# You are given three boxes

You are given three boxes: one of them contains a valuable prize, others contain nothing. You can choose any box, but you still do not know exactly where is the prize. One of the two boxes you have not selected will be opened and shown that it is empty. Currently, you can either leave the box that you originally chose, or replace it with another, unopened. What do you prefer to do (leave or replace)?

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This task is a variation of the Monty Hall paradox and was formulated in 1975 by geographic data statistician Steve Selwyn. Monty Hall was the first host of the television game show “Let’s Make a Deal.” Selvin’s mind-teaser relates to a situation similar to the final round in this television show where participants choose prizes that are behind the doors. In a letter to American Statistician, Selwyn stated that it is reasonable to pick an exchange.This option appeared so contradictory that the author was forced to defend his position later. Monty Hall contacted Selwyn and agreed with him.

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Since then, this paradox has become the subject of a vast number of negotiations. After mentioning it in 1990 by Marilyn Vos Savant in her column in Parade magazine, it gained much popularity within the general public. The following year, John Tierney from the New York Times said that this mind-teaser was “discussed both in the halls of the Central Intelligence Agency. It was analyzed by mathematicians from the MIT and developers from Los Alamos National Laboratory…”. It appeared that this task is used in the Car Talk show, which is shown on NRP as well as in the television show NUMB3RS. Moreover, this mind-teaser is used in the interviews in Bank of America and other financial companies.

The most interesting part of this challenge is the complexity level.

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During certain researches, it was found that only 12% of the people the correct answers to this question. This is a shocking result because any person trying to solve this task with guesswork should answer correctly in 50 cases out of 100. Simply put, this is a case where intuition leads you in the wrong direction.

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Most people answering the question think that there is no difference whether you leave the first box or replace it. More advanced people can state that any person who believes that they can raise their expectations with the help of the replace is mistaken.

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In any situation associated with probabilities, you should know what happens by chance and what is intentional. Imagine that your friend flips a coin 10 times, and each time it drops the eagle up. What is the chance to have an eagle at the following coin flip? Actually, you cannot state for sure as you need to find out whether it is the series of random actions or the coin has certain peculiarities.

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When Selvin created this mind-teaser, the original version of the show “Let’s Make a Deal” was still aired and became an integral part of pop culture. My grandmother watched this show and stated that Monty was a deceiver, albeit famous. Here’s how she justified it: “If Monty wants to offer you this door, he must know that there is something less valuable behind it compared to the thing the participant already had.”

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Once during the interview, Monty said that when he knew that the participant had chosen the biggest prize, he offered money in exchange for the thing that was behind the door. It added much entertainment to the show. Once a person changed a large prize for a smaller one, he turned into a loser. Actually, it triggered much more emotions among the audience.

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Now, let’s entitle the boxes. Let them be the following: chosen, opened and tempting. Originally, the chances to choose a box with a prize are equal to one to three.

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When you choose a particular box, one of the two remaining boxes opens as well. It turns out that it is empty. In order to define how this affected your chances of getting a big prize, you need to know who is opening the second box and what is the purpose.

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There are two probable options.

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The opened box was picked randomly (i.e. flipped the coin) out of those two boxes that you have not chosen. It means that there could be a prize in the opened box, however, there was nothing.

The box was opened by a person who knew what was contained there and he was previously trying to show you the empty box. Also, he could do it under any conditions.

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The initial mind-teaser of Selvyn does not leave any doubts that the second option is preferable for TV.

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This important detailing is not often taken into account. As it was mentioned above, this task given during the interview is a bit contradictory. There is mentioned a leader that can use machinations and there is no information about the way the opened box is chosen. You need to ask the interviewer to specify the details and you should state that the question allows providing different answers depending on the way the second box is chosen.

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Opening the box via the first option, you gain particular information: there is no prize in this box, albeit it could be there. It greatly increases your chances that there is a prize in your chosen box, from 1⁄3 to 1⁄2. Chances of a prize in a tempting box also change. Since both the opened box and the other one have a 50/50 chance of winning, there is no point in replacing one box with another.

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Opening the box via the second option, you will get no useful information. Monty (or any other person) knows what is contained in the boxes and can always pick an empty one and show it to you. The person’s premeditated demonstration does not increase your chances that the picked initially box will appear valuable. Put it simply, the initial chance equal to ⅓ will remain the same after opening the box.

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In other words, opening the second box did not change the probability equal to ⅔ that there is a prize in one of two boxes. Once you accept the leader’s suggestion to replace the boxes, you double the chances of getting a prize.

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If you still don’t understand why Selvin’s answer is correct, imagine there are 100 boxes. You choose box №79. Then Monty opens 98 of the remaining 99 boxes. They are all empty. After that, in addition to your box, say, box number 18 remains unopened. Monty asks you if you want to change box №79 to box №18?

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You started with a probability of 1 to 99 that the car keys are in your box. Monty behaves like an in-show leader. In fact, he does not intend to show you anything but an empty box. The chance that the prize is in your box remains the same, that is, 1⁄100. However, the chance that the prize is in box №18 after the leader has opened all the other boxes that were empty, rises to 99⁄100. So, if you are given 100, you increase your chances by 99 times (!) If you replace it with the remaining one.

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When psychologists Donald Granberg and Tad Brown gave this mind-teaser during the interview, they always heard explanations like the following:

“I would not choose another box: in case I appear mistaken while replacing the box, I will tear up mysefl more than if I lose with my chosen box.”

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“It was my first choice made instinctive and if I was mistaken, let it be this way. In case I replace the box and become looser, it will be even worse.”

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“I will definitely regret in case I replace the boxes and lose. It is better psychologically to stand by my convictions”.

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All these are syndromes of loss fear. People tend to go away from their decisions that may turn out to be bad, even if the chances of winning are favourable. Anyone who creates new products should keep it in mind. A consumer who thinks about replacing boxes or brands may be guided by something that has nothing in common with logic.

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Loss fear is also natural to mathematical geniuses. In this way, they are no different from other people. They say that the famous mathematician Paul Erdosh solved the mind-teaser wrong when first heard of this challenge. “Even physicists, Nobel laureates, frequently provide incorrect solutions, and they insist on their wrong option and are ready to argue with those people who provide right solutions” – said psychologist Massimo Pyattelli Palmarini.