Introduction The Monty Hall (MHP) problem is named afterthe game show host Monty Hall. In its popularity, the show called ‘Let’s Make aDeal’ which ran from 1963 till date is where the problem actually stems. Thescenario given in MHP is the same scenario given to contestants in the show andit follows. It operates by the contestant being asked to choose between threedoors. Behind one ofthe doors, there is a valuable prize (typically a car), whilst behind each ofthe other two doors, there would usually be an undesirable prize (typically agoat) (Frannco-Watkins, Derks and Doughtery, 2013). The contestant is then askedto select one of three doors with the hope of winning the desirable prize.
Atthis point, the host opens one of the unselected doors to reveal the undesirableprize. After the reveal, the host then gives the contestant the option to standby their initial choice, or change their mind and switch to the other unopeneddoor (Frannco-Watkins, Derks and Doughtery, 2013). Throughout the game, the hostis aware of the location of the prize and has no intention of exposing it tothe contestant. By looking at the choices of previous contestants in the game, theresults have shown that there is a strong proclivity to stay on the chosen door.This can be attributed to the probability of a 50 percent success rate ofgetting the car if either one of the remaining doors are chosen. Contestantswill usually be reluctant to take such chances, and thus, will tend to rely ontheir intuition when making a choice. TheMonty Hall problem was first posed into a mathematical problem and solved byStevie Selvin in 1975 in a letter written to a scientific journal; The AmericanStatistician. Subsequent to his first solution not being accepted, Salvin wrotea follow-up letter in which he used Bayes Theorem to further explain theprobability of this notorious puzzle.
Despite the fact that a solution hadalready been released, it remained unknown up until 1990 when Marilyn VosSavant released a solution to this problem on her popular weekly column ‘AskMarylyn’. Using theories of probability, she found that it was always in thecontestant’s best interest to switch to the remaining door. Taking this context intoconsideration, this report offers a unifying empirical explanation on the MontyHall problem and its possible mathematical solutions, using a series ofdifferent trials to prove that this problem is purely based on facts. Findings/TheoryBayesTheorem was first used as a possible solution to MHP in 1975 by Stevie Selvin.
It was devisedby English statistician, philosopher and minister Reverend Thomas Bayes,it describes the probability of an event, based on prior knowledge of conditions that might berelated to the event (Gupta, 2017).Standard notation:Pr(A?B); probability of A given BPr(B); probability of BBayes Theorem is written as follows; (Google.org, 2017)To recall the scenario from the MHP,if the contestant always chooses the first door and assuming the game show hostreveals a goat behind the second door, Bayes Theorem can be applied to thisproblem by letting A be the event that the car is behind the first door and lettingB be the event that the game show host reveals a goat behind the second door.Therefore (The Angry Statistician, 2012) This scenario follows such that if the first door is where the car islocated, a goat will be revealed behind the second door ½ the time. The gameshow host never opens the door with the car in it. Hence, If the goat is revealed to be behind the second door, either the firstor the last door is where the car is located.
Considering that the probabilitythat it is behind the first door is 1/3 and the sum of the two probabilitiesmust equal 1, the probability the car is behind Door 3 is (1?1/3) = 2/3Thus, switching isthe winning strategy. Analysis & Evaluation The probability that thecontestant choses the door with the car behind it is 1/ 3 as the car is equallylikely to be behind any of the three doors, and so, the probability that it isbehind either one of the other two doors is 2/3 (Robinson, 2013). Subsequent to the host revealing agoat, there is a predilection by contestants to discern the probability of thetwo doors left unopened to be a 50 percent success rate of getting the car andare therefore compelled to stay with door 1. This is why the Monty Hall problemcould be counterintuitive; our default algorithms produce probabilities that donot match reality and our brains are accustomed to probabilities that are basedon independent random events, such as rolling a die or tossing a coin. In suchcases, we account for the probability by dividing the specific outcome by thetotal number of outcomes (Frost, 2013). MHP is ostensibly viewed as such aprobability, nonetheless, in order for this to comply, the process must berandom and the probability must not change. Therefore, MHP contravenes bothassumptions, as the probability never changes and the process is completelysystematic. Thiscan be proven through statistics because it is empirical and therefore can beapplied to real-life situations.
There are three possible scenarios to theMonty Hall problem; in the first, the goat, the car, and the second goat are indoors one, two and three respectively. If the contestant chooses the first doorand the host always reveals the second goat, if the contestant stands by theirchoice of the first door, they will get the goat while is they switch, they wina car. Inthe second scenario, the goat is in the first two doors and the car in the thirddoor. If the contestant chooses the first door and host opens the second door toreveal the second goat.
If the contestant decides stands by their choice of thefirst door, a goat would be won whereas if they switch they win a car. In the finalscenario, the car is in the first door and the two goats are in the last two.If the contestant chooses the first door and the host reveals a goat, keepingin mind that the goats could either be the second or third door.
If they decideto stand by the first door, they will win the car and if they decide to switch,they will get the goat.In two outof three of the given scenarios, there is a 2/3 chance of getting the car ifone switches to the remaining door as opposed to a 1/3 chance of getting thecar if the initial choice remains. Consequently,not only does ‘always switching’ give the player the car with unconditionalprobability of 2/3, but no other strategy gives a higher success chance. (Morganet al.
, 1991) ConclusionWhen given the choice, individuals are morelikely to rely on their intuition rather than statistical facts as it brings aform of comfort and as a result of the small window of time contestants have tochoose a door, it is faster than a detailed analysis. Nonetheless, if a trialis taken place in which two individuals would carry out the game show scenarioof the problem fifty times each where individual A would switch to theremaining door and individual B would stay on initial door chosen. At the endof the trial, as a result of variable change and statistics, the expectedresults would be for individual A to have twice as many cars as goats which are2/3 x 50 = 33 cars and 1/3 x 50 = 16 goats. Whereas individual B would havetwice as many goats as cars which are 1/3 x 50 = 16 cars and 2/3 x 50 =33goats. This would allow individuals to be more easily convinced of this problem