This one bugged me a lot recently. An evil wizard plays a game. Everyone knows the rules of the game. It goes like this.
He picks a person at random and calls him into a dark room. He rolls a six sided die. If he rolls six, he kills that person and the game is over forever. If he rolls less than six, he sets that person free, and never bothers that particular person again, but the game continues.
If the game continues, then instead of calling in one person, the evil wizard calls in nine. He rolls a six sided die. If he rolls six, he kills all nine and the game is over forever. If he rolls less than six, he sets all nine free, and never bothers those particular nine people again, but the game continues.
If the game continues, then instead of calling in nine people, he calls in ninety people ...
The game continues in this manner until he rolls a six and the game ends (and everyone in the room is slaughtered). (The pattern for the number of people he calls into the room is: 1, 9, 90, 900, 9000, ... etc. The thing to note is that at any time after the first stage, the number of people in the room is equal to 90% of the number of people who have ever been in the room.)
Now, here's the deal. You get a call from a stranger informing you that your friend Julian entered the room. The stranger doesn't know what happened after that. He also knows the game is over, but he doesn't know when it ended. What is the probability that Julian is dead?
The paradox: there are two good answers.
Answer one: The probability is 1 in 6. We know that Julian entered the room with some number, possibly zero, of other people. At that time, the wizard rolled a die, and there was a one in six chance that he killed Julian and his cellmates.
Answer two: The probability is 9 in 10. We know that Julian was in the room, and nine in ten people who ever entered the room are dead.