"Naked statistics" - the most interesting book about the most boring science

Riddle of Monty Hall

"Riddle of Monty Hall" - the famous problem of the theory of probability, to confound the participants of the game show called Let's Make a Deal ( «to make a deal"), is still popular in some countries, which premiered in the United States in 1963 year. (I remember, every time I watched this show as a child, when you do not go to school due to illness.) In the introduction to the book, I have already pointed out that in this game show may be interesting for statisticians. At the end of its release party to reach the final, becoming with Monti Hall before three large Door: № 1, the door 2 and the door № № 3. Monty Hall explained finalist, which is very valuable prize hidden behind one of these doors - such as a new car, but for the other two - a goat. Finalist had to choose one of the doors and get what was behind it. (I do not know whether there was among the participants of the show at least one person who wants to get a goat, but for simplicity, we will assume that the vast majority of participants dreamed of new car.)

The initial probability of winning is quite simple to determine. There are three doors, with two goat hides, and for the third - the car. When the participants of the show along with Monty Hall stands in front of these doors, he has one chance in three to choose a door, behind which there is a car. But, as noted above, Let's Make a Deal lies the trick, immortalized this TV show and its lead in the literature on the theory of probability. After the finalists of the show will point out some of the three doors, Monty Hall opens one of the two remaining doors, behind which is always a goat. Then Monty Hall asks finalist, if he wanted to change his mind, that is, to abandon the previously selected them closed door to another door closed.

Let's say, for example, that the user has entered a number on the door 1. Monty Hall then opened the door number 3, behind which a goat. Two doors, door number 1 and number 2 door remains closed as before. If a prize is behind a door number 1, the finalist would have won it, but if for door number 2, he would have lost. It was at this moment Monty Hall refers to the player with the question of whether he wants to change his initial choice (in this case refuse to Doors number 1 in favor of Doors number 2). Of course you remember that both doors closed until. The only new information that the participant has received, is that the kid was behind one of the two doors, which he did not choose.

Do finalist should be abandoned in favor of the initial choice of Doors number 2?

The answer is: yes, it should. If he will stick to the original selection, the probability of winning them a valuable prize will be ⅓; if it changes its mind and will point to the door number 2, the probability of winning a valuable prize will be ⅔. If you do not believe me, read on.

I admit that such a response at first sight far from obvious. It seems that, no matter what of the other two doors have chosen a finalist, the likelihood of a valuable prize in both cases equal to ⅓. There are three closed doors. At first, the likelihood that a prize is hidden behind all of them is ⅓. Is has a value decision finalist change their choice in favor of another closed door?

Of course, since the hitch is that Monty Hall knows what is behind each door. If a Finalist chooses door number 1, and it will really be a car, Monty Hall can open any door number 2 or number 3 door, to show a goat, hiding behind it.

If a Finalist chooses door number 1, and the car will be behind door number 2, the Monty Hall opens door number 3.

If the finalist will indicate the door number 1, and the car will be behind door number 3, the Monty Hall opens door number 2.

He changed his mind after the leading open some of the doors, a finalist receives a selection advantage of two doors instead of one. I will try to convince you of the correctness of this analysis in three different ways.

The first - the empirical. In 2008, a columnist for the newspaper The New York Times, John Tayerni written material about the "phenomenon of Monty Hall." After the publication staff developed an interactive program that allows you to play this game and decide for yourself, to change their original choice or not. (The program even provides little goats and avtomobilchiki that appear from behind the door.) Program It captures your winnings when you change your initial choice, and when left to his own opinion. I paid one of his daughters for her to play this game 100 times, each time changing the initial choice. I also paid her brother, so that he, too, has played this game 100 times, each time leaving the original decision. Daughter won 72 times; her brother - 33 times. Efforts were rewarded every two dollars.

These episodes of the game Let's Make a Deal show the same pattern. According to Leonard Mlodinovu, author of The Drunkard's Walk, those finalists who changed his the initial choice of the winner is approximately two times more likely than those who remained at their opinion.

My second explanation of this phenomenon is based on intuition. Let's say the rules of the game have changed slightly. For example, finalist starts with selecting one of the three doors: Doors № 1 № Doors Doors № 2 and 3, as it was originally provided. But then, before you open some of the doors, behind which hides a goat, Monty Hall asks: "Do you agree to give up their choice in exchange for opening the remaining two doors? "So, if you choose door number 1, you can change your mind in favor of number 2 Doors & Doors number 3. If the first point to the door number 3, you can choose door number 1 and number 2 door. And so on.

For you, it would not be a particularly difficult decision: it is obvious that you should refuse the initial choice in favor of the other two doors, because it increases the chances of winning with ⅓ to ⅔. The most interesting is that it is essentially a version of the Monty Hall offers a real game, after open the door, behind which hides a goat. The fundamental fact is that if you were given the opportunity to choose two doors, behind one of them, in any case, would be hiding a goat. When Monty Hall opens the door, behind which there is a goat, and only then asks you Do you agree to change their initial choice, it significantly increases your chances of winning valuable prize! In fact, Monty Hall tells you, "The probability that a prize is hidden behind one of the two doors, that you have not chosen the first time, is ⅔, but it's still more than ⅓!»

This can be represented as follows. Say you're shown the door number 1. After that Monty Hall gives you the opportunity to abandon the original decision in favor Doors number 2 and number 3 Doors. You agree and have at your disposal two doors, which means that you have every reason to expect to win a valuable prize with probability ⅔, rather than ⅓. What would happen if, in that moment, Monty Hall opened the door number 3 - one of "your" door - and it turned out to be a goat? would shake the fact that your confidence in the decision? Of course not. If the car is hidden behind door number three, Monty Hall would have opened the door number 2! He did not show you anything.

When the game is on nakatannomu scenario, Monty Hall really gives you a choice between the door, you specified at the beginning, and the two remaining doors, behind one of which can be car. When Monty Hall opens the door, behind which a goat, it just provides you a favor by demonstrating, for which of the two other doors have no car. You have the same probability of winning in both of the following scenarios.

1. Choosing Door number 1, then the consent of "switch" on the door of number 2 and number 3 door before both will open any door.
2. Choosing Door number 1, then the consent of "switch" on the door of number 2, after Monty Hall show you goat of the door number 3 (or select Doors number 3, after Monty Hall show you a goat behind door number 2).

In both cases, the refusal of the initial solution provides you the benefit of the two doors, compared with one out and you can thus double their chances of winning: with ⅓ to ⅔.

My third embodiment represents a more radical version of the same base intuition. Suppose Monty Hall offers you to select one of 100 doors (instead of one of the three). Once you do, say, pointing to the door of number 47, it opens up the remaining 98 doors, behind which are the goats. Now closed doors are only two: your door number 47, and another, for example door number 61. Should you abandon your initial choice?

Of course yes! With 99 percent probability the car is behind one of the doors that you choose at the beginning. Monty Hall gave you a favor by opening 98 such doors, the car was not for them. Thus, there is only a 1 in 100 chance that your original choice (door number 47) will be correct. At the same time there is a 99 out of 100 chance that your first choice is wrong. If so, then the car is behind the remaining door, then there is the door number 61. If you want to play with a chance of winning 99 times out of 100, then you need to "switch" on the door of number 61.

In short, if you ever have to participate in the Let's Make a Deal game, you definitely need to give from its initial decision when Monty Hall (or the one who will be his substitute) will provide you with the opportunity to choice. More universal conclusion from this example is that your intuitions about the probability of occurrence of certain events can sometimes mislead you.

"Naked Statistics" by Charles Whelan

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