Why Switching Doors Doubles Your Chances: The Monty Hall Paradox Explained

The Monty Hall problem, also known as the Monty Hall paradox, originates from the American TV game show "Let's Make a Deal" hosted by Monty Hall. A contestant faces three closed doors, behind one is a car and behind the other two are goats. After the contestant picks a door but before it is opened, the host, who knows what is behind each door, opens one of the remaining doors to reveal a goat and then offers the contestant the option to switch to the other unopened door. Switching increases the probability of winning the car from 1/3 to 2/3.

A well‑known description of the problem was sent by Craig F. Whitaker in 1990 to Marilyn vos Savant’s column in Parade Magazine, asking whether switching provides an advantage.

The story traces back to Steve Selvin’s 1975 letter to the American Statistician, which introduced the puzzle and later coined the term “Monty Hall problem” in an August 1975 correspondence.

Monty Hall himself noted that in the actual show there was no opportunity for contestants to switch after a door was opened.

A similar problem appeared in 1959 as the “three prisoners problem” in Martin Gardner’s column, offering a clearer statement of the conditions.

The earliest known appearance may be in Joseph Bertrand’s 1889 book Calcul des probabilités , where it was called “Bertrand’s Box Paradox.”

Mueser and Granberg later clarified the problem with an unambiguous set of conditions:

There are three doors, only one hides a car; the other two hide goats.

The car is placed behind one of the doors before the game starts.

The contestant selects one door without knowing what is behind any door.

The host knows what is behind each door.

If the contestant’s chosen door hides a goat, the host must open another door that also hides a goat.

If the contestant’s chosen door hides the car, the host randomly opens one of the two goat doors.

The contestant is then asked whether to stay with the original choice or switch to the remaining unopened door.

Switching the choice increases the contestant’s chance of winning the car.

Reference: https://baike.baidu.com/item/%E4%B8%89%E9%97%A8%E9%97%AE%E9%A2%98/1242689