Your odds are, via my bullshit math, somewhat better than that. Well, not complete BS, but here goes ;)
Let's assume that we've eliminated all disease as a cause of death. Further assume we've cheated old age via a fountain of youth potion, such that death due to aging is no longer possible either. We are not, of course, immortal as a result - stepping in front of a bus will still do you in, in our brave new world. So we're left with death by some type of injury as the sole remaining cause of death for everyone - accident, suicide, or homicide.
Now, according to the fine folks at the National Safety Council ("You there, safen up!"), your odds of dying from accident or injury were 1 in 1755 in 2002. This means that you had about a 0.06% chance of dying due to misadventure in 2002, or a 99.94% chance of surviving the year - at least as far as injury is concerned. So to go forward, for the sake of convenience let's assume that this rate of 1 in 1755 continues to hold true for the next few millennia or so. Assume a fixed rate of death due to injury, in other words. This is likely to be a big source of BS here - there's no real reason to assume it'll remain the same, but what the hell.
Okay, relying on the false precision of my calculator here, your odds of survival for a given year are 1754/1755, or 0.999430199 (yeah, yeah - and I cut some off, too, but let's not have a significant digits argument here). Extrapolate those odds outward, and what you find is that you hit 50% odds at around 1216 years. In other words, you have a 50% chance of surviving to age 1216, given those odds of death due to injury.
Now, the problem is, that number doesn't really tell us all that much about you or any other given individual - your odds of surviving this year are the same as they were last year, and there's no bookkeeper up there keeping track of the odds steadily accumulating one way or the other for you specifically. But what it does tell us is something about the whole population. Basically, what we have is a fixed rate of attrition - decay, in effect - and what we can do is start thinking in terms of a half-life for the whole population, rather than an average lifespan. So given that fixed rate of decay, 1 in 1755 annually, our half-life as a population should be right around 1216 years.
Okay, the folks at CDC tell me that 4,091,063 live births occurred in 2003 (yeah, I'm mixing years - sue me). Let's assume we applied the magic elixir to them at birth, eliminating any possibility that they'll ever die of disease or "natural causes". Okay, well, we know that, statistically speaking, half of them will be dead by 1216 years of age, but what about the rest? Grinding our way through our half-life calculator, what we find is that after 1500 years, we're still left with about 42.5% of our original population of 4+ million babies. After 2000 years, we're left with about 32% of our original population. Not bad, eh? At 3000 years, we drop to about 18%, and at 5000 years, slightly less than 6% of our original population should still be hanging around, statistically speaking.
Now let's go for it. After 10,000 years, only about 0.33% of our original population will be left - somewhat less than 14,000 people out of the 4+ million we started with. After 15,000 years, we're down to 0.02% of the original population, or fewer than 800 people. After 20,000 years, we drop down to 0.001% of the original population, or fewer than 50 people.
What's it all mean? Well, our elixir would, based on the current rate of death by injury, make it possible to live for a very long time in a few cases. The vast majority of people wouldn't make it that far, but for a lucky few, they would have a lifespan that far exceeded what we currently know as the entire span of recorded human history, which is only about 5000 years so far if you go back to the early Sumerians. Something to ponder with your coffee in the morning ;)
Excellent post! I'm keeping it for future reference. =)
By the way, I suppose the "death by accident" rate would increase from its present value, because if you knew you were immortal, you'd possibly take some more risks from time to time -- skydiving, bungee jumping, maybe even dueling, etc.