Monty Hall problemi
Problemin tarihi

Monty Hall problemine benzer birkaç olasılık bulmacasından en eskisi Bertrand'ın kutu paradoksudur. Bu 1889'da Joseph Bertrand tarafından Calcul des probabilités adlı kitabında ortaya atıldı (Barbeau 1993). Bu bulmacada üç kutu bulunur: bir kutu iki altın madeni para, bir kutu iki gümüş madeni para ve bir kutu da her birinden birer tane madeni para içerir. Bir kutunun rastgele seçilmesinin ve bir madeni paranın rastgele çekilmesinin ardından, çekilen paranın altın olması durumunda geriye kalan paranın altın olma olasılığı nedir. Monty Hall probleminde olduğu gibi burada da sezgisel yanıt 1/2'dir, halbuki olasılık 2/3'tür.
Martin Gardner'ın Scientific American'daki Mathematical Games köşesinde 1959'da yayınlanan üç mahkûm problemi (1959a, 1959b), Monty Hall probleminin izomorfudur. Problem, rastgele içlerinden biri gizlice seçilip affedilecek olan üç mahkûmu içerir. Mahkûmlardan biri, diğer mahkûmlardan idam edilecek olanın adını söylemesi için gardiyana yalvarır. Bu mahkûmun kendi kaderi hakkında hiçbir bilgiyi açığa çıkarmasa da kendi affedilme şansını 1/3'ten 1/2'ye çıkarır. 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. This problem is equivalent to the Monty Hall problem; the prisoner asking the question still has a 1/3 chance of being pardoned but his unnamed cohort has a 2/3 chance.
Steve Selvin 1975'te American Statistician dergisine gönderdiği bir çift mektup ile Monty Hall problemini ortaya attı (1975a, 1975b). İlk mektup, 15 yıl sonra Parade dergisinde yayınlanan gösterime yakın bir problem içeriyordu. İkici mektup "Monty Hall problemi" teriminin ilk kullanımı gibi görünmektedir. Problem aslında yarışma programının bir ekstrapolasyonudur. Monty Hall heyecan yaratmak için yanlış bir kapıyı açar ve yarışmacıya, kapı değiştirme fırsatı yerine 100 $ gibi daha küçük bir ödül teklif eder. Monty Hall'ün Selvin'e yazdığı bir mektuba göre:
And if you ever get on my show, the rules hold fast for you—no trading boxes after the selection. (Hall 1975)
A version of the problem very similar to the one that appeared three years later in Parade was published in 1987 in the Puzzles section of The Journal of Economic Perspectives (Nalebuff 1987).
Phillip Martin's article in a 1989 issue of Bridge Today magazine titled "The Monty Hall Trap" (Martin 1989) presented Selvin's problem as an example of what Martin calls the probability trap of treating non-random information as if it were random, and relates this to concepts in the game of bridge.
A restated version of Selvin's problem appeared in Marilyn vos Savant's Ask Marilyn question-and-answer column of Parade in September 1990 (vos Savant 1990). Though vos Savant gave the correct answer that switching would win two-thirds of the time, she estimates the magazine received 10,000 letters including close to 1,000 signed by PhDs, many on letterheads of mathematics and science departments, declaring that her solution was wrong (Tierney 1991). Due to the overwhelming response, Parade published an unprecedented four columns on the problem (vos Savant 1996:xv). As a result of the publicity the problem earned the alternative name Marilyn and the Goats.
In November 1990, an equally contentious discussion of vos Savant's article took place in Cecil Adams's column The Straight Dope (Adams 1990). Adams initially answered, incorrectly, that the chances for the two remaining doors must each be one in two. After a reader wrote in to correct the mathematics of Adams' analysis, Adams agreed that mathematically, he had been wrong, but said that the Parade version left critical constraints unstated, and without those constraints, the chances of winning by switching were not necessarily 2/3. 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 (Tierney 1991) in which Monty Hall himself was interviewed. He appeared to understand the problem, giving the reporter a demonstration with car keys and explaining how actual game play on Let's Make a Deal differed from the rules of the puzzle.
Over 40 papers have been published about this problem in academic journals and the popular press (Mueser and Granberg 1999). Barbeau 2000 contains a survey of the academic literature pertaining to the Monty Hall problem and other closely related problems.
The problem continues to resurface outside of academia. The syndicated NPR program Car Talk featured it as one of their weekly "Puzzlers," and the answer they featured was quite clearly explained as the correct one (Magliozzi and Magliozzi, 1998). An account of the Hungarian mathematician Paul Erdős's first encounter of the problem can be found in The Man Who Loved Only Numbers—like many others, he initially got it wrong. The problem is discussed, from the perspective of a boy with Asperger syndrome, in The Curious Incident of the Dog in the Night-time, a 2003 novel by Mark Haddon. The problem is also addressed in a lecture by the character Charlie Eppes in an episode of the CBS drama NUMB3RS (Episode 1.13) and in Derren Brown's 2006 book Tricks Of The Mind. Penn Jillette explained the Monty Hall Problem on the "Luck" episode of Bob Dylan's Theme Time Radio Hour radio series. The Monty Hall problem appears in the film 21 (Bloch 2008). Economist M. Keith Chen identified a potential flaw in hundreds of experiments related to cognitive dissonance that use an analysis with issues similar to those involved in the Monty Hall problem (Tierney 2008).
Diğer iki kapının şansı 2/3 iken, yarışmacının seçtiği kapının şansı 1/3'tür.
Yarışmacının seçtiği kapının şansı 1/3, diğer iki kapının şansı 2/3. Bu 2/3'lik şans, hâlâ açılmamış olan kapınınki 2/3  ve sunucunun açtığı kapınınki 0 olmak üzere paylaştırılır.
Yarışmacının 1. kapıyı seçmesi hâlinde tüm olası sonuçları ve bunların gerçekleşme ihtimallerini gösteren karar ağacı
Monty Hall probleminin 30 sonucunun oyun kartlarına uyarlanması.
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