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The Monty Hall problem is based off of a paradox that is very commonly misunderstood. The Monty Hall problem originated from the popular game show called Let’s Make a Deal. The host, Monty Hall shows the contestant three doors and explains that behind two of the doors await goats, and behind the third door awaits a new car. If you choose the door with the car behind it, you get to keep it. What’s the catch? Well, the same applies to the other two doors; if you choose a door with a goat behind it, you are stuck with a goat. One needs to fully understand the basic rules of Let’s Make a Deal in order to completely go over the problem. The most commonly misunderstood rule deals with the fact that Monty does not randomly select a door to open and show the contestant. Monty Hall knows which door hides the car and which doors hide the goats. Monty will never open the door with the car behind it (until the game is over). Nor will he ever open the door chosen by the contestant (again, until the game is over). Individuals who have not correctly solved the problem at hand, may not have completely understood these assumptions/rules. (Just a side note)To start the game, Monty Hall will ask the contestant to choose a door. Once they make their choice, Monty will open a different door and reveal one of the goats. He’ll then give the player the option of changing doors. What should they do? A common response, even among brilliant mathematicians, is to say that there is no difference between switching and not switching. You’d come to the conclusion that since there are two doors left, the probability of the car being behind each remaining door is now fifty percent. This statement is wrong. The correct strategy is to switch doors. In fact, the odds of winning the car double once the switch is made. By sticking to your guns and not switching, you have a ? chance, or 0.33333333333 (11 3’s) chance of winning. However, by switching from your original answer after Monty reveals one of the goats, you have a ? chance, or 0.66666666666 (11 6’s) chance of winning. Of course, the results won’t be exactly ? (0.66666666666) or ? (0.33333333333). Just in case you might be wondering, it doesn’t take much to explain. Let’s start by looking at a different example, like flipping a coin. If you were to flip a coin exactly 100 times, you would expect to get around 50 heads. You wouldn’t even think twice if afterwards you counted 58 heads, or even 43. Well, this is the basic idea that lays behind a confidence interval. The confidence interval comes up with a range of acceptable values for your outcome, instead of one, single expected value.The first instance of the Monty Hall (or Three Door Problem) goes back to around 1959, when Martin Gardner introduced a (more confusing) version of it in Scientific American called the Three Prisoners Problem.In the Three Prisoners Problem, there are three prisoners, #1, #2 and #3. Each prisoner is in a separate cell and sentenced to death. In this situation, the governor has selected one of them at random to be executed. The warden knows which one is to be executed, but is not allowed to tell. Prisoner #1 begs the warden to let him know the identity of one of the others who’re on the chopping block. He says something along the lines of, “If prisoner #2 is going to be executed, give me prisoner #3’s full name. If prisoner #3 is going to be executed, give me prisoner #2’s name. And lastly, if I am going to be executed, go ahead and flip a coin to decide whether to name prisoner #2 or prisoner #3.” You understand why Monty couldn’t air that on TV, right?The Monty Hall Problem is also quite similar to work dating even farther back called Bertrand’s box paradox (started by Joseph Bertrand in his 1889 work). In Bertrand’s box paradox, there are three boxes. One box contains two gold coins, another box contains two silver coins, and the final box contains both one gold coin and one silver coin. The “paradox” in this situation is all in the probability. After choosing one box at random and withdrawing one coin at random, if that coin happens to be a gold coin, then the next coin drawn from the same box will most likely be a gold coin as well (which is confusing too).There aren’t very many ways to apply this to real life scenarios, but the best one is in certain card games like Bridge. Here, instead of calling it the Monty Hall Problem, it’s called the Principle of Restricted Choice. The principle of restricted choice states that when you play a particular card, it decreases the probability that its player holds any of the same card. Basically, the principle helps other players guess the locations of unobserved matches to the card played. Professional Bridge players rely on the Principle if Restricted Choice by noting how the numbers work out the same. Say your two opponents hold the King and Queen of Hearts, but you don’t know which has it. If you see Opponent #1 play the King of Hearts, what’s the chance that Opponent #2 has the Ace? You’re aware that it was about equally likely that Opponent #1 started with the King alone, or both the King and Queen, so it seems to be about a 50% chance. But if Opponent #1 held both, he could have just as easily played the Queen. Therefore, you should only count half of the cases where he holds both. This makes it a 1/3 (or 0.33333333333) chance he has the Queen, and a 2/3 (or 0.66666666666) chance that Opponent #2 does.Now that I’ve confused you once again, let’s go over it using a simple (or possibly more confusing) metaphor. Consider this approach. You have three items, one being extremely valuable, and two being completely worthless. Once you’ve chosen a door, you split the objects into Theirs and Mine. They have two items and you have one, so they probably have the extremely valuable object (In this context, “probably” means “more likely than not”). At this point, they show show you one of their objects, it being one of the worthless items. You always knew that they had at least one of the worthless items, so you have learned nothing new and still think that they probably have the valuable object. The only difference now being that you know where the item is if they have it. So you make an inference: they probably have the valuable item, and because they’re hiding it, the chance of it being behind the other door would certainly make sense. Because of this, you jump to conclusions and believe that the valuable item is behind their door. Therefore you must change your choice, and switch doors with them (I hope that made it easier).In summary, I’ve explained the Monty Hall problem (or at least attempted to), and worked to show you the paradox behind it. The concept of restricted choice is the main key to comprehending the Monty Hall problem and other similar situations.

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