Let's not forget that there really aren't that many possible outcomes in this game, so a skeptic can easily construct a truth table.

For example, if you assume the prize is behind door #1, write a truth table of all possible outcomes if you stay with your original choice in every case. When you choose door #1, you win. When you choose door #2 or #3, you lose. So when "staying," you have a 1/3 chance of winning.

On the other hand, if you switch every time and do the truth table, choosing #1 loses, but choosing either #2 or #3 wins, so the player has a 2/3 chance of winning when switching in all cases. Once you realize this, the same odds apply whether the prize is behind doors #1, #2, or #3.