You participate in a game show where you must choose between 3 doors. Behind 2 doors is a goat, but behind 1 door is a car. You pick a door, say door no. 1, but before you open the door, the host opens another door, say door no. 3, which has the car behind it. "Oops," says the host.
You lose.
Wow, people in that thread are so stubbornly and arrogantly wrong, it's insane. Normies having opinions slightly outside of normie-norm is even worse than normies being normal normies.
A lot of people are leaving out the point that the host KNOWS which door has the car behind it and is presenting you a much higher-quality choice by opening all the other doors. If the host doesn't know himself, it doesn't make sense and you get an outcome like the above.
Well, yes, a host would never choose randomly, for that reason (and that would be hilarious), but...even if he did choose randomly, but "randomly" chose a non-prize...it does still make sense to switch. Wild, but true.
It's not the knowledge that changes the odds.
I guess what I mean is that a key feature of the simulation is the host always reveals a non-prize.
These days, since I already have a car without government mandated shut-off switches, I'd rather the goat.
True.
No, if he chose randomly and didn't reveal the car by pure chance then both doors remain equally likely to have the car behind them and switching doesn't matter. Most teachers are bad at explaining/don't understand the monty hall problem themselves and they take shortcuts when explaining it so that's an understandable confusion though.
The point is that the host choosing to reveal a door has more knowledge than you and he divulges new information by elimination when he opens a door that lets your make a more favourably informed choice in the second round.
But lot of people try to explain the system where the probability of your first choice is just frozen in time to before the outcome is revealed, and you get to chose a second time at 1/2 odds simply because there happen to be only two doors left. The leads to the nonsense outcome where if host knows nothing and accidentally reveals the car at random, then you definitely shouldn't chose to switch doors because there's still a 1/3 chance you win a car if you stick, but a 0/2 chance if you switch. When in reality the choice is meaningless.
No, actually both explanations you’ve given are wrong as I understand it. Here’s the solution as I know it:
When you picked, you had a 1/3 chance to pick “right.” Since probabilities sum to 1, that means there is a 2/3 chance that the prize is behind a door you did not pick. One of those doors is then eliminated and you are shown that door did not contain the prize. It doesn’t matter whether that door was eliminated at random or purposefully—what matters is that you know the prize was not behind that door, and that out of the selection categories remaining, [door you originally picked, 1/3rd chance of having a prize] and [all doors you did not pick, 2/3rd chance of having a prize] you get to repick your category. If it helps, you may think of repicking as opening every door in the latter category, since that’s effectively what will happen.
If you switch, you have a 2/3rd chance of hitting the right door, because that’s the necessary inverse of your original 1/3rd chance, now that the other possibility of a dud door has been removed.
Yeah, I think I was wrong. I was thinking about it before I left for work, but didn't have time to edit. It's weird to think about, and still hard to completely grasp, but I do believe intent matters. It's just like how something crazy improbable can happen, and it doesn't affect the next result. You can flip 99 coins, and have them all turn up heads...you're already in the crazy universe, but the next flip is still a 50/50 chance.
So, you could have ten doors, and the host could randomly remove eight losers...the remaining two doors are still equally likely, you just got a really unlikely outcome...which I suppose means logically if not mathematically, you actually have more likelihood of the car being behind the already chosen door.
So, if the host chooses randomly, and you know that, I suppose you're actually better sticking with your choice. The more doors he reveals (again, if randomly, and you know it's random) without revealing the car, the more likely the car isn't in the doors that were unchosen.
Ugh. This gets more and more confusing, because you do have to know he's not choosing randomly, to be able to reach the "answer." If you don't know how he's choosing, it gets into more mind games than logic or math.
So I suppose I was a little hasty. I was still right that all those people on Twitter were retards and didn't even get the basics, but there were more assumptions you have to make than I took into account, initially. You have to know how the host is removing doors to be able to calculate whether or not you switch.
The odds only change if what's behind the doors are fixed ahead of time.
If the host chooses a door, the game show rolls 1 out of ndoors chance of car behind it, and then puts the car or goat behind it before opening it then the odds don't change at all even if the host didn't know.
Switching your choice is the classical hidden-information answer, staying with your first choice is the quantum changes when-observed answer.
Yeah, but the assumption is that there is a right door. You're changing the fundamental rules of the problem and saying "guys, when you say the whole thing works differently, it works differently!" That's true, but it's not exactly insightful or useful.
Yup.
If everything is fake and gay, it doesn't matter if you switch or not, the funniest outcome will happen, if someone has their hand in it. i.e., you switch, and the car was "always" behind the door you initially chose. Oops, you lost!
It's also pointless, though, because if everything is in flux, your choice doesn't matter. So the fixed version is the only one that really makes sense, if the "problem" is expected to be solved.
Yeah, but—and I say this having read the comments and knowing people aren't getting it—shouldn't you assume that, unless you're a moron? You thought they set up a game show so there was a chance they reveal the big prize and then stand there sheepishly going "so... uh.. do you want to pick a new door, or...?" Why would that ever be part of the design?
Also, as u/Kienan pointed out, even if it was random, you'd still switch.
I will direct to myself for the argument that switching doors when the host has no knowledge and just got lucky is meaningless. It's only smarter to switch doors if the host knowingly makes the correct choice.
It depends on how much you see the problem as a contrivance and how much as an actual scenario. Either way, the point is not to see it as a one-off event, but to run the simulation many times in order for the statistics to manifest. In every iteration of the simulation, the host must select a non-prize door.
Or that the car and goat are already behind a door. There may be nothing behind any of the doors until you choose; they're always wheeling stuff around behind the scenes in game shows.
The answer in the original Parade magazine column way back relied on assumptions not stated in the description, and the smart people coming up with the 'wrong' answer just had different assumptions.
The author of the column went by a pretentious "vos Savant" assumed name and lied about her credentials; she claimed to be in the Guinness Book as highest IQ, but was never the record holder at the time of publication and the one year she was listed was for a test taken when she was like 3 years old. The whole column was a sham.