This is a very good answer for a limit game, but if you are playing in a no-limit game against players who have your $1 trillion covered, the probability of you busting out approaches 1 as the number of trials approach infinity.

In this situation, it is possible to bust out in a single hand. You're unlikely to go all in, but it will happen eventually. Most of the times when you go all in, you will be a large favorite but not a lock. If you change your strategy such that you will only go all in on the river with the nuts, you will be beaten very easily. That strategy can very easily be defeated by an opponent who pushes preflop on every hand--even if you had AA, you'd have to fold, because you'd only be a 9:1 favorite.