Do you mean that you're 99% confident in your reasoning that it comes up tails 99.9999% of the time? If so, you'd be much less than 99.9999% sure of heads in the first place.
You generally use beta distribution for coinflips. One beta distribution that would get your certainty is alpha = 999999, beta = 1. Landing on heads would get you a posterior of alpha = 999999, beta = 2, which would give you a certainty of about 99.9998% of landing on tails.
My problem is that your confidence isn't well specified. If you could give me a standard deviation, that would work better. Also, with something like this, a beta distribution isn't actually a very good prior. The most likely reason for it to land on heads is that you messed up, and the probability is more like 99.9%, which would be crazy unlikely under the prior I gave.
Do you mean that you're 99% confident in your reasoning that it comes up tails 99.9999% of the time? If so, you'd be much less than 99.9999% sure of heads in the first place.
You can be 99.9999% sure of heads - and 99% confident of that - if you memorised your confidence - but then subsequently could not remember for sure if there were six "9"s - or maybe seven.
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.)