Pirate Game and Coupon Collector's Problem

Solve these two classic brain teasers: **(a) The Pirate Game:** Five pirates (ranked 1 through 5 by seniority, with 5 being the most senior) must divide 100 gold coins. The most senior pirate proposes a split. All pirates vote, and the proposal passes if at least half vote in favor (the proposer can vote for himself). If it fails, the proposer is thrown overboard and the next most senior pirate proposes. Each pirate is perfectly rational, wants to maximize gold, and prefers survival above all else. What split does Pirate 5 propose? **(b) The Coupon Collector:** There are $n$ distinct coupon types. Each time you buy a product, you receive one coupon uniformly at random. What is the expected number of purchases needed to collect all $n$ types?