What helped me is the fact that the only way you WON'T win by switching, is if you correctly choose it the first time. And that only happens 1/3 of the time
The problem I always had with this (and still have) is that you're not asked about the probability (i.e. what trend you expect to see in a string of repeated experiments), but rather you're asked:
"Is it to your advantage to switch your choice?"
Well that depends on where the prize is. If the prize is behind the other door, then yes it's to your advantage to switch. Otherwise, it isn't. How can there be any more to it than that? You're not going to be given the opportunity to play the game 100 times. This is your one shot and the location of the prize is already fixed.
Well if you don't switch then your odds of winning are a simple 1 out of 3. Switching inverts your initial guess, so instead of losing 2 out of 3 times because your initial pick was wrong, switching makes you win 2 out of 3 times when your initial pick is wrong.
Imagine there are 100 doors. You pick one at random. You have a 1 in 100 chance of getting the right answer.
If the host opens all the doors that contain the booby prizes bar one (in this case 98 goats) and leaves one door remaining, plus your original guess, it's pretty obvious you are unlikely to have guessed correctly on your first try.
Scale it down to 3 doors and the host always revealing a goat and you have the same situation except there's a 1 in 3 chance you'll win by sticking with your first guess and a 2 in 3 chance if you swap.
Right, it's your one shot. So what gives you better odds (that is, is to your advantage). If you don't switch, you're selecting the lower odds chance of winning, putting yourself at a disadvantage. Since you don't know where it is, but do have a chance to double your chance of winning, then switching is advantageous.
"Is it to your advantage to switch your choice?"
Well that depends on where the prize is. If the prize is behind the other door, then yes it's to your advantage to switch. Otherwise, it isn't. How can there be any more to it than that? You're not going to be given the opportunity to play the game 100 times. This is your one shot and the location of the prize is already fixed.