The Monty Hall problem, and why it fools nearly everyone
You are on a game show with three closed doors. Behind one is a car and behind the other two are goats. You pick a door. The host, who knows where the car is, opens one of the other two doors to show a goat, then offers you the chance to switch to the last closed door. Should you?
Yes. Switching wins the car two times in three, and sticking wins it one time in three. Almost everybody who meets the puzzle for the first time thinks the odds must be even, because there are two doors left and one car. The reason they are not comes down to what the host knows.
Where the puzzle came from
The problem is named after Monty Hall, the host of the American game show Let’s Make a Deal, although the show never ran the game in exactly this form. The statistician Steve Selvin posed it in a letter to The American Statistician in 1975. It became famous in 1990, when Marilyn vos Savant answered it in a column in Parade magazine and said to switch. The column drew a large number of letters insisting the answer was wrong, many of them from people with mathematical training. The answer was right.
Walking through every case
The quickest way to see the answer is to list every possibility. Say you pick door 1. The car is equally likely to be behind any of the three doors, so there are three cases, each with a chance of one in three.
- The car is behind door 1. The host opens door 2 or door 3. If you switch, you lose.
- The car is behind door 2. The host cannot open door 2, and cannot open your door, so the host opens door 3. If you switch, you get door 2 and win.
- The car is behind door 3. The host has to open door 2. If you switch, you get door 3 and win.
Switching wins in two of the three equally likely cases. Sticking wins only in the first. Nothing about which door you picked changes this, so the same holds whatever door you start with.
Why intuition gets it wrong
The mistake is to treat the host’s choice as if it were random. It is not. When your first pick is a goat, which happens two times in three, the host is forced to open the only other goat door, and the door left closed must hide the car. The host’s action is carrying information about where the car is, and switching is how you use it.
Your first pick, on the other hand, was made with no information at all, so it keeps its one in three chance. Opening a goat door does not improve your door. It concentrates the remaining two thirds onto the one door the host chose not to open.
A larger version makes this easier to feel. Imagine 100 doors. You pick one, and the host, who knows where the car is, opens 98 of the others to show goats. Your door had a one in 100 chance when you picked it and nothing has changed that. The single door the host carefully avoided is now a much better bet.
The rules that make the answer true
The two in three figure depends on exactly how the host behaves, and changing the rules changes the answer. It assumes three things: the host always opens a door, the host always opens a door with a goat behind it, and the host always offers you the switch.
If the host does not know where the car is and opens one of the other doors at random, sometimes the car will be revealed and the game is spoiled. In the games where a goat happens to appear, switching and sticking each win half the time, because the host’s choice no longer carries any information. If the host offers a switch only when your first pick was the car, switching always loses. This is why the puzzle is so often argued about: people who disagree about the answer are frequently picturing different hosts.
With the standard rules, though, the answer is settled, and it is easy to check. Three playing cards, one of them marked as the car, and a friend playing the host will show the pattern within a few dozen rounds.
