My favorites are the unexpected hanging:
A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.
Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the "surprise hanging" can't be on Friday, as if he hasn't been hanged by Thursday, there is only one day left - and so it won't be a surprise if he's hanged on Friday. Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.
He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn't been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.
The next week, the executioner knocks on the prisoner's door at noon on Wednesday — which, despite all the above, was an utter surprise to him. Everything the judge said came true.
and the Monte Hall one:
You are in a game show. There are three doors. Behind 2 are goats, and behind one is a shiny car. If you pick the one with a car.
after you pick a door, the host opens one of the others, revealing a goat. You are then given the option to change doors. Is it in your best interest to do so? At this point, most people would say it doesn't matter, you now have a 50/50 chance with either door. They are wrong. It is really in your best interest to switch, in which case you will have a 50% chance of winning, where as if you keep the same choice you only have a 33% chance.
Im not going to explain the mathematics behind the second one, as it is one of the most popular paradoxes, and there's a good chance that everyone here already knows why your chances would improve if you switched. If you dont, hink about it, it's really quite interesting.