alt Hacker NewsMy favorite example is Monty Hall problem. "Smart" people often use it as the evidence of how bad general people are at probability.
It really isn't. The problem is usually given in this form:
> Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice? [1]
The only correct ansewr to this question is "I don't know." People who answer yes are just taking way too many standardized tests.
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But it is in your advantage! Code a simulator and do some numerical experiments. The simulations where you switch end up winning more.
Edit: from your other reply I see your beef is more with the wording.
Edit edit: although I disagree with your point even then. But it seems to lead to rather fruitless argument, so let's leave it here.
I guess I don't understand why you think "I don't know" is the only correct answer. It's clearly not, which is the point of the problem in the first place. It's a bit hard to grok, but once you do, it's clear what the right answer is.
Perhaps there's another formulation you meant to write, but every time I've seen it, it's been equivalent to the thing you just asked, and the only potentially ambiguous part of the question is "to your advantage". If you stay then you have a 1/3 chance of getting a car, and if you switch you have a 2/3 chance.
Yes, it's also correct that you don't "know" the result, and you might prefer goats to cars (even then you should probably sell the car and buy several goats), but there's a reasonable enough interpretation of "advantage" that you shouldn't dismiss the problem outright.
Perhaps it'll be more intuitive to you if you scale the number of doors up. If there are 100 doors, only one containing a car, you pick one, the host reveals 98/99 remaining doors as goats, it's obvious the correct choice is to switch. The correct answer is a mathematically provable probability. 1% chance you picked the right door, 99% chance the door was in the remaining pool, therefore 99% chance the last remaining door is the correct door.
Can you explain the mistake (of people taking the standardized tests)? I don't understand it from the message, and the Wikipedia article seems to detail exactly the "normal, smart" switching solution.
The difference here is that the Monty Hall problem has an explanation that while counterintuitive, is statistically sound. You should always switch, because the probability you picked the correct door is locked in at the time you made the choice between 3 doors. It is 1/3 that you picked correctly, and 2/3 that you picked incorrectly. The counterintuitive part is that if you switch, you are effectively selecting all the doors you did not pick originally. It's your original door, vs the field.
The green hat problem hinges on subjective interpretations of the meaning of both "liar" and the different ways in which the liar's sentence may be false. It may be false because the liar owns many hats, none of which are green. Or they own many hats, only some of which are green. Or they own no hats. These are all reasonable interpretations of how the sentence might be false, and the answers presented are not necessarily mutually exclusive.