Heuristics and biases teach us that humans systematically make errors in reasoning, these errors are common in lay people and scientists alike.
The monty hall problem:
Theres a gameshow you’re on, theres three doors (one with a car, and two with goats). you choose door1. Monty opens door 3 to show you a goat.
What are the chances you win the car if you choose door1 (your door) or door2 (the unopened) door. Most people think its either 1/3 (with either door) or (1/2 chance) with either door. But its not. Door1 is 1/3 chance to win. door2 is a 2/3 chance to win.
I came across this problem and I actually solved it. Which isn’t surprising because its a brutally simple reason why door2 is a 2/3 chance to win, but because every single person i’ve talked to since, is convinced* that its the same probability with either door.
The reason is because you’re dealing with different subsets and the goat reveals information which was previously unknown. (this is all only true if the gameshow host knows the location of the car, btw.)
2/3 times the car is going to be in the two doors that you didn’t pick (by chance) so that when the goat is revealed, picking the *other door, has a higher probability to win.
If anyone is doubtful of how in turns out in practice, grab a deck of cards, (use an ace for the car and 5’s for the goats)
I pass you one card face down, and I have two cards, (2/3 times i’ll have the ace) I show you a 5 and all thats left is two face down cards. 2/3 times, i’ll have the acec.
to put it simply the reason the second door is 2/3 is because monty knows where the car is and the fact that he decided to show you the goat in the 3rd doors means it is more likely there is not a goat in the 2nd door. the way you worded the problem in the first post fails to mention that monty knows where the car is and deliberately showed a goat, without this information I assumed that monty chose to open the 3rd door without knowing where the car was, in which case the two remaining doors would have an equal chance 1/2. I don’t think this is a problem in human reasoning but more like a lack of information and failure to properly state the problem or perhaps my failure to assume the actions of monty were deliberate, either way it’s kind of a stupid problem, at least in my opinion
The part that I have a hard time with is, what if we add another contestant or viewer partway through?
Suppose the scenario all plays out just as described in the example, and after that I turn on the TV and start watching. For me, I see 2 doors and a goat. From the two of them talking, I get the general idea that there’s a car behind one of the two doors, and a goat behind the other. It seems like in that situation, I’d have a 50/50 chance of picking the right door, if I didn’t know which one the contestant picked originally. Right?
Can someone explain to me how discovering which door the contestant picked would provide information that makes one door more likely than the other? What is it about the contestant picking door 1 that makes me go, “Ah, it’s probably door 2!”
Has anyone read ‘The Man Who Only Loved Numbers’? Some good stories in there about how some of the finest mathematical minds of our century couldn’t grasp this. For my part, I understand the proof I saw, but its still very counter-intuitive. If Paul Erdos didn’t understand it, I don’t feel so bad.
Be simple enough to test with 2 people and a deck of cards, wouldn’t it? The stats aren’t just numbers in space, if the example is right, switchng really should get you the car (or ace) 66% of the time.
someoneatwhatever is always making absurd and nonsensical claims that he doesn’t even try to justify. Of course his beliefs are irrelevant, its not just numbers in space, which anyone intelligent enough to operate a deck of cards should be able to find out for themselves. His beliefs won’t alter any probability when making the choices or the statistical outcome. Don’t expect any coherency besides complaints about my repeated use of the word ‘absurd’ sad and pathetic.
people don’t believe it only because they don’t ‘get’ it. It works out that way in the real world, regardless of personal opinion. thier lack of belief doesn’t change the probability difference between picking thier card or the non revealed. if they don’t believe, they don’t get it, because thats just how it works out. again master the playing card and get back to us.
How can a prediction be “unjustified”? It’s either going to turn out accurate or inaccurate between subjective opinions based upon the criteria it self-describes.
Men only gather around the predictions that keep proving themselves true, which is the error of taking science as a faith…
That’s exactly the definition of ambiguity – whatever a person is compelled to do is whatever a person is compelled to do.
The situation says nothing until beforehand or afterward in context, until it is discussed and made sense of, otherwise we’re just being. Actions speak louder than words, yes, but what are we doing here on this forum – acting or writing? Where do word acts begin and end? Is this even the problem at hand?
The hypothetical is a memory of something that may or may not be real, according to fiction or nonfiction. Predicating events based on hypothetical contexts are what men instinctively do in order to anticipate the exchange of gunfire. The hypothetical can be true when what is “true” is definitely going to happen (like that I’m going to drive my car within the next two hours).
I want some clarification…
Skepticism & Nihilism continually beg the question. Pragmatism ends it, so it also forces men to decide what is “valid” and what is “invalid”.
It’s only as flawed as flawed men allow it to be. The fallacy of logicians (and most other logically predicated sciences) is that they don’t know how, where, or when to update their language after mistaking it as an absolute authority. I won’t speak for others, but my authority rests in the same place I put my faith…