Part I: The Monty Hall Problem
The Monty Hall Problem (explained below) is one of those math results that strikes most people as not making intuitive sense. The problem is often illuminated by restating it with 100 doors instead of 3 doors. This makes many people go, “Ah, now I get it,” and concede that their intuition must be wrong. Nevertheless, for many of them the 3-door scenario continues to be counterintuitive.
This leads many to ask, “Why don’t I understand the Monty Hall Problem?” Like this person at Quora: Why doesn’t the “Monty Hall problem” make sense to me? The usual response is to try to demonstrate to the person why the correct answer is correct—to try to get it to click. But, even when this works (sometimes it seems to), it doesn’t address why the problem’s solution feels so counterintuitive, nor why the standard wrong answer feels so right. I think I have an idea of what’s going on. We’ll see. (I’ll read this again later to see if I still think so.)
First, a summary of the problem.
Suppose you’re playing a game in which you are faced with three closed doors. The doors are numbered 1, 2, and 3. You are told by the game-master (who does not lie and only speaks the truth) that behind one of the doors there is a car, and that behind each of the other two doors there is a goat. You are not told which door has which item. (The game-master need not know which door has which item, by the way, though the game goes better if she does. See the End Note, however, for how the game-master’s knowing could affect a player’s credence in her guess.) The arrangement of goats and car will not be changed throughout the course of the game. Continue Reading