If you're having a bit of a slow day (or share at least a fraction of my interest in such things), Lawyers, Guns and Money have a brief piece up on the Monty Hall problem, a probability problem that seems to confuse an awful lot of very smart people.
I can confirm Campos' claim that the four explanations are in the right order both in terms of abstraction and difficulty. Over the years I've tried various ways to explain the problem, but all of them are variations on 2 (I grant that 3 might work better, but it requires pen and paper to write down the list, and that seems to immediately make people regress to childhood homework traumas). Certainly I'd never attempt to use the most abstract explanation 1 with anyone but another mathematician, and even then I wouldn't expect a particularly high success rate amongst geometers and numerical analysts.
I also confess to sharing in Campos' frustration regarding the near impossibility of persuading certain people to abandon their intuitive position, irrespective of how many different explanations are employed to prove it incoherent. Already the comment thread over there is overflowing with misunderstandings that it's taking all my self-restraint to not correct.
16 comments:
I sympathise; I failed to resist and spent last weekend trying to explain quantum randomness, and will confirm that letting go of intuitions, or at least being open to other possibilities, is bloody hard.
Still, it could be worse; I've seen arguments rage for weeks and users be given impending ban warnings over the 'aeroplane on a conveyor belt' problem, which in theory is conceptually far simpler but comes with a massive sting in how you set up the problem.
Yeah, that one came up in the maths coffee room a couple of weeks ago. Everyone ended up like puppies with chew toys, including me.
By "that one" I mean the plane/conveyor belt discussion.
A layman's view: I don't like 2 much. Both 3 and 1 are better for combatting my intuition (though I can't say why - I think 2 just feels like a different problem, and that gets me worrying about loopholes).
Another interesting question is why it is so counter-intuitive. Is it an evolutionary wiring thing, or the way we were taught probabilities at school? Maybe because the situation is quite contrived so we can't draw on real experience to think about it?
In my mind the problem stems from thinking of probabilities as fixed quantities that exist "out there", when in reality they are dependent on the knowledge of the situation and may change when the amount known changes. If you can convince someone of that up front, Monty Hall won't seem half as bizarre.
"A layman's view: I don't like 2 much... I think 2 just feels like a different problem, and that gets me worrying about loopholes)."
I think that's entirely fair. What I would usually do is give the 99 goats example, then go down to 98, then 50, then 10, 5, and so on. That way, by the time you get back to the original 2 goats, people have to consider why that particular situation varies from a general rule that can be (comparatively) easily grasped.
"Another interesting question is why it is so counter-intuitive."
It's a good question; which I find hard to answer precisely because the Monty Hall problem doesn't seem even remotely counter-intuitive to me. The first time I saw it I wanted to stick where I was, not because of the probabilities that underlined it, but because I'd rather risk picking the wrong box and keeping it than risk picking the right box and then changing my mind (which, though idiotic, still governs my behaviour with answers at pub quizzes). I think that reaction might be a common one, but I would doubt it really factors into why people find it so hard to comprehend the problem when it's explained to them.
I think your answer is the best one. People believe that because the car could be in any box, their choice has to be as valid as any other. The problem comes in persuading people that their belief may need updating in light of new information. This was shown most clearly by a commenter in the LG&M thread, who asked why the problem's answer wasn't wrong in the same way that claiming a coin that comes down tails 50 times in a row is more likely to come down tails is wrong. Aside from not realising that the difference is between unknown events and known events in an unknown order, the truth is any rational person who saw a coin come down tails 50 times in a row would assume tails is the more likely event, since the chance of a fair coin managing that run is about 0.00000000000009%.
Sorry; forgetting my own rants. Safer to say there's a 0.0000000000002% chance of a run of 50 heads or tails, and that such an event would certainly cause one to reject the null hypothesis that the coin was fair.
Also, dare I ask what is the plane/conveyor belt problem?
I haven't read the explanations yet, but I would like to gloat that I got it right first time, by spending 30 seconds working out the three options (if I chose the wrong door, I should switch, if I chose the right door I shouldn't and the likelihood of having chosen the wrong door is double that of having chosen right, so I should switch).
Of course, Psychologists do have to learn basic probility as UGs...
Hm, turns out I used method 3, while rejecting method 1 (my first attempted angle) on the grounds that starting from scratch with two options means my original choice could still equally be right (50:50). The idea that changing in itself ups the odds from 1/3 to 1/2 is rather cognitively dissonant for me I'm afraid and seems to enter into the realms of cat-killing physicists and philosophers.
So I picked up a pen and worked it out, which made me much happier.
However, I can offer the valuable insight into why people resist this partly - it's a bit like the concorde fallacy. Once you're invested in X you feel a strong urge to keep investing lest that first investment be lost, regardless of the fact that it will likely cost you more to keep going. We find it difficult to disengage from things like that.
And as for pub quizes, the no-second guessing also works by preserving instinctive recall against conscious uncertainty.
Damnit, I've re-read 1 and now understand. Kinda.
Hm, sorry for the spam!
The problem is that most people try to solve the extensive form solution (start with prior of 1/3 for choosing the car, update on Monty's information) but don't think hard enough about the updating.
Solving the problem in the normal form is much easier. You have two strategies, to switch or not to switch. It is trivial to work out the probability of victory for each strategy, since a switcher will win iff he chooses a goat, and a sticker will win iff he chooses the car.
Normal form wins again.
Related, the best title for a maths/logic/philosophy seminar I've been to is "Judy Benjamin is a Sleeping Beauty, modulo Monty Hall". It was a good talk too.
@ Tomsk
The aforementioned problem asks whether or not a plane which is moving forward along a conveyor belt that can match the plane's speed exactly will take flight when it reaches the necessary velocity, even if the conveyor belt means the plane isn't moving forward relative to the Earth.
That's just a trick: the "even if" makes it 2 questions, as in "even if we accept the following false statement..."
I believe the question is a trick, yes. Doesn't a conveyor belt mean you aren't moving relative to the Earth, though?
I hate physics.
Only if you're in a car. A plane works by pushing air around, which the conveyor belt has no effect on. The wheels will go round twice as fast, but otherwise there's no effect.
Gotcha. The wheels aren't powered. In fact, edenspresence already explained that to me, and I'd forgotten it.
Did I mention I hate physics?
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