31 Dec A wonderfully confusing little problem (LEONARD MLODINOW) | Part B’
In the Monty Hall problem you are facing three doors: behind one door is something valuable, say a shiny red Maserati; behind the other two, an item of far less interest, say the complete works of Shakespeare in Serbian. You have chosen door 1. The sample space in this case is this list of three possible outcomes:
Maserati is behind door 1.
Maserati is behind door 2.
Maserati is behind door 3.
Each of these has a probability of 1 in 3. Since the assumption is that most people would prefer the Maserati, the first case is the winning case, and your chances of having guessed right are 1 in 3.
Now according to the problem, the next thing that happens is that the host, who knows what’s behind all the doors, opens one you did not choose, revealing one of the sets of Shakespeare. In opening this door, the host has used what he knows to avoid revealing the Maserati, so this is not a completely random process. There are two cases to consider.
One is the case in which your initial choice was correct. Let’s call that the Lucky Guess scenario. The host will now randomly open door 2 or door 3, and, if you choose to switch, instead of enjoying a
fast, sexy ride, you’ll be the owner of Troilus and Cressida in the Torlakian dialect. In the Lucky Guess scenario you are better off not switching—but the probability of landing in the Lucky Guess scenario is only 1 in 3.
The other case we must consider is that in which your initial choice was wrong.
We’ll call that the Wrong Guess scenario. The chances you guessed wrong are 2 out of 3, so the Wrong Guess scenario is twice as likely to occur as the Lucky Guess scenario.
How does the Wrong Guess scenario differ from the Lucky Guess scenario?
In the Wrong Guess scenario the Maserati is behind one of the doors you did not choose, and a copy of the Serbian Shakespeare is behind the other unchosen door.
Unlike the Lucky Guess scenario, in this scenario the host does not randomly open an unchosen door. Since he does not want to reveal the Maserati, he chooses to open precisely the door that does not have the Maserati behind it. In other words, in the Wrong Guess scenario the host intervenes in what until now has been a random process.
So the process is no longer random: the host uses his knowledge to bias the result, violating randomness by guaranteeing that if you switch your choice, you will get the fancy red car. Because of this intervention, if you find yourself in the Wrong Guess scenario, you will win if you switch and lose if you don’t.
To summarize: if you are in the Lucky Guess scenario (probability 1 in 3), you’ll win if you stick with your choice. If you are in the Wrong Guess scenario (probability 2 in 3), owing to the actions of
the host, you will win if you switch your choice.
And so your decision comes down to a guess: in which scenario do you find yourself? If you
feel that ESP or fate has guided your initial choice, maybe you shouldn’t switch. But unless you can bend silver spoons into pretzels with your brain waves, the odds are 2 to 1 that you are in the Wrong Guess scenario, and so it is better to switch. Statistics from the television program bear this out: those who found themselves in the situation described in the problem and switched their choice won about twice as often as those who did not.
The Monty Hall problem is hard to grasp because unless you think about it carefully, the role of the host, like that of your mother, goes unappreciated.
But the host is fixing the game.
The host’s role can be made obvious if we suppose that instead of 3 doors, there were 100. You still choose door 1, but now you have a probability of 1 in 100 of being right. Meanwhile the chance of the Maserati’s being behind one of the other doors is 99 in 100.
As before, the host opens all but one of the doors that you did not pick, being sure not to open
the door hiding the Maserati if it is one of them. After he is done, the chances are still 1 in 100 that the Maserati was behind the door you chose and still 99 in 100 that it was behind one of the other doors. But now, thanks to the intervention of the host, there is only one door left representing all 99 of those other doors, and so the probability that the Maserati is behind that remaining door is 99 out of 100!
For the earliest known statement of the problem (under a different name) didn’t occur until 1959, in an article by Martin Gardner in Scientific American.
Gardner called it “a wonderfully confusing little problem” and noted that “in no other branch of mathematics is it so easy for experts to blunder as in probability theory.” Of course, to a mathematician a blunder is an issue of embarrassment, but to a gambler it is an issue of livelihood.
The Drunkard’s Walk