But in the explanation it pays to swtich 2/3 of the time. and the trials the schools have run have shown that it pays to switch.
Think about it this way. Under your assumption, it does not pay to take ANY action since you say the door in front of you now has a 50/50 chance of having a car there. So under your assumption, you should be winning 50% of the time. OK. So if we are to run the game multiple times, and you always stick with choice 1, arey ou saying you'll win 1/2 the time or would you acutally win 1/3 of the time?
Switching door can bring you advantage, but not up to 2/3, should be 1/2.
Please read my reasoning.
Surely, if you don't swith door, your probability of winning the car is always 1/3. But what if you do switch door, does your probability goes up to 2/3?
No.
Why is that? Because if you switch, then you enter a totally new game in which there are two doors for you to choose---a car and a sheep. So you have a 50% probability of winning. If you don't switch, you are still stick to the previous game in which you only have 1/3 chance to win. The fact that the host opens doors for you is actually giving you the opportunity whether you want to enter a new game or not.
That is to say, the chance of winning by switching door is not 1 minus 1/3, but 50%. Switching door or not decides it's a one game or two separte games.
Wiki is wrong and you are wrong.
In Wiki's example, if you have 10000 doors, then switching door brings you 99.99% winning probability, which is obviously wrong. Probability is not continuous in two separate games so that you can not 1-0.01%=99.99% to get the answer.
Malak's mistake lies at that he thinks if he don't switch, he can still enter the new 50-50 game, which is not correct.
Wiki's mistake lies at that if we choose to switch, probability condition changes but we are not entering a new game.