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So you pick a door, you have a 1/3 chance of picking the correct door. Now they open one of the other wrong doors (if you picked a wrong door, they show you the other one, if you picked the right door they could show you either wrong one) and then give you the chance to repick, Now, there are two doors and one is right, so your chances would appear to be 50%, but if you picked that door out of three, they really are still 33%. Now you are given the chance to pick the other door if you want to, upon making a choice of weather to switch or not, it would appear that your odds are now 50%, yet how did the odds suddenly change if you decide to stay with the door you were originally on?

Also, my teacher says that if do choose the other door, you will be correct 2/3 of the time, how is this true? Since you are picking again from two, it would seem that you only have a 50% chance of choosing the right door. Can anybody prove that it would be 66% of the time mathematically?

Thanks for your help.