- The problem (in case you don't know it): 3 doors, 1 with a prize, 2 with a goat. You pick a door but according to the rules the host then has to open one of the doors you didn't pick (the door with the goat). You are given the option of switching your choice to the remaining closed door. Should you switch?
- The solution: When you made your original choice your chances of picking the prize were 1/3. If you don't switch they remain 1/3. By switching, if you had an empty door you will always switch to the door with a prize (by elimination). So since your chances of choosing an empty door were 2/3, by switching your chances of getting a prize are 2/3. You should therefore switch because you double your chances of prizehood. More here, here, here, here and here.
- The reaction: This originally appeared in a US column and the columnist answered correctly. She got tens of thousands of letters to the contrary, including thousands of mathematicians and academics protesting her innumeracy. It became a very heated debate and one of the most famous problems in recent history. In fact, some are quite addicted to this problem.
So what does this mean? Many treatments describe this problem as a paradox: the solution is supposed to be unintuitive (the intuitive solution being "it makes no difference if you switch since you have 2 closed doors and your chances are 50-50"). My response is twofold: "no it isn't!" and "so what?". Here it is in reverse order:
- So what? - There are many, many facts that we don't find intuitive. This is normal: our brains are evolution's solution to a very specific problem: the human being. It was not "trying" to create a creature to ponder the mysteries of the universe. Many fields of science have left the realm of the intuitive decades ago -- the most famous examples being relativity and quantum theory. Interestingly, quacks often still rail against these because they can't imagine them. Imagination is weak -- sometimes it's better stick to other faculties. Of course imagination can be trained. We can't "properly" imagine how big a googol is but we can do better than people even 100 years ago. By getting the facts your imagination becomes attuned to them. This is the case with Monty Hall too.
- No it isn't! - I've never been able to see how anyone would see this problem as weird. People have come up with lots of ways of "visualising" the result, but I'm not sure if anyone's come up with this before. Let's call it the Multiple Doors Version -- there are 1000 doors and only 1 has the prize. You pick say door 1 and then the host opens a whopping 998 doors. So all you're left with is the one you picked and say door 223. Do you now switch? Although our intuition is NOT designed for probability I think this version passes the intuition test. And it also tells us something interesting about how our brains work in terms of using exaggeration to make some things stand out.
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So why, after eliminating 998 out of the original 1000 doors, is the chance of the prize being behind door 1 different than behind door 223?
When you picked 1 you had a 1/1000 chance of picking the prize. So when the host “magically” leaves door 223 closed (of all the doors) it has a 999/1000 chance of having the prize.
Michael, re-read the Wikipedia article you linked to - under ‘Aids to understanding’, there’s a paragraph called ‘Increasing the number of doors’ which illustrates the solution in much the same way you have.
Yep, I was pretty sure I wasn’t exactly inventing the wheel here. On the other hand if you look at comment 1, it still doesn’t workin helping some people “feel the answer”
I tried the experiment with 52 cards and a person who had not been exposed to it (my wife). She also replied “why would I switch cards at the end?”. This proves that the decision is not intuitive at all: you are better off using decision-tree theory. I remember reading the original New York Times article 16 years ago, still fresh from my operating research class, and yet it took pencil and paper to figure it our correctly.
Did you ask your wife to try find a particular card (such as the seven of diamonds)?
Although it is unintuitive, I think in your wife’s case the fact that she was seing 50 different KINDS of cards threw in a red herring — because she was seeing all different cards it might have been easier to see the 2 closed cards as equivalent. Maybe it might have worked better if she was searching for say a single blue card in a sea of red.
But probably you’re right, even that wouldn’t have made a difference.
Michael,
I asked her to look for the Ace of Hearts. My conjecture is that, if I increased the number of choices, going up to say 1,000,000, at some point she would change her strategy from “No-switch” to “Switch”. I can’t do that now with her because I already explained to her the logic of the puzzle.
It would be interesting to see what is the number that makes people change their strategy, depending on conceptual abilities in math, probability or decision theory.
Your exaggeration method might well be more intuitive, but is that the only intuitive extension of the problem? What if one’s intuition says “no! that’s not the same problem! the same problem is where I’ve selected one door of 100, and they tell me it’s not in another one…”?
You’re right, but an extension needs to exaggerate otherwise it’s no different to the original problem. I’m not sure if I’ve seen a reaction like the one you described — have you?
In this case it might do to get the person to appreciate the exaggerated version even though they think it’s different and then argue backwards (what if only 97 doors were opened? then 96 etc.)
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