My favourite one of these brain-$%**s is the
Monty Hall problem. Despite the fact that I can't myself figure out why it is true (I mean, WHY do the odds change... just , they..don't!!), I know it is true because I've tried it out with cards.
0.9r is difficult to test practically.
It is a good example of a strange concept, that's not always immediately intuitive for most people.
It comes about because the host KNOWS where the prize is.
When you pick a door, you have a 1 in 3 chance of picking the right one.
The host knows whether you have picked right or wrong. If you have picked right, then he can show you either of the other doors, as they are both wrong. But if you have picked wrongly, he can only show you the other wrong door. He obviously can't open the door with the prize behind.
So at the start there is a 1 in 3 (33%) chance of the prize being behind any door.
The door you initially choose has a 1 in 3 (33%) chance of being right.
After the host opens one of the OTHER doors, you now know that the prize is behind either your door, or the other one.
But the host hasn't done anything to change the fact that you had a 33% chance of picking your door in the first place. Nothing has changed to affect what you already chose.
So your door still has a 33% chance of being right (like it did when you first picked it).
But now the prize is DEFINITELY behind either your door, or the only other remaining one. So if your door has a 33% chance of being the right one, the other door must be 67%, so you are better to swap.
But I agree, it's not something that's very obvious at all.