It seems some information is missing, so I'll try reformulating the problem my way:
A jar contains 1 fair coin (equal odds for heads and tails), and 99 unfair coins (99.9% chances of tails, 0.1% of heads). You pull out a coin, flip it, and it comes up tails. What are your expectations for the second flip?
P(fair & heads) = 0.01 * 0.5 = 0.005
P(unfair & heads) = 0.99 * 0.001 ~= 0.001
so P(heads on first throw) = 0.006
so P(fair | heads on first throw) = 5/6
so P(heads on second throw | heads on first throw) = 5/6 0.05 + 1/6 0.001 ~= 0.42
... at least, that's one way of interpreting "you are 99% confident in your results", I consider that the remaining 1% is "your analysis was completely wrong and the coin is just as likely to land on heads or tails". A more realistic situation would be one where your confidence would be distributed among possible coins in coinspace, something like "90% confident of less than 0.01% odds for heads, 99% confident in less than 1% odds for heads, 99.9% confident in less than 10% odds for heads, etc.".
I like the reinterpretation of the problem, but is
P(unfair & heads) = 0.99 * 0.0001 ~= 0.001
a typo? Just running the numbers through SpeedCrunch gives 0.99 * 0.0001 = 0.000099, and 0.000099 ~= 0.0001, which seems intuitively right as 0.99 is "almost" 1.
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.)