Sorry Astroslav, I don't agree.
"Can someone please exaplain to me why changing your mind about which door you choose to win the car makes a difference to the odds?
(3 doors, 2 with goats behind them, one with the car prize)"
See if I have the ground rules straight: If I pick the car on my first guess, I win the car. If I pick a goat on my first guess, then I get to pick another door; if I pick the car thia time, I also win the car. Nobody moves the goats or car around between first and second guesses.
Then it does not make any difference whether you change "your" door or not. The probability of winning this game is 1/3 in a single guess, and (2/3)x(1/2) in two guesses, for a total probability of 1/3 + 1/3 = 2/3.
To draw the same conclusion a different way, the probability of LOSING the game is the probability of choosing a wrong door twice in a row. Then (wrong 1st choice) x (wrong second choice) = 2/3 x 1/2 = 1/3.