Monty Hall Problem - The Problem

The Problem

Steve Selvin wrote a letter to the American Statistician in 1975 describing a problem loosely based on the game show Let's Make a Deal (Selvin 1975a). In a subsequent letter he dubbed it the "Monty Hall problem" (Selvin 1975b). The problem is mathematically equivalent (Morgan et al., 1991) to the Three Prisoners Problem described in Martin Gardner's Mathematical Games column in Scientific American in 1959 (Gardner 1959a).

In 1990 the same problem was restated in its well-known form in a letter to Marilyn vos Savant's Ask Marilyn column in Parade:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice? (Whitaker 1990)

There are certain ambiguities in this formulation of the problem: it is unclear whether or not the host would always open another door, always offer a choice to switch, or even whether he would ever open the door revealing the car (Mueser and Granberg 1999). The standard understanding of the problem (resolving these ambiguities) is that the host is indeed constrained always to open an unchosen door revealing a goat and always to make the offer to switch. Very often it is also assumed that the car is initially hidden completely at random, and that if the host has a choice of door to open (which happens if the player initially picked the car) then this choice is completely random, too. (Krauss and Wang, 2003:9). Some authors, either instead of or together with these probability assumptions, assume that the player's initial choice is completely random (for instance, Selvin 1975a).