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i thought it would be helpful to make a comment on why the probability after switching is 66%, since it's very counterintuitive. the main reason has to do with why a certain door was chosen to be opened.
in more detail: it helps to instead think about a situation where there are 100 doors, 99 of which have a goat behind them. (original problem was about goats and a car). you choose one of 100 doors. there's a 1% chance you picked the door with the car. in other words, there is a 99% chance the car is behind a door you did not pick. put differently, 99 times out of 100, you are in a situation where the car is behind a door you did not choose.
afterward you pick your door, the host picks 98 doors with goats behind them and opens them. (this is another crucial detail, the host can only open doors that don't have a car behind them.) it is still true that 99% of the time, the car is behind a door you did not choose. this is because the doors were opened after you made your choice. but now, there is only one door you did not choose, so that door has a 99% chance of having a car behind it.
Your hypothetical strapped 495 people to train tracks, you absolute monster! Statically, about forty of those 495 have serious chronic back pain, too. You obviously don't care about the disabled. Be more careful next time when changing the number of doors in a Monty Hall problem!
On the other hand, statistically, the chance of anyone dying is much much lower
their back pain is in service of the greater good. with an extra 485 people strapped to the tracks the chances of anyone dying goes from 33% to 1%. i did it for the people
I can kind of understand the logic behind it, if you assume your door can't be affected by the probability of it, but the thing that still stumps me about this is how the probability for your door is "locked in."
You picked a door out of a set, and by opening any number of doors, the host has altered the set. The other door remaining went from being a 99/100 chance of having a goat behind it to being in a set of 98 knowns, and 2 unknowns. While the host can't choose it if it has a car, he also can't choose yours. You wind up with 2 identical doors and X number of open doors, with each door having a 50/50 chance given the re-evaluation.
I know this is supposed to be the wrong answer, but I can't see why it's wrong. If you have an explanation, I'd love to finally be able to understand this problem.
It's because the host has knowledge of the situation. Each door has 1% chance. You lock one door closed that the host can't touch, goat or car. The other rule for the host is they have to open all of the doors except one and they also can't open the car.
The door they leave is all possibilities from all doors at the begining minus your door (since the host couldn't mess with it), which is 1%. Your original choice determined that door would stay closed by the host so the host can't effect it's odds with their knowledge. There is only a 1% chance you locked the car behind that door picking randomly. Either you got the 1% and locked the car door or you didn't. The host will remove 98 goats without random chance and which will leave a car or a goat left. There's a 99% chance you made them leave the car with your first choice being 1% and a 1% chance you made them leave a goat. It's still the original odds, but it flips because the host has knowledge and either was forced to removed all possible bad choices and leave the car or you managed to hit the 1/100 chance at the start. There's a 99% chance you didn't, so switching has a 99% chance the host left it there.
I think this might have been the answer that helped me the most. Most of all, it's that the Monty Hall problem isn't about you, it's almost entirely about the host's action of revealing doors.
There's a 98/99 chance he left that door because it's the car, or 1/99 because it's the goat (assuming the one left out of calculation is your door which he can't choose). Your original choice, whether or not you picked the car, is largely irrelevant. His actions can't affect your door because he can't choose it
You're not betting on a new set of 1/2, you're not even betting on the door itself having a new probability. You're betting on the act of the host revealing doors.
that's a really good question. the probability getting locked in is very counterintuitive. i think it helps to think about what happens in each case.
let's say you pick one of the 99 doors with goats behind them. this happens 99% of the time. the host is tasked with opening 98 doors. of the 99 doors you didn't choose, one has a car behind them. the host does not have a choice about which doors to open, his hand is forced. in this sense, the 99 other doors are tied together: since you originally chose a door with a goat behind it, the host is forced to leave only the door with a car unopened. so, you switch and you get a car.
next, let's say you get lucky and pick the door with a car behind it. this happens 1% of the time. now the host gets a choice about which doors to open: he gets to pick one door with a goat to leave unopened. in this scenario switching gets you no car.
so, 99% of the time, switching gets you a car. i hope this is helpful!
This definitely helped me look at it as a whole, and definitely started me down the right path of getting it. Thanks!
Way easier to explain it this way:
Switching turns a correct guess into an incorrect one, and an incorrect one into a correct one. Your initial guess was more likely incorrect.
I never understood why in the 100-door case, the host opens 98 doors, and not just one door. That feels like changing the rules.
I fully understand the original problem with 3 doors; I know the win probability is 2/3 if you change. But whenever I hear the explanation for 100 doors case, it just makes everything confusing. By opening 98 doors, it feels like the host wants you to switch to the other door. In 3 doors case it's more natural.
Because the problem is explicitly about the choice between two doors. You have to eliminate all but two choices.
But even then, you'd still have a better chance by switching.
Your intuition about the change is the whole point - it exposes why the result is what it is.
In both cases the host opens every door but one.
i like to use the 100 door case to highlight how the probability "transfers" to the remaining unopened door. i understand what you mean about it feeling like the rules are changing though. in some ways opening only 1 door would be more natural. that being said, if 98 doors open it stays true to the original game in the sense that after the doors get opened, you're left with two closed doors. at least to me, this makes it clearer how the probabilities of the 98 doors get "transferred" to the remaining unopened door.
i also think part of the beauty of the 100 door explanation is that it does make it feel like the host wants you to change. in that way, increasing the number of opened doors brings out a "hidden truth" about why you should change doors.
but at the end of the day it's all just a different way to understand the problem. if the traditional way makes more sense to you, then there's value in that as well.
It's really, really simple. Just counter-intuitive.
Each door at the start has a 33.33% chance - and obviously this doesn't ever change.
What changes is that one door opens without your impact, and that door always has people laying on track. Which means the choice is now between two: your initial doors (doesn't change, so 33.33%), and NOT your initial door (100% - 33.33% = 66.66%).
And no, this is not a loophole. The chance is impacted because one door (the automatically opened one) was opened by someone who knew what was behind all doors.