Introduction

The Monty Hall (MHP) problem is named after

the game show host Monty Hall. In its popularity, the show called ‘Let’s Make a

Deal’ which ran from 1963 till date is where the problem actually stems. The

scenario given in MHP is the same scenario given to contestants in the show and

it follows. It operates by the contestant being asked to choose between three

doors. Behind one of

the doors, there is a valuable prize (typically a car), whilst behind each of

the other two doors, there would usually be an undesirable prize (typically a

goat) (Frannco-Watkins, Derks and Doughtery, 2013). The contestant is then asked

to select one of three doors with the hope of winning the desirable prize. At

this point, the host opens one of the unselected doors to reveal the undesirable

prize. After the reveal, the host then gives the contestant the option to stand

by their initial choice, or change their mind and switch to the other unopened

door (Frannco-Watkins, Derks and Doughtery, 2013). Throughout the game, the host

is aware of the location of the prize and has no intention of exposing it to

the contestant. By looking at the choices of previous contestants in the game, the

results 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 of

getting the car if either one of the remaining doors are chosen. Contestants

will usually be reluctant to take such chances, and thus, will tend to rely on

their intuition when making a choice.

The

Monty Hall problem was first posed into a mathematical problem and solved by

Stevie Selvin in 1975 in a letter written to a scientific journal; The American

Statistician. Subsequent to his first solution not being accepted, Salvin wrote

a follow-up letter in which he used Bayes Theorem to further explain the

probability of this notorious puzzle. Despite the fact that a solution had

already been released, it remained unknown up until 1990 when Marilyn Vos

Savant released a solution to this problem on her popular weekly column ‘Ask

Marylyn’. Using theories of probability, she found that it was always in the

contestant’s best interest to switch to the remaining door. Taking this context into

consideration, this report offers a unifying empirical explanation on the Monty

Hall problem and its possible mathematical solutions, using a series of

different trials to prove that this problem is purely based on facts.

Findings/Theory

Bayes

Theorem was first used as a possible solution to MHP in 1975 by Stevie Selvin.

It was devised

by English statistician, philosopher and minister Reverend Thomas Bayes,

it describes the probability of an event, based on prior knowledge of conditions that might be

related to the event (Gupta, 2017).

Standard notation:

Pr(A?B); probability of A given B

Pr(B); probability of B

Bayes 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 host

reveals a goat behind the second door, Bayes Theorem can be applied to this

problem by letting A be the event that the car is behind the first door and letting

B 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 is

located, a goat will be revealed behind the second door ½ the time. The game

show host never opens the door with the car in it. Hence,

If the goat is revealed to be behind the second door, either the first

or the last door is where the car is located. Considering that the probability

that it is behind the first door is 1/3 and the sum of the two probabilities

must equal 1, the probability the car is behind Door 3 is (1?1/3) = 2/3

Thus, switching is

the winning strategy.

Analysis & Evaluation

The probability that the

contestant choses the door with the car behind it is 1/ 3 as the car is equally

likely to be behind any of the three doors, and so, the probability that it is

behind either one of the other two doors is 2/3 (Robinson, 2013). Subsequent to the host revealing a

goat, there is a predilection by contestants to discern the probability of the

two doors left unopened to be a 50 percent success rate of getting the car and

are therefore compelled to stay with door 1. This is why the Monty Hall problem

could be counterintuitive; our default algorithms produce probabilities that do

not match reality and our brains are accustomed to probabilities that are based

on independent random events, such as rolling a die or tossing a coin. In such

cases, we account for the probability by dividing the specific outcome by the

total number of outcomes (Frost, 2013). MHP is ostensibly viewed as such a

probability, nonetheless, in order for this to comply, the process must be

random and the probability must not change. Therefore, MHP contravenes both

assumptions, as the probability never changes and the process is completely

systematic.

This

can be proven through statistics because it is empirical and therefore can be

applied to real-life situations. There are three possible scenarios to the

Monty Hall problem; in the first, the goat, the car, and the second goat are in

doors one, two and three respectively. If the contestant chooses the first door

and the host always reveals the second goat, if the contestant stands by their

choice of the first door, they will get the goat while is they switch, they win

a car.

In

the second scenario, the goat is in the first two doors and the car in the third

door. If the contestant chooses the first door and host opens the second door to

reveal the second goat. If the contestant decides stands by their choice of the

first door, a goat would be won whereas if they switch they win a car.

In the final

scenario, 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, keeping

in mind that the goats could either be the second or third door. If they decide

to stand by the first door, they will win the car and if they decide to switch,

they will get the goat.

In two out

of three of the given scenarios, there is a 2/3 chance of getting the car if

one switches to the remaining door as opposed to a 1/3 chance of getting the

car if the initial choice remains. Consequently,

not only does ‘always switching’ give the player the car with unconditional

probability of 2/3, but no other strategy gives a higher success chance. (Morgan

et al., 1991)

Conclusion

When given the choice, individuals are more

likely to rely on their intuition rather than statistical facts as it brings a

form of comfort and as a result of the small window of time contestants have to

choose a door, it is faster than a detailed analysis. Nonetheless, if a trial

is taken place in which two individuals would carry out the game show scenario

of the problem fifty times each where individual A would switch to the

remaining door and individual B would stay on initial door chosen. At the end

of the trial, as a result of variable change and statistics, the expected

results would be for individual A to have twice as many cars as goats which are

2/3 x 50 = 33 cars and 1/3 x 50 = 16 goats. Whereas individual B would have

twice as many goats as cars which are 1/3 x 50 = 16 cars and 2/3 x 50 =33

goats. This would allow individuals to be more easily convinced of this problem