You can: Assuming that Powerball simply collects all earnings and pays them to the winner (they don't and keep a share), simply regard your lottery ticket as your share and deposit the price you pay in your savings account.

You seem to deliberately ignoring the fact that Facebook is a publicly traded company.

You can't go up to the lottery winner and offer to buy part of their winning ticket. You can buy shares of Facebook today (in fact, at a nice discount currently).