(Disclaimer: I'm not an educated mathematician. Feel free to double check my results and tell me if I'm wrong)
So, to explain the lottery question and the very unintuitive answer..
Every time you don't win, someone else wins. The number of people who have won the lottery once will very quickly grow significantly larger than the number of people you are. Since they are more than you, the odds of one of them winning are higher than the odds of you winning, which means that there will eventually be at least one person who has won the lottery twice.
That person has the same chance of winning as you do, so each drawing, you both have a one-in-a-hundred-million chance of winning. If neither do, either someone who has never won will win, or someone who has won once will win, and there will suddenly be twice as many people who have won twice than you are.
Because of the high number of participants, this gets iterated maaaany times before you're likely to win (50 million times on average), and the pool of winners who aren't you has plenty of time to grow.
Even in a lottery of just 1000 people, your odds of winning before someone else wins three times is only about 15%.
The odds of you winning a fair 100 million person lottery before someone else wins TEN times in total is about 50%.
This, my friends, is the power of the pigeon hole principle.