MONTY HALL PROBLEM

Monty Hall is the name of the host of an American television program, "Let's Make a Deal" initially aired on NBC in 1960s.

The series broached one of the famous probability related conundrums.

The riddle was posed by Steve Selvin (an American statistician and professor emeritus of biostatistics at the University of California, Berkeley), via a letter to the scientific journal of " The American Statistician ".

The problem was also resolved by Steve Selvin himself, later on.

THE RIDDLE

To broach the details of the riddle, first let's elaborate on the general setup and agenda of the TV Show.

At the end of each episode of the show, Monty Hall would call the contestant to come to stage and face 3 closed door.

Behind one of the doors, a valuable prize has been hidden, and a live goat behind each of the other two. The contestant takes whichever was behind the door that he chose. Let's assume the a brand new car, is the prize.

The initial probability of winning the car is straightforward; one in three (1/3).

Then, as twist, Monty would open one of the other 2, always revealing a goat. Then once more asks a question from the participant around his decision, to see if he is interested in switching his chosen door, to the other closed one.

• Let's assume the participant has originally chosen door #1.
• Monty opens door #3, revealing goat.
• Then asks the contestant about their interest in switching their decision from door #1 to door #2.

Should they switch? Should they stick to their initial respond?

" Yes, they should change the selected door to door #2 "

In fact, intuitively, chances may seem to remain identical, but that would be untrue.

In fact this scenario, in terms of probability of success is similar to have a 2/3 chance of winning from the beginning.

By switching the contestant leverages the benefit of choosing two doors, instead of one.

As an intuitive justification, if you play the simulation of the game 100 times, you will always see that the number of times you win, is close to twice as you get goats, when you switched your chosen door at the end.

One other way to explain is that once the contestant has been asked if they prefer to switch, after revealing door with goat behind it, a new information as been fed to the context that alters the probabilities, since the host (Monty) knows the answer.

Evidently, the above explanation does not encounter the Monty's intention to deceit the participant, if their original guess was true. The elaborations, relies solely on mathematics, statistic and probabilities.