For those that think it's a 50-50 proposition when choosing a new door AFTER 1 of the 3 doors is revealed, here's another explanation:
Assume there are 1,000 doors instead of 3 doors. Choose a door. Your odds of picking the door with the car behind it are 1/1,000. Now, would you rather stick with that pick or get to open ALL the other 999 doors to find a car. Of course, you would switch your pick and take all the other 999 doors, and your odds of winning the car would be 999/1,000. Simple, right?
Just because someone opens 998 of the doors in the 999 door group, which they know have a goat behind them, before you switch your pick, does not change the fact that you are getting to pick the whole 999 door group instead of the 1 door you initially chose. Whether the host opens 998 of the 999 doors or leaves all 999 doors closed--before you switch your pick--the host is offering you the whole group of 999 doors.
Dear Jay Helfert, if you try your bet at home with a friend and as few as 10 beer cups with a coin under one of them, it will take you less than 5 minutes to realize you are the BIG sucker by sticking with your initial choice. The probability that you initially pick the right beer cup is 1/10, and not switching your pick means that you are stuck with that probability, yet you are only getting even money if you win. You say, "It's a 50-50 bet with only 2 cups remaining, so I can stick with my original choice and I will win 50% of the time." Wrong! You made your initial choice when there were 10 cups, so you only have a 1/10 probability that you chose the winning cup! With 10 cups, you need 9:1 odds to break even on your bet if you stick with your initial choice. Damn, it would have been fun to play that game for your ebony Ed Prewitt. With 10 cups, I could have given you 2:1 odds and robbed you.
Finally, if any of you 50-50 believers wants to bet, I'll bet any amount under $10k per game and I'll pay you 10 time your bet if you win. Here are the rules: 1 million cups per game with a penny under one of them. You make your initial pick, I'll turn over 999,998 of the remaining cups, and you have to stick with your initial pick. Heck, I'll even turn over all 999,999 of the other cups, but you have to stick with your initial pick.