Essay Example on Monty Hall Problem

Introduction

The Monty Hall problem is a brain teaser game in the form of a probability puzzle based on an American television show called 'Let's Make a DEAL' and was named after its original host (Behrends, 2008). The problem was initially being created and solved through a letter sent to the American Statistician by Steve Selvin in 1975. The game became famous after it was published in Marilyn Vos Savant's "Ask Marilyn" column in a 1990 Parade magazine.

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How to play The Monty Hall Problem game

Suppose you are on a game show, and you get to be given to choose between three doors. Behind two doors, there are goats, and behind the third door, there is a car. You are then told to select one door between the three doors, and the host who already knows what is behind those days opens another door, say maybe door number three, which has a goat and then asks you whether you want to pick door number two (Tierney, 1991). Would it be an advantage to switch your side? Surprisingly the odds of either switching or staying are not undecided as many people would expect them to be. The game is all about evaluating and re-evaluating one's decisions as a lot of new information emerges.

The problem

After the participant in the game show chooses one daor and the host opens another, he or she is asked to make a decision either to stay or switch to the other door.

Reasons to switch

Let us say during The Monty hall problem game, you pick a door and hold, and then you automatically have a one-third chance of taking a win. If you decide to stick with your first choice no matter what, then you have the advantage of maybe improving your chances of winning. Even if Monty decides to add fifty more doors instead of the three or even blow the other doors up, it doesn't matter (Adams, 1990). This is because the best you can do with the original initial choice is one out of three, while the remaining doors take the remaining chances, which is two out of three.

This explanation of why switching works may seem to make more sense, but still, it does not explain why the odds get better on the other side. However, the key to the solution showing that the probability of winning the car after switching is 2/3 is dependent on the behavior of the host (Barbeau, 1993). This involved the emergence of various standard assumptions. The first assumption is that the host must always open the door revealing the goat and not the car. Another assumption is that the host must always open the door that the contestant did not pick. Lastly is that the host must always give a chance to the contestant to switch between the initially chosen door and the rest of the closed doors.

Another reason that makes switching attractive instead of just the regular game involves imagining a variant. Let's imagine there are a hundred doors to pick in the beginning, and you decide to pick one door, and then Monty looks at the other remaining ninety-nine doors (Vazsonyi, 1998). Monty looks for the goat in those ninety doors except the one chosen by the contestant. Would you stick with your original door or the one which was filtered from the ninety-nine doors? What Monty is doing is taking a set of ninety-nine choices and changing them by removing ninety-eight goats. When he's done, he has one top door out of ninety-nine for the participant to choose. It is now your decision to make whether you want a random door out of the one hundred doors or the best one of ninety-nine random chances.

Assuming that two choices mean equal chances, that is 50-50 in this situation, is our biggest hurdle (Williams, 2004). We assume that when you know nothing about either choice, then the two choices are equally equal. However, more information is much better. For instance, let us say in a basketball league you are told to choose between a player A, a rookie, has less experience, and Player B, who won the "Most Valuable Player" award the last ten years in a row. This would not change your guess, and you will choose Player B with near certainty, while others who are not informed would still call it a 50-50 situation. The general idea here involves that information matters, and the more you know, the better the idea. In the players' comparison scenario, you know better than the other uninformed people and yet have better chances (Bell, 1992). Although there is a chance the rookie may be the best player in the league, but we take probabilities.

Reasons to stay

At first, when we choose one of the three doors at random, our probability of winning the car becomes 1/3. If after the host opens another door and we stay with our initial choice. Nothing will have changed even though it may seem like it has after opening the door; hence the probabilities remain the same as well (Seymann, 1991). You can decide to think about it this way if we are committed to the initial decision. The game automatically becomes equivalent to the one where we are only getting to choose one time and then get subsequently stuck or locked to our choice. In this scenario, even if the host opens all the doors at once or he decides to open them slowly one by once, which is the rule in The Monty Hall problem, the problem of the contestant winning or the host getting the car will always remain to be 1/3.

When the door that has a goat behind is opened, and we are given a chance to choose again, instead of taking it as a continuation of the game, we should otherwise take it as a new game Tierney, John (21 July 1991). In the new game, we have only two doors that we are choosing from; thus, the probability of getting the car say by flipping a coin will become . We only get this boost again in probability if we make truly and repeatedly make a random choice. Even though our original choice is one in the two remaining possible choices, if we truly stick with our original pick, then our probability of acquiring the car is still 1/3.But if we decide to make a new pick randomly among the remaining two doors which we still might land on our original pick, we end up with a new 50-50 cat, so the solution is to choose again, but the actual correct solution is to make another pick, and in this pick, you will pick the other door that is not the one that you had initially chosen. This makes the probability of acquiring the car to go up to 2/3.

Results in a coded simulator in a python for the Monty Hall Problem, which was run 900,000 times, and then the results analyzed. They should know that if you have a 1/3 chance of getting to choose the right door, you have a 2/3 chance of getting to choose the wrong door. It could almost mean the same to say that switching the selection of the doors after the host opens one is like effectively getting to choose two doors. A clearer way of understanding this problem is to rethink the Monty Hall problem as a slightly different game (Stibel, Dror, Ben-Zeev, 2008). Rather than playing against two opponents, all of you are assigned randomly a different door which you get to keep whatever you pick in the assigned door.

Conclusion

The Monty Hall problem is and has been a very applicable problem not only in study fields such as statistics and probability theory but also in different areas of academia. My understanding of this problem is just on the surface, and there are many variations. Still, more investigations are being done to be able to dive into them, especially the cases where it is more disadvantageous than advantageous to switch.

In particular, the simulation and change of perspective only improve the performance of the participants if presented in the psychological dependency. However, if you view it the way it has been a long tradition of attempts in vain to practically and theoretically explain the problem, the progress of present experiments and studies was to try to understand these versions. It is also clear that the two choices are 50-50 when you don't know anything about them. The Monty Hall Problem is a choice of a random guess, but also more information means you are re-evaluating your options. I also think that the reason a lot of people fail to understand it is because it keeps on changing plans and is not straight. It is also easier to conclude that the host of the game show will most likely never open a door with a prize to keep the show entertaining.

References

Adams, Cecil (2 November 1990). "On 'Let's Make a Deal,' you pick door #1. Monty opens door #2 - no prize. Do you stay with door #1 or switch to #3?". The Straight Dope. Retrieved 25 July 2005.

Barbeau, Edward (1993). "Fallacies, Flaws, and Flimflam: The Problem of the Car and Goats". The College Mathematics Journal. 24 (2): 149-154. doi:10.1080/07468342.1993.11973519.

Behrends, Ehrhard (2008). Five-Minute Mathematics. AMS Bookstore. p. 57. ISBN 978-0-8218-4348-2.

Bell, William (August 1992). "Comment on 'Let's make a deal' by Morgan et al". American Statistician. 46 (3): 241.

Seymann, R. G. (1991). "Comment on Let's make a deal: The player's dilemma". American Statistician. 45 (4): 287-288. doi:10.2307/2684454. JSTOR 2684454.

Stibel, Jeffrey; Dror, Itiel; Ben-Zeev, Talia (2008). "The Collapsing Choice Theory: Dissociating Choice and Judgment in Decision Making" (PDF). Theory and Decision.

Tierney, John (21 July 1991). "Behind Monty Hall's Doors: Puzzle, Debate and Answer?". The New York Times. Retrieved 18 January 2008.

Vazsonyi, Andrew (December 1998 - January 1999). "Which Door Has the Cadillac?" (PDF). Decision Line: 17-19. Archived from the original (PDF) on 2014-04-13. Retrieved 16 October 2012.

Williams, Richard (2004). "Appendix D: The Monty Hall Controversy" (PDF). Course notes for Sociology Graduate Statistics I. Retrieved 2008-04-25.