The more I think about it--and I'm a whopping thirty years old, ya know--I don't think it makes sense to define probability. I think you either know what "1 in four chance" means, or you don't.
(1) It doesn't mean, "If you did this 1 billion times, then..." It's statistically possible that you could get 1 billion heads in a row with a fair coin. So you can't talk like that to define 1/2 probability.
(2) I used to think it could mean, "You would be willing to give x odds in a wager on the subject." But no, that doesn't work either, because of both incentive problems and "diversification" issues (for lack of a better term). E.g. if I say there's a certain probability I'll miss my connecting flight (we were in the hotel between legs of the trip home when I thought of this), that doesn't translate into the odds I'd need to wager against myself.
Obviously, I would always be willing to bet that I'd miss my connecting flight, because I can just walk very slowly. On the other hand, I might not want to bet the other way, because who wants to get a big payoff if he makes his connection, but lose money if he misses it??