, for your own safety, take a bomb with you: after all, the likelihood that two guys with bombs will meet at once on the same flight is extremely small.

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Finally, hands and other organs have reached the next article.

So, meet the next guest in our studio - Player error or false Monte Carlo inference. The term invented not by me, although it sounds somehow poppy, without the abstruse words characteristic of highbrow guys. This distortion is very simple in understanding, nevertheless, it lives everywhere, both in the thin bluish substance of the lumpen, who have come down to the letter E in the study of the alphabet, and in the dense thickets of raisins, wise with experience with a bunch of knowledge of gray-haired sages. Here's what Vicki has to say about this:

The gambler's fallacy or Monte Carlo fallacy reflects a widespread misconception of randomness. It is connected with the fact that, as a rule, a person is not aware at the intuitive level of the fact that the probability of the desired outcome does not depend on the previous outcomes of a random event.

For example, in the case of tossing a coin many times in a row, such a situation may well occur that 9 "tails" in a row come up. If the coin is "normal", then for many people it seems obvious that the next time you throw it, the probability of getting heads will be higher: it is difficult to believe that "tails" can come up ten times in a row. However, this conclusion is erroneous. The probability of hitting the next heads or tails is still 1/2.

It is necessary, however, to distinguish between the concepts: the probability of falling "heads" or "tails" in each specific case and the probability of falling "tails" ten times in a row. The latter will be equal. However, the probability of any other fixed sequence of "heads" and "tails" at 10 coin tosses will be the same.

What does this mean in translation into our pihara-trading language?

The simplest and most well-known example is the classic flat dogon. Those. popan pays tb2.5 no matter in which match by odds + -2, merges, doubles the bet on another match tb 2.5 with odds of about two, merges, doubles the rate again, etc. Well, or Martingale, call it what you want, not the point. And if you offer him to shove the total less at the third or fourth iteration, he will probably get angry with the mega-argument "Thou Thou, because there was already 3 tm, right now the probability of tb is higher." And it turns out to be absolutely right. But only in his imaginary universe, in real life, everything is somewhat different. The likelihood of a future event, all other things being equal, does not depend in any way on the past, at least one even a million. Axiom.

Into the account of a million. Recently we spoke with a Kent about this (¡Hola senor Alejandro!). At some moment, a person who perceives this world absolutely adequately to the simple question "Before that, a million heads have fallen. What is the probability that it will come up tails?" replies that a little bit, but still higher. We quickly eliminated this point, but the situation is indicative.

Moved away from the topic. So what should a person do who has fallen into dogon (of which I am a tough opponent)? The most important thing is not to think red or black, total over or under total, fish or chicken, nothing depends on you. Just pkhni for any outcome and hope in front of the TV, but rather go in for sports, sex, fishing, emphasize the necessary. So you will burn fewer calories from the "wrong choice", which, in fact, did not exist. Now mathematics (gods, fortune, mastiushka, call it what you want) has turned its face or ass towards you, and nothing can be done about it. There is no need to catch up with seven iterations of the total more, boldly phai the total is less, this does not affect the result in any way. More precisely, it only affects the fact that the catch-up will eventually put you on your shoulder blades, the mathematics cannot be fooled, the margin will do everything for you. For many years I watched the tops of the pikhars on the pump-room, among the successful at a solid distance there was not a single catch-up, but now this is not about that.

Let's take another example. At one time I talked online in trading sessions with one well-known equestrian trader, I will not voice his name. So, he, too, was caught in the web of this cognitive error. The train of his thoughts proceeded along the following lines: 3 times in a row the favorite mare came first, which means that the next fava race should be laid. Won - hsn, leim fava in the next race with doubled rage, then tripled, etc. And this "system" gave profit for a certain period of time. But at one crappy moment the inevitable happened: mathematics won him over, he got into such a sum that he left our slender, albeit not stable, ranks for a long time. He could not believe that such a thing was possible, it took him a long time to accept, understand and rethink, he caught such a depression that a massage with Australian koalas would not have helped him at that moment. I think this is not an isolated case.

I had a case when I myself got into such a thing. I vaguely remember the details, the case is old. The long-standing Italian championship is a dull show, catenaccio, draws are frequent guests. In one of the rounds there were no draws, and my fragile brain tells me that the trend will return in the next round. Stupidly took draws in all matches and ... megaposos, again not a single draw. But I’m a tough pepper, you won’t take me so easily, in the next round I again take pricks with a doubled rate (hello Illusion of control) - and only one draw in the whole round. According to the classics of the genre, I had to shove and fight back, well, now everything will definitely be good. But reality dug deeper, I stupidly ran out of money. I’ll answer your question: I don’t know what happened in the next round, I didn’t watch the cuts, I thought I’d go crazy if I saw an ocean of nothing. An expensive lesson, but it turned out to be very useful.

I'll wrap up, 3 a.m. I make a riddle to consolidate, self-analyze and improve the absorption of the above. What is the likelihood that Barcelona will not win at home against, say, Malaga two times in a row? Kef on p1 - 1.2. And how soon can this come? The first person to answer correctly with me is a nishyachok, say, I will write an article on the topic chosen by him.

So, in summary. Don't look at what happened before, it doesn't matter. If you looked, draw conclusions, they are subjective. Draw conclusions - do not make predictions from them, they are unreliable. Nevertheless, they built a prediction - be ready to easily change it, do not cling to it as the only true one (one of my favorite cognitive errors, let's talk about it another time). If you grabbed and can't let go - go to the factory, get in a taxi, a pizza delivery man, choose any other pick, games with probabilities are, alas, not for you yet. But do not despair, read, work on yourself, improve your understanding of the processes taking place in your head, brown your brain. Having passed the oil-bearing and coal layers, sooner or later you will get into states of mind that are not so ossified and compressed, and someday, with a certain degree of probability, you will be able to set foot again on the ornate path of non-kyle dough.

This is how the boy decided that the flashlight is the cause and salvation is the effect, while in reality the flashlight will only illuminate his escape route.

False conclusion Monte Carlo

The players are no doubt aware of the Monte Carlo false conclusion. Some, however, will be surprised to learn that this is a false conclusion - they somehow consider it "the strategy of Monte Carlo". Well, this is exactly what the croupiers are counting on.

We all know that there are half black and half red sections on the roulette wheel, which means that we have a 50% chance that when the wheel is turned, red will fall out. If we spin the wheel many times in a row - say, a thousand - and at the same time it will be in good order and there will be no cunning devices on it, then red will fall out about 500 times. Accordingly, if we spin the wheel six times, and all six times it comes up black, we will have a reason to think that by betting on red we will increase our chances of winning. After all, red should fall, right? No it is not true. On the seventh time, the probability that red will fall out will be the same 50%, as well as every Next time. This is true regardless of how many times black is dropped in a row. So here's a very reasonable piece of advice based on the Monte Carlo error.

If you are going to fly on an airplane, for your own safety, take a bomb with you: after all, the likelihood that two guys with bombs will meet at once on the same flight is extremely small.

Vicious circle in proof

A proof vicious circle is a situation in which the statement itself is used to prove a statement. Often this logical error in itself becomes a real anecdote: the narrator does not even have to invent colorful details.

Autumn. The Indians on the reservation ask the new leader if the coming winter will be cold. The leader, however, was modern man and knew nothing of how his ancestors knew whether the winter would be warm or cold. Just in case, he ordered all the Indians to store firewood and prepare for the cold winter. A few days later, it occurred to him, albeit belatedly, to call the National Meteorological Service and inquire about the forecast for the winter. Meteorologists said that the winter is indeed expected to be very cold. Then he told his people to be even more active in the preparation of firewood.

After a couple of weeks, he decided to clarify the forecast with the meteorologists.

- You are still predicting us cold winter? He asked.

- Oh sure! - answered him. - Winter looks like it will be extremely frosty!

After that, the leader ordered the Indians to drag every chip that they could pick up into the reserves.

And again, a couple of weeks later, he called the National Meteorological Service in order to find out more precisely what experts think about the coming winter.

- We assume that this winter will be one of the coldest on record! - answered him.

- Really? - the leader was amazed. - How do you know?

- Yes, the Indians are stocking up on firewood like crazy! - answered the meteorologists.

So, as proof of the need to collect as much firewood as possible, the Indian chief eventually gave his own instruction to store as much firewood as possible. A vicious circle in proof got the Indians to file great amount wooden round. Fortunately, by that time they already had circular saws.

Assertions backed by links to higher power, loved by all bosses without exception. However, argumentation based on authority in itself is not a logical error: expert opinion is no worse than other types of proof and has every right to life. The mistake, however, is to hold on to the authority’s opinion as a straw to support your case, despite compelling evidence to the contrary.

Ted, meeting his friend Al, exclaimed:

- Al! I heard you died!

- That is unlikely! - Al laughed. - As you can see, I am quite alive!

“It's impossible,” Ted said back. - The person who told me about your death, I trust much more than you.

When appealing to expert opinion, you always need to understand who exactly you consider to be the authority.

A customer at a pet store asks to show him the parrots. The salesman brings him to two beautiful birds.

“One of these parrots costs $ 5,000 and the other costs $ 10,000,” he says.

- Wow! - the buyer gasps. - What can one that costs 5 thousand do?

- He sings all the arias from all the operas by Mozart!

- And second?

“It reproduces Wagner's Ring of the Nibelungen in its entirety. Oh yes, I have another parrot, it costs 30,000.

- Wow! And what can he do?

- Personally, I haven’t heard anything from him yet. But these two call him "maestro"!

In our own expert opinion, some authorities deserve much more credibility than others. The problem, however, is that the person you are talking to may have different authorities than you.

The four rabbis regularly engaged in theological controversies, during which three usually united against the fourth. Once an elderly rabbi, as always, left alone and unable to withstand an argument with three rivals, decided to appeal to higher powers.

- God! He cried. - My heart tells me that I am right and they are wrong! Please give me a sign so that they can be convinced of my correctness!

It was a beautiful summer day. However, after the rabbi finished his prayer, a black cloud appeared in the sky, just above the heads of the four “colleagues”. Thunder rumbled, and the cloud disappeared without a trace.

- Here it is, God's sign! I knew it! Now do you understand that I am right? The old rabbi exclaimed.

However, three of his comrades disagreed with him, saying that on hot days such clouds are by no means uncommon. And then the rabbi prayed again:

- Lord, I need a clearer sign that would show that I am right and they are not! Lord, give me a more impressive sign!

This time, four black clouds appeared in the sky at once. They instantly merged together, and lightning struck the top of the nearest hill.

- I told you I was right! Cried the rabbi.

But his friends reiterated that everything that happened can be explained by completely natural reasons. The rabbi was already ready to ask God to give him a huge, undeniable sign, but as soon as he managed to say: "Lord! ..", the sky turned black, the earth shook and a powerful thunderous voice rumbled:

- OH PRRRRAAAAV!

The old rabbi, akimbo, turned triumphantly to his companions:

- Well, now you see ?!

“Well,” one of the rabbis shrugged. - Now there are three of us against two!

Zeno's paradox

A paradox is a reasoning that seems to be quite sound and is based on supposedly adequate evidence, however, in the end it leads to contradictory or frankly false conclusions. If you tweak this sentence a little, it becomes a ready-made definition of an anecdote - at least, most of the anecdotes in this book will fall under it. There is something absurd about how true statements turn false - and absurdity always makes us laugh. If you try to keep two opposing ideas in your head, you cannot avoid dizziness. But more importantly, with the help of a paradox, you can make the company laugh at any party.

Player error (gambler "s fallacy)

O. and., Or the false conclusion of Monte Carlo, reflects a widespread misunderstanding of the randomness of events. Suppose a coin is flipped many times in a row. If there are 10 heads in a row, and if that coin is correct, it would seem intuitively obvious to most people that tails are delayed. However, this conclusion is false.

This error has received the name "negative recency effect" in the special literature and consists in a tendency to predict an early termination of what often happened in recent times developments. It is based on a belief in local representativeness (that is, on the belief that a sequence of randomly occurring events will carry the characteristics of a random process even when it is short). Thus, according to this misconception, the generator random events For example, a coin toss should lead to outcomes in which - even after a short time - there will be no significant predominance of one or the other of the possible outcomes. If a series of identical outcomes occurs, the expectation arises that the random sequence will correct itself in the near future, and the deviation in one direction will thus be subject to the obligatory balancing of the deviation in the other. However, randomly generated sequences, especially if they turn out to be relatively short, turn out to be completely unrepresentative of the random process that generates them.

Player error is more than just a reflection of ordinary statistical ignorance, as it can be observed in privacy even sophisticated people in statistics. It reflects two aspects of human. cognitive function: a) strong and unconscious motivation of people to find order in everything that they observe around them, even if the sequence of outcomes they observe arises as a result of a random process, b) universal human. a tendency to ignore calculated estimates of probabilities in favor of intuition. While logic can convince us that random processes have no control over their outcomes, our intuitive responses can be overwhelming and overwhelming at times. Reed, who explored the comparative power of logical and intuitive thinking, argues that the latter is often more coercive than the former, probably for the reason that such conclusions come to mind suddenly, therefore, do not lend themselves to logical analysis and are often accompanied by a strong sense of their rightness. In contrast to the fundamental impossibility of tracing the process by means of which such intuitive "decisions" are found, the process logical reasoning open to analysis and criticism. Therefore, people rule logical thinking, and from intuitive thinking they simply get results, to-rye fill the latter with a strong sense of a sense of righteousness.

O. and. most common in a situation where outcomes are generated purely by chance. If some skill factor is involved in the development of events, a positive recency effect is more often observed. An observer is more likely to view a streak of successes (eg, a billiards player) as evidence of skill, and will build their predictions of subsequent outcomes in a positive rather than a negative direction. Even throwing the dice can lead to a positive effect of novelty to the extent that the individual is convinced that the outcome of the event is somehow influenced by the "art" of the thrower.

See also Barnum Effect, Player Behavior, Statistical Inference

What is a paradox? A paradox is called two incompatible and opposite statements, each with convincing arguments in its own direction. The most pronounced form of paradox is antinomy - reasoning that proves the equivalence of statements, one of which is an explicit negation of the other. And it is the paradoxes in the most exact and rigorous sciences, such as, for example, logic, that deserve special attention.

Logic is known to be an abstract science. It has no place for experiments and any specific facts in their usual sense; it always presupposes an analysis of real thinking. But discrepancies in the theory of logic and the practice of real thinking still take place. And the most obvious confirmation of this is logical paradoxes, and sometimes even logical antinomy, which personifies the contradictions of the logical theory itself. This is what explains the significance of logical paradoxes and the attention paid to these paradoxes in logical science. Below we will introduce you to the most striking examples logical paradoxes. This information will certainly be of interest both to those who study logic in depth and to those who simply like to learn new and interesting information.

Let's start with the paradoxes compiled by the ancient Greek philosopher Zeno of Elea, who lived in the 5th century BC. His paradoxes are called "Zeno's Aporias" and even have their own interpretation.

Zeno's aporias

Zeno's aporias are outwardly paradoxical arguments about motion and multitude. In total, Zeno's contemporaries mentioned over 40 aporias (by the way, the word “aporia” is translated from the ancient Greek language as “difficulty”) of his authorship, but only nine of them have survived to this day. If you wish, you can familiarize yourself with them in the writings of Aristotle, Diogenes Laertius, Plato, Themistius, Philoponus, Aelius and Symplicius. We will give an example of the three most famous.

Achilles and the tortoise

Imagine that Achilles is running at a speed ten times the speed of a turtle, and is a thousand paces behind him. While Achilles runs a thousand steps, the turtle will only take a hundred. Until Achilles overcomes another hundred, the turtle will have time to do ten, etc. And this process will continue indefinitely and Achilles will never catch up with the turtle.

Dichotomy

In order to overcome a certain path, you must initially overcome half of it, and in order to overcome half, you need to overcome half of this half, etc. Based on this, the movement will never begin.

Flying arrow

The flying arrow always remains in place, because at any moment in time it is at rest, and since it is at rest at any moment in time, it is always at rest.

Here it will be appropriate to cite another paradox.

The Liar's Paradox

The authorship of this paradox is attributed to the ancient Greek priest and seer Epimenides. The paradox sounds like this: “What I am in this moment I say - a lie, ”that is, it turns out: either "I am lying" or "My statement is false." This means that if a statement is true, then, based on its content, it is a lie, but if this statement is initially false, then its statement is a lie. It turns out that it is false that this statement is a lie. Therefore, the statement is true - this conclusion brings us back to the beginning of our reasoning.

Nowadays, the Liar's paradox is seen as one of the formulations of Russell's paradox.

Russell's paradox

The Russell paradox was discovered in 1901 by the British philosopher Bertrand Russell, and later independently rediscovered by the German mathematician Ernst Zermelo (sometimes this paradox is called the "Russell-Zermelo paradox"). This paradox demonstrates the inconsistency of Frege's logical system, in which mathematics is reduced to logic. Russell's paradox has several formulations:

• Omnipotence Paradox - Is an omnipotent being capable of creating anything that can limit its omnipotence?
• Suppose some library has set the task of compiling one large bibliographic catalog, which should include all and only those bibliographic catalogs that do not contain references to themselves. Question: Do I need to include a link to this directory in this directory?
• For example, in some country a law was passed that the mayors of all cities are forbidden to live in their city, and are allowed to live only in the "City of Mayors". Where, then, will the mayor of this city live?
• The paradox of the barber - there is only one barber in the village, and he is ordered to shave everyone who does not shave himself, and not to shave those who shave himself. The question is: who should shave a barber?

The following paradoxes are no less interesting and amusing.

Burali-Forti paradox

The assumption that the idea of ​​the possibility of a set of ordinal numbers can lead to contradictions, which means that set theory, in which it is possible to construct a set of ordinal numbers, will be inconsistent.

Cantor's paradox

The assumption about the possibility of a set of all sets can lead to contradictions, which means that the theory according to which it is possible to construct such a set will also be contradictory.

Hilbert's paradox

The idea that if all the rooms in a hotel with an infinite number of rooms are occupied, then in any case more people can be accommodated in it, and their number can be infinite. This paradox explains that the laws of logic are absolutely unacceptable to the properties of infinity.

False Monte Carlo Inference

The conclusion is that, when playing roulette, you can safely bet on red if black fell ten times in a row. This conclusion is considered false for the reason that, according to the theory of probability, the occurrence of any subsequent event is not influenced by the event that preceded it.

The Einstein-Podolsky-Rosen paradox

The question is whether processes and events developing far from each other are capable of influencing each other? For example, does the birth of a supernova in a distant galaxy in any way affect the weather in Moscow? As an answer, we can cite the following: based on the laws of quantum mechanics, such an influence is impossible due to the fact that both the speed of light and the speed of information transfer are finite quantities, and the Universe is infinite.

The twins paradox

The question is: will the twin traveler who returned from space travel in a superluminal starship be younger than his brother, who remained on Earth all this time? If we proceed from the theory of relativity, then more time has passed on Earth (according to the Earth's course of time) than in a starship flying at superluminal speed, which means that the twin traveler will be younger.

The murdered grandfather paradox

Imagine being in the past and killing your grandfather before he met your grandmother. The conclusion follows that you will not be born and will not be able to return to the past to kill grandfather. The presented paradox clearly demonstrates the impossibility of traveling to the past.

The predestination paradox

For example, a person finds himself in the past, has sexual contact with his great-grandmother and conceives her son, i.e. his grandfather. This becomes the cause of a succession of descendants, including the parents of this person, as well as himself. It turns out that if this person had not made a trip into the past, he would never have been born.

These are just a few logical paradoxes that occupy the minds of many people today. It will not be difficult for an inquisitive mind to find more than a dozen of similar ones (for example,). A considerable amount of time and effort can be devoted to studying, refuting or proving each of them. And, quite likely, about each paradox, you may form your own personal original inferences. But this tells us that, despite the prevalence of the laws of logic and cause-and-effect relationships in our life, not everything in our life depends on them. Sometimes contradictions similar to logical paradoxes arise in Everyday life each person. In any case, this is great food for the mind and cause for thought.

By the way, with regards to reflections: on the topic of logical paradoxes there are very interesting book under the title "Gödel, Esher and Bach". Its author is the American physicist and computer scientist Douglas Hofstadter.

Dear readers, it would be great if in your comments you gave several examples of logical paradoxes that are familiar to you. And also we will be interested in your opinion on the meaning of logic in our life - Vote for one of the statements below.