Here's the thing: in Bayesian updating, probability mass moves from one hypothesis to another. That means you need another hypothesis. For Bayes' theorem, you need three terms: the prior probability of your hypothesis (99%), the probability of seeing the experimental result assuming your hypothesis is true (0.0001%), and the probability of seeing the experimental result whether or not the hypothesis is true. You need another piece of information, and that will depend on your state of knowledge. For instance, the coin could be perfectly fair, slightly balanced in favor of tails (40:60), slightly balanced in favor of heads (60:40), 90:10 (either direction), etc. You need another hypothesis, or probably a bunch of them.
If you want to make the math simple, you can just consider two possibilities: your hypothesis, and the null hypothesis (the coin is perfectly fair, 50:50). If that's the case:
P(A|X) = P(A)P(X|A) / (P(X|A)P(A) + P(X|N)*P(N))
P(A|X) = (.99)*(.000001) / ((.99)*(.000001) + (.5)*(.01))
which works out to: P(A|X) = 0.000197961... (Nota bene: Check my math. I haven't slept much recently, and I could easily have made any number of mistakes)
Again, this is a really simplistic calculation, because there are so many other plausible hypotheses, and I haven't studied quite enough probability theory to do all the calculations, even if I had more information.
That's pretty much what I got. I used a probability distribution for the null hypothesis, but it still works out to 50% for the first flip.
This isn't intended as a full discussion, I'm just a little fuzzy on how a Bayesian update or any other kind of probability update would work in this situation.
You have a coin with a 99.9999% chance of coming up tails, and a 100% chance of coming up either tails or heads.
You've deduced these odds by studying the weight of the coin. You are 99% confident of your results. You have not yet flipped it.
You have no other information before flipping the coin.
You flip the coin once. It comes up heads.
How would you update your probability estimates?
(this isn't a homework assignment; rather I was discussing with someone how strong the anthropic principle is. Unfortunately my mathematic abilities can't quite comprehend how to assemble this into any form I can work with.)