People are used to believing that what seems obvious is right. That’s why they often fall into trouble, misjudging the situation, trusting their intuition and not taking the time to critically reflect on their choice and its consequences.
What is the Monty Hall paradox ? This is a graphic illustration of a person’s inability to weigh their chances of success in the conditions of choosing a favorable outcome in the presence of more than one adverse outcome.
Monty Hall's Paradox Formulation
So what kind of beast is this? What, in fact, are we talking about? The most famous example of the Monty Hall paradox is the television show, popular in America in the middle of the last century, entitled "Let's Bet!" By the way, it was thanks to the leader of this quiz that the Monty Hall paradox got its name later.
The game consisted of the following: the participant was shown three doors, seemingly identical. However, for one of them the player was waiting for an expensive new car, but for the other two impatiently languished for a goat. As it usually happens in the case of quiz shows, which was behind the door chosen by the contestant, it became his win.
What is the trick?
But not so simple. After the choice was made, the host, knowing where the main prize was hidden, opened one of the two remaining doors (of course, the one behind which the artiodactyl lurked), and then asked the player if he wanted to change his mind.
The Monty Hall paradox, formulated by scientists in 1990, is that, contrary to intuition, which suggests that there is no difference in making a leading decision on the basis of a question, you must agree to change your choice. If you want to get a great car, of course.
How it works?
There are several reasons why people do not want to give up their choice. Intuition and simple (but incorrect) logic say that nothing depends on this decision. Moreover, not everyone wants to follow the lead of another - this is the real manipulation, isn’t that so? No not like this. But if everything was immediately intuitive, then they would not be called a paradox . There is nothing strange in doubting. When this puzzle was first published in one of the major magazines, thousands of readers, including recognized mathematicians, sent letters to the editorial office stating that the answer printed in the issue was not true. If the existence of probability theory was not news for the person who got on the show, then perhaps he could have solved this problem. And thereby increase the chances of winning. In fact, the explanation of the Monty Hall paradox comes down to simple mathematics.
The first explanation, more complicated
The probability that the prize is located behind the door that was originally selected is one of three. The chance to find it behind one of the two remaining is equal to two out of three. Logically, isn't it? Now, after one of these doors is open, and a goat is found behind it, in the second set (the one that corresponds to 2/3 of the chance of success) there is only one option. The value of this option remains the same, and it is equal to two out of three. Thus, it becomes obvious that by changing his decision, the player will double the probability of winning.
Explanation number two is simpler
After such an interpretation of the decision, many still insist that there is no sense in this choice, because there are only two options and one of them is definitely winning, and the other definitely leads to defeat.
But probability theory has its own view on this problem. And this becomes even clearer if you imagine that the doors were originally not three, but, say, a hundred. In this case, the opportunity to guess where the prize is located , the first time is only one in ninety-nine. Now the participant makes his choice, and Monty excludes ninety-eight doors with goats, leaving only two, one of which was chosen by the player. Thus, the option chosen initially retains the chances of winning equal to 1/100, and the second offered opportunity - 99/100. The choice should be obvious.
Are there any rebuttals?
The answer is simple: no. Not a single sufficiently substantiated refutation of the Monty Hall paradox exists. All the "revelations" that can be found on the Web are reduced to a misunderstanding of the principles of mathematics and logic.
For anyone who is familiar with mathematical principles, the nonrandomness of probabilities is absolutely obvious. Only those who do not understand how logic works can disagree with them. If all of the above still sounds unconvincing - the rationale for the paradox was checked and confirmed on the famous program "Legend Destroyers", and who else to believe, if not them?
The ability to make sure clearly
Well, let it all sound convincing. But this is only a theory, is it possible to somehow look at the work of this principle in action, and not just in words? Firstly, no one has canceled living people. Find a partner who will take on the role of lead and help play the above algorithm in reality. For convenience, you can take boxes, boxes or even draw on paper. Repeating the process a few dozen times, compare the number of wins in the event of a change in the initial choice with how many victories stubbornness brought, and everything will become clear. And you can do even easier and use the Internet. There are many Monty Hall paradox simulators on the Web, in them you can check everything yourself and without unnecessary props.
What is the use of this knowledge?
It may seem that this is just another puzzle, designed to strain the brain, and it serves only entertainment purposes. However, the Monty Hall paradox finds its practical application primarily in gambling and various sweepstakes. Those who have extensive experience are well aware of the common strategies for increasing the chances of finding a value bet (from the English word value, which literally means "value" - a forecast that will come true with a higher probability than was estimated by the bookmakers). And one of these strategies directly involves the Monty Hall paradox.
An example in work with a tote
A sports example will not differ much from the classic one. Let's say there are three teams from the first division. In the next three days, each of these teams should play one decisive match. The one that scores more points than the other two will remain in the first division, the rest will be forced to leave it. The offer of the bookmaker is simple: you need to bet on maintaining the positions of one of these football clubs, while the odds are the same.
For convenience, conditions are accepted under which rivals participating in the choice of clubs are approximately equal in strength. Thus, it is impossible to uniquely identify a favorite before the start of the game.
Here you need to remember the story about goats and a car. Each team has a chance to stay in place in one of three cases. Any of them is selected, a bet is made on it. Let it be Baltika. According to the results of the first day, one of the clubs loses, and the two have yet to play. This is the same “Baltika” and, say, “Shinnik”.
Most will retain their original bet - in the first division will remain “Baltika”. But it should be remembered that her chances remained the same, but the chances of “Shinnik” doubled. Therefore, it is logical to make another bet, a larger one, on the victory of “Shinnik”.
The next day comes, and the match with the participation of “Baltika” is a draw. The next one is Shinnik, and his game ends in a 3-0 victory. It turns out that he will remain in the first division. Therefore, even though the first bet on Baltika is lost, this profit is blocked by the profit on the new bet on Shinnik.
It can be assumed, and most will do, that the win of “Shinnik” is just an accident. In fact, taking probability as an accident is the biggest mistake for a person participating in sports sweepstakes. After all, a professional will always say that any probability is expressed primarily in clear mathematical laws. If you know the basics of this approach and all the nuances associated with it, then the risks of losing money will be minimized.
Benefits in Forecasting Economic Processes
So, in sports betting, the Monty Hall paradox is simply necessary to know. But the scope of its application is not limited to just sweepstakes. Probability theory is always closely associated with statistics, therefore, in politics and economics, understanding the principles of the paradox is equally important.
In the conditions of economic uncertainty, which analysts often deal with, it is necessary to remember the following conclusion arising from the solution of the problem: it is not necessary to know the only correct solution. The chances of a good prognosis always increase if you know what exactly will not happen. Actually, this is the most useful conclusion from the Monty Hall paradox.
When the world is on the verge of economic turmoil, politicians always try to guess the right option to minimize the effects of the crisis. Returning to the previous examples, in the economic sphere, the problem can be described as follows: there are three doors to the leaders of countries. One leads to hyperinflation, the second to deflation, and the third to the treasured moderate economic growth. But how to find the right answer?
Politicians say that one or another of their actions will lead to an increase in jobs and economic growth. But leading economists, experienced people, including even Nobel Prize winners, clearly demonstrate to them that one of these options will definitely not lead to the desired result. Will politicians change their choice after this? It is extremely unlikely, since in this respect they are not much different from the same participants in the television show. Therefore, the probability of error will only increase with an increase in the number of advisers.
Does this cover information on this topic?
In fact, so far only the “classic” version of the paradox has been considered here, that is, the situation in which the presenter knows exactly which door has the prize, and only opens the door with the goat. But there are other mechanisms of the leader’s behavior, depending on which the principle of the algorithm and the result of its execution will differ.
The influence of the leading behavior on the paradox
So what can a facilitator do to change the course of events? Assume different options.
The so-called "Devil's Monty" is a situation in which the leader will always offer the player to change his choice, provided that he was initially correct. In this case, a change in decision will always lead to defeat.
On the contrary, “Angelic Monty” refers to a similar principle of behavior, but in the event that the player’s choice was initially incorrect. It is logical that in such a situation, a change in decision will lead to victory.
If the leader opens the door at random, having no idea what is hidden behind each of them, then the chances of winning will always be fifty percent. At the same time, a car may also be behind an open leading door.
The host can 100% open the door with a goat if the player has chosen a car, and with a 50% probability if the player has chosen a goat. With such an algorithm of actions, if a player changes his choice, he will always win in one of two cases.
When a game is repeated again and again, and the probability that a certain door will turn out to be advantageous is always arbitrary (as well as which door the leader will open, he knows where the car is hiding, and he always opens the door with a goat and offers to change choice) - the chance to win will always be one of three. This is called Nash equilibrium.
As well as in the same case, but under the condition that the host does not have to open one of the doors at all, the probability of victory will still be 1/3.
While the classical scheme is verified quite easily, it is much more difficult to experiment with other possible algorithms of behavior for the facilitator. But with the proper meticulousness of the experimenter, such a thing is possible.
And yet, why all this?
Understanding the mechanisms of action of any logical paradoxes is very useful for a person, his brain and an understanding of how the world can actually be arranged, how much its structure can differ from the usual idea of the individual about him.
The more a person knows about how what works around him in everyday life and what he is not used to thinking about at all, the better his consciousness works, and the more efficient he can be in his actions and aspirations.