@Jonny @MeBigbrain @matty @pepsi_man @petra @sickburnbro You don't understand the scenario, or the solution?

The explanation is straightforward, just counterintutive to most people's understanding of probability. Probability does not represent a discrete fact about the world, it represents a measure of your knowledge about something's outcome.

If you pick 1 door out of the three, you have a 1/3 chance of getting it right.

If you picked it correctly, then you win, you go home, and there's no problem to solve.

If you did not pick it correctly -- the 2/3 chance -- then you are shown a non-winning door, and asked if you want to change it.

Now, if you are in this situation, which you would be in 2/3 times, the host will now give you a choice to switch to one of the other ones.

Now, you KNOW that the 1 door you picked is wrong. You aren't picking one door out of three, you are picking one door out of two. So your second choice is not to find 1 winner out of 3, it's a choice to find one winner out of the 2 remaining.

Which means that because you now have more information, the chance of either of the remaining two being winners is 50% -- because you've eliminated one of the choices that was wrong.

If the game started by the host simply saying "Pick one of these three doors, the prize can be in any of them -- oh, but we're going to throw out one of the wrong ones, and you can only choose between two, one of which is the winner" -- it would be obvious that it's a 50/50 choice without the third door.

The intuition is that each door has an inherent 1/3 chance that's a property of the door. That's not, that's just a property of your knowledge about the door. Note that the host does not think each door has a 1/3 chance, he knows which door has a 100% chance and which doors have a 0% chance.