In honor of Monty Hall, who passed away this past weekend, this classic article solving his namesake probability problem. Because of comments about adjusting the rules (see below), I have made some additions.
Many of you will already know the answer, but read on anyway because it turns out to be an excellent example to demonstrate fundamental ideas in probability.
Incidentally, I just did this yesterday to a group of surgical residents [this was in 2012]: you might be happy to know that none of them got the right answer. One even insisted—for a while—that I was wrong.
Here’s the problem.
Setup: Monty Hall shows you three doors, A, B, and C, behind one of which is a grand prize, behind the others is nothing. Monty knows which door hides the grand prize. You want the prize.
Rule: You pick a door behind which you think is the prize. You are free to change your mind as often as you like.
The Pick: I’ll suppose, since I can’t quite hear you, that you settled on A (the resident yesterday took that one) and did not change your mind.
Question: Given the information we have, what is the probability that the prize is behind door A?
Answer: Almost everybody will–correctly—say 1/3. Why?
Well, conditional on the evidence we have, which is that there are three doors and the prize is behind one of them, we deduce the probability to be 1/3.
Probability is always logically assigned conditional on evidence of some kind. It is not subjective. Here, the evidence is simple to articulate. But it isn’t always so easy.
What Monty does next: If you did not pick the prize, Monty opens a door; not the one you picked, and obviously not the one with the prize behind it, and asks if you would like to keep your original door choice our would you like to switch?
Question: Which strategy, staying or switching, maximizes the chances of you winning the prize? We can state this another way, but first let’s suppose Monty opened door B. There is no prize behind door B.
Same question: This is equivalent to the one we just asked. Given the evidence we have, what is the probability that the prize is behind door A (or C)?
Your Answer: What’s that? You said 1/2? Why?
Evidence: Did you say to yourself, since you well learned the lesson that probability is conditional, that “There are just two doors and the prize is behind one of them”? If you said that, then conditional on that evidence, the probability is 1/2.
But that is not the best answer. What you should do is switch, because the probability that the prize is behind door C is 2/3!
Conditional on all the evidence we have, that is. What other information could there possible be? Let’ see.
We know that the prize is either behind A or C. Let’s assume it’s behind C.
When Monty was picking a door to open, to show there was nothing behind it, could Monty have opened door C? No. He was forced to open B.
So in this case, it would be better for you to switch—obviously, because the prize is behind C.
Suppose that instead of A you had first picked door B. Which doors could Monty have opened? Right: only A. And again you should have switched.
Now suppose you had first picked C. Which doors could Monty have opened? Right: A or B. And in this case you should stay.
If the prize is behind C, that makes two situations where you were better off switching and one where you should stay. Do you agree?
Then we have the answer, because the exact same analyses can be made if the prize were behind A or B (try it yourself). Thus, conditional on all the evidence, the probability of winning is 2/3 if you switch and 1/3 if you stay.
What was the evidence that you failed to condition on when you said the probability was 1/2? You forgot that Monty’s choice of which door to open was constrained. He couldn’t open the door you picked and he couldn’t open the door that hid the prize.
That very simple information radically changed the probability structure of the answer.
Update In a note to me, Roy Spencer rightly points out that the problem changes if Monty is also ignorant of what is behind the doors (these are new rules). In step two, if Monty opens the door with the prize, which could happen since he doesn’t know originally where the prize is, then it depends on the new rules. Are you still allowed to switch? If so, the probability the prize is behind the open door which you see is, of course, 1, and you should switch. If you are not allowed to switch, then you lose, end of story.
But Monty, since he does not know where the prize is in this updated scenario, might open the empty door. If he does, and if you know Monty is ignorant of the prize location, then the probability the prize is behind the unopened door is 1/2 and it doesn’t matter if you switch or stay.