After I wrote this, I looked up the problem in Wikipedia. Sure enough, there's an article on this with many of my insights, and much more.
The problem: Monty Hall (a game show host in the 70's) had a show "Let's Make A Deal." A typical episode had this scenario:
There are 3 rooms on the stage with doors marked A, B, and C. Monty tells you that behind one is a $50,000 prize and behind the other two are nothing. Pick one. You pick door A.He then asks his assistent to open one of the others, say B- it's empty! Then he says, "Do you want to keep door A or switch to C?" Which do you choose?
There's no right answer to this, because you don't know what Monty is thinking:
But in our "Monty Hall Math Puzzle", you choose one, then he always opens one of the other two that has nothing behind it. Then he always gives you the choice of switching.
I first encountered this in the 1980s at work- someone had heard it, but didn't have the answer. My colleagues were guessing different answers and giving reasons which didn't seem compelling. So I drew a branching diagram and figured it was better to switch.
To review, we chose A and Monty opened B.
When we chose A and Monty opened B, my diagram basically showed:
Since B was opened, we must have hit case 1 or case 3. Once B is opened, we know cases 2 and 4 did NOT happen. Case 3, where the prize was behind C, happens 1/3 of the time and case 1 happens 1/6 of the time, with the prize behind A. So when B is opened it's twice as likely the prize is behind door C, so we should switch.
In 2003 while studying to be a math teacher I took a problem-solving course and we were given this problem. Remembering the answer, I solved it again with no trouble. But some people weren't convinced and I thought about it more deeply.
Why were people not convinced? They remembered in the beginning each room has an equal chance of holding the prize. In the final situation there are two rooms. Equal chance + 2 rooms seems to equal 50-50 chance. My class had one very sharp engineer who just couldn't get past this. I didn't know how to explain it to him at the time, but I figured it out later:
The key is that after opening a door, the other doors are not the same. Monty never opens the door that you pick, but he must know where the prize is to know whether to open B or C. His action adds information to the puzzle. When you choose A, odds are 2 out of 3 that you're wrong. If C has the prize, Monty does not randomly open A or B- he always opens B.
This is the basic asymmetry added by Monty's action. If C has the prize, after you choose A, he always opens B, never A. If A has the prize, he only opens B half the time. So if Monty chooses to open B, then more often C has the prize.
Put another way, if Monty did sometimes open the door you pick, showing it to be empty, and let you pick either of the others, then the odds in this situation would be 50-50. But he doesn't, and they're not.
I did realize it boiled down to the initial probabilites.
A simpler way of looking at it is this, since we know in the
beginning there's 1/3 of a chance the prize is behind each door:
Since we chose room A:
1/3 of the time the prize is A, so if I keep A I win 1/3 of the time.
2/3 of the time the prize is B or C so if I keep A I lose 2/3 of the time.
He agreed with this, but still couldn't see why the end-situation wasn't 50-50. I thought up a variant of the problem to illustrate. Imagine this:
Say I lay out 52 playing cards face down on a glass table. If you can choose which is the ace of hearts, you win $100. Here's what we'll do. You pick one, then I'll look under the table, see where the ace of hearts is, get back up and turn over 50 cards, leaving just your card and one other.
Imagine we do it- you choose one card, knowing there's 1 chance in 52 that you picked the ace at the beginning and 51 chances in 52 that you were wrong and the ace is one of the other 51 cards. Then I look underneath and turn over 50 cards so the Ace of Hearts is either your card or the one that's left.
Will you want to switch?
Whether it's 3 cards, 4 cards or 1000 cards, if you choose one and I turn over all of the rest except one, your initial choice cuts the cards into 2 sets- your card and the rest. If your strategy is to switch, then you're always choosing the rest, the bigger set of cards.
We saw how that one man (and others as well) were troubled by the end situation (2 doors left) plus the isolated notion that in the beginning each room had equal chance of holding the prize. This brings up the whole topic of how our other ideas confuse us. Mathematical thinking is as much about ignoring extraneous ideas as it is about paying attention to the "facts".
Another math-teaching concept this illustrates is that "knowing" or really, "feeling" gets in the way. To problem-solve well, one needs to pick apart what one feels one knows and figure out what one really knows. We know that in the beginning the prize had an equal chance of being behind any door (although this might not have been true in the actual game show). We don't really know they're equal at the end (and indeed they're not.)
When I brought the problem to non-mathematicians, other ideas got in the way. One is our familiarity with card tricks. Magicians fool people reliably. Instead of concentrating on the math, we wonder what the trick is- did he switch the cards (or is someone moving the prize between rooms)? Most people are thoroughly trained (by life, if not by teachers) that being clever about people is much more powerful than being clever about math. (And in most situations they're right!)
Another problem was revealed when I asked a 12-year old boy this different question: Say I had 3 cards, an Ace, 2 and 3, upside down. I shuffle them and lay them face down on a table. You try to pick the Ace. Say we do this a hundred times- about how many times do you think you'll get the right answer? He said, "50." When I asked him why, he said, "Because I'm a pretty good guesser." Partially this is superstition. Partly, he has simply learned to trust his judgement.
Thus another impediment to choosing the right strategy (to switch) is our bias to stick to our original decision. (This might be bolstered by a defensive feeling he's offering me a choice so I give up "my" prize.)
So I changed the part about him picking/guessing: "We'll do the same thing, but each time, you just turn over a random card. About how many times will you pick the Ace?" He thought for a moment and said, "30." In response to me again asking why he said, "It'll be about a third of the time, but things aren't perfect, so it'll be a little less than that." (Finally- a math mistake!)
I once tried to teach a precocious 11-year old boy about the Monty Hall problem and he just couldn't accept the answer or the reasoning. I had a hunch I knew why.
So we did a trial. I wrote 1,2,3 a hundred times on two pieces of paper. Then on my paper, without him seeing, I put a square around one on each line, indicating it had the prize. Then I gave him the pen and we did his first row. He picked one by drawing a circle around it, then I crossed one of his out. Then we did the next row. At the end, we saw his choice was correct 35 times, and the one he could have switched to was right 65 times! It didn't teach him the math, but it opened up his experience to the possibility of accepting the truth- allowing him to learn. I could see the gears working in his brain.
Then I said, "Let me ask you something. What if instead of first picking a door you think it's behind, you first pick a door you do NOT think it's behind. Then I open a door. Now do you think it's behind the one you picked or the other one?" The other one. He was simply very attached to his first decision.
Two more issues are worth bringing up. The first is about answers and (especially in California) the very poor math education delivered in public schools which teaches that math is all about getting the right answers. If a mathematician plays the Monty Hall game and doesn't win the prize, it's because being right 2/3 of the time is no guarantee of being right in a single trial. But for many people, they instead conclude the strategy is wrong. We tell kids that understanding and showing their work is important, but almost every math class emphasizes and grades the answers, drilling into them much more powerfully that it's about the answers. Understanding concepts and developing ability to solve problems fall by the wayside.
So it's very hard for most people with this kind of training to think about the odds. They want the right answer. Having picked one at the outset weighs heavily in its favor. Many, many people learn in school to trust their instincts rather than their thinking.
Finally, there's stress. I remember watching "Let's Make a Deal". The contestants were excited and we were all stressed.
I once took a Myers-Briggs test at work, where they identify ones dominant location on the relationship/thinking/doing triangle. For us, they gave us two profiles, when we were relaxed and when stressed- they were different! This is true for most of us. When faced with a decision about winning or losing thousands of dollars, can we retreat to a methodical personality? When I'm stressed, my mind becomes more intuitive, less methodical. And as I've aged, I more easily become stressed (making interviews harder.)
Competition is inherently stressful. We did an exercise once in math-teaching class. After announcing the first person to finish would win a mechanical pencil, we were given a puzzle. Afterwards we were told to think about how we did, how we felt, the whole experience. The next day we were given a puzzle of similar difficulty and told as a reward for working on it, we'd each get a mechanical pencil at the end. I still felt competitive pressure, but not as much, and I did better. Many kids are extremely stressed out in math class. Some kids do much more poorly on the tests.
The first moral of the story is that for most people, this isn't a math problem- it's more about Monty's motivation, being tricky or tricked, having confidence in your first choice and whether you trust yourself or have learned you're error-prone. It's about your ability to focus on the problem and not see yourself as an actor in it, not feeling attachment to your choice or feeling pressured to be clever or win the prize.
In other words, math is all about perspective. With bare numbers, it's pretty easy for kids to do math. But once the numbers start becoming long and the procedures are onerous, other thoughts and feelings kick in and "doing math" (really just calculating) is no longer about the numbers and procedures- it's about stress and remembering. The solution is to have kids learn number sense so numbers are not intimidating and the procedures make sense, to have them work problems to gain insight rather than be rewarded and punished for their answers. The solution is to teach them to think and reason.
Interestingly, modern math teachers learn something about this. Many learn how to teach number sense and understandings about probability. But they can't practice this in California, where the overfull curriculum forces them to teach "content" at the expense of understanding. Then the state (and country) tests their ability to regurgitate it and grades their teachers on these scores, penalizing teachers for working at lower-performing schools...
Luckily for our problem in California there's a ready solution. The NCTM has a good set of standards that could be adopted in a heartbeat.