Monty hall problem solution

If the decision to open another door was made before the contestant chose, then the above analyses apply, but Monty did not ALWAYS open another door after a contestant chose. If Monty is free to make that choice then these probabilities no longer apply, as his decision may not be random at all. He could elect to open another door only when the contestant has chosen correctly.

I like your answer. Your selection makes two partitions: your door and the other two. So according to your link, the contestant obviously a potential door-trader can only trade doors during trading hours, namely:. The probabilites are sensitive to your assumptions. Monte knows which door holds the prize Monte always shows a door with no prize. The contrast between the Monte hall problem to the Deal or No Deal problem is a nice illustration.

He died 17th April in Tunbridge Wells, Kent. Suppose you did not know how the game worked. After Monty opens one door to show that nothing is behind it, you realize that you do not know if he opened the door BECAUSE nothing was behind it or if he just randomly selected one of the 2 possible doors. Now that is an interesting dilemma. It seems likely that Monty would always chose a door with nothing behind it, because the game is more interesting that way.

So you should always switch. Then you should always stay. Perhaps he could even have opened the door that you initally chose! Obviously, if the opened door had the prize the question would be moot. Why then would it matter if he had chosen randomly? If Monty fails to open door C, what would have changed that makes it different from Monty restrained from opening C?

Note that C appears twice on the right and only once on the left. I think I may have the answer.

Monty hall problem solution

Assume X is the opened door randomly selected whatever that means by Monty. Obviously, it helps to count ALL of the outcomes. If we were to put people in this situation and test whether they knew about the stats of this problem or not, would they switch doors anyway. My guess again, no data on this — does anyone have any? Besides, we all know that if you choose a wrong door, Monty Hall realizes this, and then proceeds with the game.

But there ARE times when you can predictably profit. An example would be a game where if you have an even number of dollars, you gain 3 dollars, but if you have an odd number of dollars, you lose it all. But it gives the wrong answer. Or does it? I just figure that my initial pick is probably wrong. Once Monty has eliminated a room the remaining one is probably right.

I try to keep in mind his dogged persistence in taking a puzzle or problem, once solved, to the next level, by asking new questions. It was the classic story of the gambler who slowly went broke betting that he would throw at least one double-six on twenty throws of a pair of dice. Unfortunately, not many of the Martin Gardner titles I remember so fondly are available for my Kindle yet.

I always did that boy-answering-door as a hamster-meister reaching blindly into a bag containing two randomly selected hamsters from the hamster barrel. So there are four equiprobable events in the original population. To make a judgement in the hamster case you must first know that the barrel contains two sex-matched pairs. Or am I being stupid?

The contestant would would be told that there is a car monty hall problem solution one of the doors, but the other doors could be a prize or a no-prize. Furthermore, Monte does not alwasy open a door not chosen before allowing a final switch. It makes the results more difficult to model. Regarding the boy opens the door. The hamster problem, once you are told taht one of the hamsters is male you have satified one of the necessary conditions toward having a mixed pair.

Back to the house with the boys. Suppose you had been told that there was at least one boy in the house. Excuse me, but I think that the surgical resident was correct. Looking back, the surprise is that using such an approach gives a clear-cut result — unambiguous. I had this problem in Yr 11 Maths. My most memorable Monte Hall moment occurred at the end of one show.

They went to a final commercial break while the woman opened her purse. When they got back from break, Monte announced that the woman had up pulled out thousands of dollars in medical bills that she just happened to have stuffed into her purse on her way out the door. Chet, as someone explained upthread, the only time you are wrong by switching is when your first choice was right.

This is because Monty always eliminates the wrong door of the two remaining. Does Monty want you to switch doors? What makes his show more entertaining? People who switch and win, or people squirming on stage trying to make a decision? I listen to a radio station on the low end of the AM dial. I also visit wmbriggs. Those two data points are likely to peg me psychologically.

The fascinating thing about the low end of the dial is the advertisements. Investment classes abound. I choose not to go. The best way to make money in investing though is to write a book on investing OR, possibly, sell a course in investing. Writing a book on statistics might work, but only if you get people to start buying it. Buying the book of our host and learning deeply from its pages will save one lots of money.

I am not so sure it will make one lots of money. The best way to make the Monty Hall problem make you money is to start a television game show that gets people tuning in. We should always switch when given the chance at the Monty Hall problem. I have never gotten the chance to use that knowledge! I am yapping in the comment section of wmbriggs.

This is not intuitive. It is clear that Monty opening a door changes the game and it is now 1 in 2 rather than 1 in 3. For reasons that are not yet clear to me, to take advantage of these new odds one must make a new choice, and there is only one choice at this point. Join using Google Join using email. Sign up with Facebook or Sign up manually.

Jimin Khim Eli Ross. Then the possible outcomes can be seen in this table: In two out of three cases, you win the car by changing your selection after one of the doors is revealed. Stick Switch Doesn't matter. Many people have probably heard of the Monty Hall problem. Which action would give you the highest chance of winning the car? Monty Hall decides to perform his final episode of the Let's Make a Deal series with a little twist, and Calvin is elated when he is the first contestant to be called to the stage.

Hall keeps talking, explaining the rules: of the doors covers a pig, while the last covers the most amazing super special magnificent supercar of your dreams. Calvin will pick 1 of the doors, after which I will open doors which cover a pig, leaving two doors for Calvin to choose from. Calvin will then have to choose whether to open the door he originally chose or the door I left closed.

Switch doors Stay at your door Both staying and switching are the same Indeterminate None of the above. In a game show, there are 3 doors. Which choice is better, if you want the brand new shiny red monty hall problem solution This is not an original problem. A host places boxes in front of you, one of which contains the holy grail.

What is the probability that the holy grail is in box 42? Suppose you're a contestant on a game show and the host shows you 5 curtains. He then gives you the following options: 0 stick with the curtains you initially chose, 1 swap either one of your curtains for one of the remaining curtains, or 2 swap both of your curtains for the remaining 2 curtains.

The game proceeds as follows: You pick a door. The game show host then opens a door you didn't choose that he knows has only farm animals behind it. You are then given the option to switch to a different, unopened door, after which he opens another door with farm animals behind it. Assume that you prefer the car over any of the farm animals! Cite as: Monty Hall Problem.

Join Brilliant The best way to learn math and computer science. Sign up to read all wikis and monties hall problem solution in math, science, and engineering topics. This is likely due to network issues. Please try again in a few seconds, and if the problem persists, send us an email. Very few raised questions about ambiguity, and the letters actually published in the column were not among those few.

Determining the player's best strategy within a given set of other rules the host must follow is the type of problem studied in game theory. For example, if the host is not required to make the offer to switch the player may suspect the host is malicious and makes the offers more often if the player has initially selected the car. In general, the answer to this sort of question depends on the specific assumptions made about the host's behavior, and might range from "ignore the host completely" to "toss a coin and switch if it comes up heads"; see the last row of the table below.

Morgan et al [ 38 ] and Gillman [ 35 ] both show a more general solution where the car is uniformly randomly placed but the host is not constrained to pick uniformly randomly if the player has initially selected the car, which is how they both interpret the statement of the problem in Parade despite the author's disclaimers. Both changed the wording of the Parade version to emphasize that point when they restated the problem.

They consider a scenario where the host chooses between revealing two goats with a preference expressed as a probability qhaving a value between 0 and 1. This means even without constraining the host to pick randomly if the player initially selects the car, the player is never worse off switching. As N grows larger, the advantage decreases and approaches zero.

A quantum version of the paradox illustrates some points about the relation between classical or non-quantum information and quantum informationas encoded in the states of quantum mechanical systems. The formulation is loosely based on quantum game theory. The three doors are replaced by a quantum system allowing three alternatives; opening a door and looking behind it is translated as making a particular measurement.

The rules can be stated in this language, and once again the choice for the player is to stick with the initial choice, or change to another "orthogonal" option. The latter strategy turns out to double the chances, just as in the classical case. However, if the show host has not randomized the position of the prize in a fully quantum mechanical way, the player can do even better, and can sometimes even win the prize with certainty.

After choosing a box at random and withdrawing one coin at random that happens to be a gold coin, the question is what is the probability that the other coin is gold. This problem involves three condemned prisoners, a random one of whom has been secretly chosen to be pardoned. The warden obliges, secretly flipping a coin to decide which name to provide if the prisoner who is asking is the one being pardoned.

The question is whether knowing the warden's answer changes the prisoner's chances of being pardoned. The second appears to be the first use of the term "Monty Hall problem". The problem is actually an extrapolation from the game show. As Monty Hall wrote to Selvin:. And if you ever get on my show, the rules hold fast for you — no trading boxes after the selection.

A version of the problem very similar to the one that appeared three years later in Parade was published in in the Puzzles section of The Journal of Economic Perspectives. Nalebuff, as later writers in mathematical economics, sees the problem as a simple and amusing exercise in game theory. A restated version of Selvin's problem appeared in Marilyn vos Savant 's Ask Marilyn question-and-answer column of Parade in September After a reader wrote in to correct the mathematics of Adams's analysis, Adams agreed that mathematically he had been wrong.

Now you're offered this choice: open door 1, or open door 2 and door 3. In the latter case you keep the prize if it's behind either door. You'd rather have a two-in-three shot at the prize than one-in-three, wouldn't you? If you think about it, the original problem offers you basically the same choice. Monty is saying in effect: you can keep your one door or you can have the other two doors, one of which a non-prize door I'll open for you.

Numerous readers, however, wrote in to claim that Adams had been "right the first time" and that the correct chances were one in two. The Parade column and its response received considerable attention in the press, including a front-page story in The New York Times in which Monty Hall himself was interviewed. In the article, Hall pointed out that because he had control over the way the game progressed, playing on the psychology of the contestant, the theoretical solution did not apply to the show's actual gameplay.

He said he was not surprised at the experts' insistence that the probability was 1 out of 2. By opening that door we were applying pressure. We called it the Henry James treatment. It was ' The Turn of the Screw '. Caveat emptor. It all depends on his mood. Contents move to sidebar hide. Article Talk. Read Edit View history. Tools Tools. Download as PDF Printable version.

In other projects. Wikimedia Commons Wikidata item. Probability puzzle. Paradox [ edit ]. Standard assumptions [ edit ]. Simple solutions [ edit ]. Savant and the media furor [ edit ]. Scott Smith, University of Florida [ 3 ]. Confusion and criticism [ edit ]. Sources of confusion [ edit ]. Criticism of the simple solutions [ edit ].

Solutions using conditional probability and other solutions [ edit ]. Refining the simple solution [ edit ]. Conditional probability by direct calculation [ edit ]. Bayes' theorem [ edit ]. The question: Should the contestant stick with their original choice or switch to the other door to maximize their chance of winning the car? These probabilities play a central role in solving the problem.

When Monty reveals a goat, he provides information that influences your chances. This confirms that switching is the better choice. This thinking is not correct. Here are some key points to understand:. The Monty Hall problem has significant implications in decision theory, particularly in how humans assess risk and make choices:.

To illustrate the Monty Hall problem further, consider a practical scenario involving multiple rounds of the game:. This empirical evidence reinforces the mathematical reasoning behind the Monty Hall problem and highlights the reliability of the probabilities over many trials. The Monty Hall Problem is an interesting study in probability. It shows how our instincts can be wrong.

By analyzing the problem, we can see better ways to decide. Understanding probability is important. It helps us see how information affects our choices. These lessons apply not just in games, but in daily life. Making informed choices can lead to better results. This is true even in uncertain situations.