Question: How many people do you need to have the odds be in favor (at least 50% chance) of two people having the same birthday?
Answer: At least 23 people.
For an explanation visit http://en.wikipedia.org/wiki/Birthday_problem
Question: Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a sports car; behind all of the others is bicycles. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 2, revealing a bicycle. He then says to you, "Do you want to pick door No. 3?" Is it to your advantage to switch your choice?
Answer: Yes, by changing your answer your chances of winning actually goes up from 1/3 to 2/3.
This becomes obvious when expanding the example. Suppose there was 100 doors rather than 3. You pick one and the host shows you that the car is not behind 98 of the doors then asks you to switch to the remaining door or keep the door you picked. Of course you would switch your door because chances are you didn't pick the correct door initially.
For more explanation go to http://en.wikipedia.org/wiki/Monty_Hall_problem#Solutions
Question: Three guests check into a hotel room. The clerk says the bill is $30, so each guest pays $10. Later the clerk realizes the bill should only be $25. To rectify this, he gives the bellhop $5 to return to the guests. On the way to the room, the bellhop realizes that he cannot divide the money equally. As the guests didn't know the total of the revised bill, the bellhop decides to just give each guest $1 and keep $2 for himself. Each guest got $1 back: so now each guest only paid $9; bringing the total paid to $27. The bellhop has $2. And $27 + $2 = $29 so, if the guests originally handed over $30, what happened to the remaining $1?
Hint: Make a list of all of the people involved and how much money they ended up with/spent.
Answer: The $9 paid by each guest accounts for the $2 that went to the bellhop. So rather than adding $27 to the $2 kept by the bellhop, the $27 accounts for the bellhops money. The $27 plus the $3 kept by the guests does add up to $30.
Question: There are 100 coins scattered in a dark room. 90 have heads facing up and 10 are facing tails up. You cannot tell which coins are which. How do you sort the coins into two piles that contain the same number of tails up coins?
Answer: The piles don't need to be the same size, so make a pile of 10 coins and a pile of 90 coins, flip all of the ten coins and it is guaranteed that the piles have the same number of tails.
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