Imagine an infinite sequence of games such that you are nearly certain to win each one, but you're also certain to lose all but finitely many of them. This seems really absurd. But given an infinite fair lottery, it can be easily arranged. Suppose a secret natural number N is chosen in our infinite fair lottery. Let Gn be the following game, for n a natural number:
- You win if N>n and you lose otherwise.
Imagine placing bets on this game. If it costs a penny to play and the payoff is a dollar, you'll think it's a great deal: after all, you're nearly certain you will win. But if you play all the games, you will make only finitely many dollars, and lose infinitely many pennies.
Conclusion? I suppose the best one is that infinite fair lotteries are impossible.