## Thursday, June 12, 2014

### The Monty Hall Problem – A Brain Teaser

The Monty Hall Problem which is a brain teaser got its name from the TV game show known as `Let Make a Deal’, which was hosted by Monty Hall and has fascinated mathematician with the possibilities presented by the `Three Doors’.

This has led to the mathematical urban legend surrounding the `Monty Hall Problem’. The problem was first posted in a letter to the American Statistician to Steve Selvin in 1975 and it became a popular quest from a reader letter quoted in Marilyn vos Savant’s, `Ask Marilyn’, column in Parade magazine in the year 1990.

The scenario is such that one is given the choice to choose one closed door out of the three doors placed before them with one having a prize of hidden car behind it while the other two have goats behind them.The contestant is not aware of where the car is, but Monty Hall knows where the prize lies. The contestant selects a door and Monty opens one of the remaining doors, the one he knows does not have the car behind it.

If the contestant has already chosen the correct door, then Monty is likely to open either of the two remaining door revealing that it does not contain any prize and goes on to ask the player if he would like to switch the selection to another unopened door or stay with the original selection. The player here is at a crossroad in considering his choice on the selection of the door, wondering on the probability of winning the car if they happen to stick to their original choice and what would be the chances of switching selection.

The Monty Hall Problem Difficult to Grasp

The Monty Hall problem is difficult to grasp and unless the player thinks carefully about the deal, the role of the host goes unappreciated. The Monty Hall problem has drawn much academic interest from the surprising result with its simple formulation.

Variations has also been made by changing the implied assumptions which has created drastic different consequences where for one variation, if Monty offer the contestant a chance to switch while he had initially chosen the door with the prize of a car behind it, then the contestant should not think of switching on options. For another variation, if another door is randomly opened and happens to reveal a goat, then it does not make any difference to the contestant.

The problem is a paradox like the veridical types since the correct result is counter intuitive and seems absurd but true. This problem is closely related to the earlier Three prisoner’s problem as well as the older problem known as Bertrand’s box paradox.

Switching can turn Loss into Win/Win into Loss

According to Jason Rosenhouse, James Madison University mathematics professor, who has written an entire book on the subject – The Monty Hall Problem – The Remarkable Story of Math’s Most Contentious Brainteaser, states that the contestant can double their chance of winning on switching doors when the three conditions are fulfilled.

In the first place, Monty never opens the door which the contestant selects, secondly he always opens a door concealing a goat and thirdly when the first two rules leave Monty with a choice of doors to open, he makes his choice at random. Switching can turn a loss into a win and a win into a loss according to Rosenhouse.