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A player who stays with the initial choice wins in only one out of three of these equally likely possibilities, while a player who switches wins in two out of three.Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. The Monty Hall Problem is a very (to me at least) counter-intuitive probability mind-experiment which contorts my brain and fascinates me at the same time, I have been mulling it over the last few weeks and wanted to write a little simulator to see if the numbers come out as predicted (if not expected, and indeed they do! Now which door do you think is most likely to hide the car?

The solutions in this section consider just those cases in which the player picked door 1 and the host opened door 3.

El problema fue bautizado con el nombre del presentador de dicho concurso, Monty Hall. The other two doors hide “goats” (or some other such

A player chooses a door at random. This simulator allows you to experiment with more then 3 doors for this reason.If the player sticks with their door then their chance of winning the car should be 1/3.
Simulador Monty Hall.

In contrast most sources in the field of probability calculate the conditional probabilities that the car is behind door 1 and door 2 are 1/3 and 2/3 given the contestant initially picks door 1 and the host opens door 3 (Selvin (1975b), Morgan et al. ).

After you choose your door (1/1,000,000 chance of hiding the car) Monty opens up 999,998 doors that hide goats to leave one door still closed. A key insight is that, under these standard conditions, there is more information about doors 2 and 3 than was available at the beginning of the game when door 1 was chosen by the player: the host's deliberate action adds value to the door he did not choose to eliminate, but not to the one chosen by the contestant originally.

It seems obvious to me that the other door that Monty left un-opened has a massively higher chance of hiding the car than your original choice!

Monty opens one of the other doors to show that there is a goat behind it. Another insight is that switching doors is a different action than choosing between the two remaining doors at random, as the first action uses the previous information and the latter does not.

If the player switches door then their chances of winning the car increases to 2/3!!

The simple solutions above show that a player with a strategy of switching wins the car with overall probability 2/3, i.e., without taking account of which door was opened by the host (Grinstead and Snell 2006:137–138 Carlton 2005).

!There are 3 doors, behind one lies a car, while behind the other two are goats. Other possible behaviors than the one described can reveal different additional information, or none at all, and yield different probabilities. The one you choose, or the one that Monty avoided opening while he opened all 999,998 other doors?! Address: 43 Churchgate, Wicklow, Co. Wicklow, Ireland.If you are having problems understanding the outcome, I find it helps to imagine that there are a million doors rather than 3. Is it to your advantage to switch your choice?Then, if the player initially selects door 1, and the host opens door 3, we prove that the conditional probability of winning by switching is:The conditional probability table below shows how 300 cases, in all of which the player initially chooses door 1, would be split up, on average, according to the location of the car and the choice of door to open by the host.And if you ever get on my show, the rules hold fast for you – no trading boxes after the selection.The given probabilities depend on specific assumptions about how the host and contestant choose their doors.