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.
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.