I've recently came across very interesting question and I'm looking for the simplest solution. It is also very interesting to see calculations for the expected amounts for both cases.
Alice wants to join her school's Probability Student Club. Membership dues are computed via one of two simple probabilistic games.
The first game: roll a die repeatedly. Stop rolling once you get a five followed by a six. Your number of rolls is the amount you pay, in dollars.
The second game: same, except that the stopping condition is a five followed by a five.
Which of the two games should Alice elect to play? Does it even matter?