Here is a nice probability problem I saw in class about a week ago. It involves playing a game.
Consider a game where the player has initially dollars. The player tosses an unfair die (or flips an unfair coin, or whatever) so that the probability of winning is . If he wins he earns a dollar, if he loses he loses a dollar. This is played until one of two situations occurs: either he reachers a certain amount or he is bankrupt i.e. he loses all his money. Given find a formula that will give the probability of succes (meaning he reaches ).
According to our textbook this is a classic going all the way back to Bernoulli (I just forgot which one it is ).