For some reason the internet still has weird arguments about the Monty Hall "paradox". I think it might be because sometimes the way it is explained is a bit confusing to some people? Idk, but I do remember that when I first heard about it the explanation was kinda weird to me until I rephrased it in simpler terms.
Here is the thing. You have 3 doors, one of them has a car train behind, the others have goats. You pick a door at random, the host doesn't open it, but he eliminates one of the other doors by showing it has a goat behind it, and asks you if you want to change your choice. So perhaps you picked door A first, he eliminated door B, and now your choice is between door A and door C. If you change your initial choice, it turns out that you will be correct 2/3 times, whereas if you don't you will only be correct 1/3 times. This confuses some people because they expect that since it becomes a choice between 2 doors it should be 50-50.
The simple explanation is that if you picked the door with the train the first time, then by changing you will always lose. If you didn't pick the correct door the first time, then changing always wins. But you will only get it right the first time 1/3 times, whereas you will get it wrong the first time 2/3 times, so changing is advantageous. That's it. That's all there is to it. There doesn't need any more mystification.
If it is still not intuitive, imagine if you had a million doors, you picked one, and the host eliminated all the others except one. There is (almost) no way you picked the right one the first time, it is literally a one in a million chance. So if it is not that one, and it almost certainly isn't, it must be the other which the host practically hand picked for you.
EDIT: For the reasoning to work, the important assumptions are two. One, the host always eliminates a goat, never a train. Two, the host always reveals one of the other two doors, not the one you picked. Both of these are significant, without the first one you have only a 1/3 chance of winning regardless of whether you change or not, and without the second one it becomes a regular 50-50 choice.
These explanations are good, but you have to be a little careful because I think there's a risk of them also giving you the wrong intuition.
For example, imagine Monty has slightly different rules- instead of always eliminating a door with a goat, he randomly picks one of the doors the contestant didn't choose and eliminates it, even if it's the one with the train (in which case they're shit outta luck). Today you're on the show, you pick a door, he eliminates one at random and- good news, he eliminated a goat. Now do you switch or stay with your current door?
In this case, it actually doesn't matter whether you switch or not, it really is 1/2 either way. But it superficially sounds like your reasoning for the real Monty Hall problem should still work- if you initially picked a goat then the remaining door is a train, and vice versa.
I don't think so, because the fact that if you change you win hinges on the fact that the train is still there to be found and hasn't already been revealed by the host. If the host revealed the train then obviously you've lost immediately. So it's not even 50-50 really, 1/3 times you will just lose immediately by having the train revealed.
You're right that before he gets involved it's 1/3. But I'm talking about the situation after Monty eliminates his door and reveals that it's a goat... which doesn't always happen- sometimes it's a train! But let's say you're actually on the show, and this has happened, and in this particular case he revealed a goat.
Yeah, what I am saying though is that if the host doesn't know which door has the train behind and he just reveals one of the doors, your total chance of winning either by changing or not changing is just 1/3, because you don't get any new information, if the host accidentally reveals the one with the train you just lose. You can just do away with the host entirely, you can just do it yourself.
Just being pedantic: There's three doors, the train has to be behind one of them, so the probabilities for each door have to sum to 1. He just revealed one door has a goat, so the probability of that door having the train is 0, so the other two can't both be 1/3, otherwise you're missing a bit of probability. You're right that the probabilities are equal for the two remaining doors, but that makes 1/2 each, not 1/3. You can use the million door analogy here- he opens 999,998 doors at random and there's goats behind all of them. You're now a lot more confident than when you started that you picked the right door, so your chances definitely feel better than 1 in a million now!
But my (hopefully less pedantic) point is that in this situation (three doors, you picked one, then Monty eliminated one at random and showed that it's a goat), you could still say: "The simple explanation is that if you picked the door with the train the first time, then by changing you will always lose. If you didn’t pick the correct door the first time, then changing always wins.". That's all still true: if you originally picked a goat and Monty eliminates a goat, you win by switching, if you originally picked a train and Monty eliminates a goat, you win by staying. But this time that would lead you to the wrong conclusion (that switching is better than staying).
IF he revealed it has a goat, yes. I am talking about the full game though. It is equivalent to a problem without any host at all. Basically it is like picking one door at random, then looking at another one and deciding you're never gonna pick it, and then rethinking which of the other two you're gonna pick. You don't learn anything new, so it is just 1/3. You are right that it is 50-50 if he has just happened to reveal the goat, but 1/3 times he just reveals the train and you're fucked. So 1/3 times you're immediately fucked, and 2/3 times it becomes a 50-50 thing (although that needs a bit of explaining for why it is the case which I won't go into), so in total you have a 1/3 chance of winning overall.
Yep, I agree with most of this, I just think it may not be intuitively obvious that looking at a door picked randomly and seeing it's a goat gives you less information (or less useful information) than being told by somebody who already knew it was a goat.
Yes, it is slightly more confusing. The key to understanding why that is is realizing that if he happened to reveal a goat, then that lends slightly more credence to the hypothesis that you picked the right door in the first place. That's because of you picked the wrong door, the host revealing one of the other two at random has a 1/2 chance of revealing a goat. On the other hand, if you picked the right one, then the host will always reveal a goat. If you work it out, 1/3 times in total he reveals the train and you lose, 1/3 times he reveals a goat and your initial choice is incorrect, and 1/3 times he reveals a goat and your initial choice is correct. So if you see that he revealed a goat, whether you switch or you don't doesn't make a difference.