How many observations needed to reach 90% confidence? Let's say I have a pool of 1,000 marbles. Most likely they are all red. Maybe a few black. How many marbles do I need to look at (assuming each outcome is red) before I'm 90% confident that 99% of the marbles are red? What is this type of analysis called? Have I defined all the necessary parameters?
 A: The way you worded your question sounds a bit off. Taken literally, we never have a confidence that a parameter equals a point value. Usually, we talk about confidence intervals, or the probability that a parameter lies within a given range. As you know, the larger our sample, the smaller that internal will be (the more precise our estimate will be).
So the problem with your working is that we can draw large samples and get a confidence interval from, say [80%-100%], a larger one gives us [90%-100%]. A really large one gives [95%-99%] maybe a huge one [96%-98%] (all using 90% confidence). So only that last draw with a huge sample gives us a range that excludes your query of 99%. But that last example also suggests that you have more than 1% black marbles. 
I wasn't sure whether you meant to ask "90% confident that at least 99% of the marbles are red"? 
To pursue your analysis (which is known as a statistical power analysis), we do need to make an assumption about the true proportion of black marbles. If we assume it's zero, then the power analysis will simply conclude all samples of any size are 100% red. That does us no good.
So we pick a different starting assumption: 1% black. Any sample we draw has an expected proportion of 99% red, but we have to construct a sampling confidence interval around that that depends on the degree of confidence (90%) and the true proportion of red marbles we're assuming (99%). Again, we use the "true" or asserted proportion because we're asking a hypothetical question. 
The usual Normal approximation to the confidence interval would be .99 +/- z(.95)*sqrt(p(1-p)/n), where z(.95) is the inverse normal CDF that leaves 5% in each tail = 1.64. But this leaves you with a confidence interval that will likely exceed 1.0 on one side and that ought to tell you something's fishy. 
Turns out the normal approximation isn't great when p is very small or large (near 0 or 1); some people use the Poisson distribution in this case, and there are other approximations to the confidence interval outlined in this Wikipedia article: https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval
My experience is that the Poisson is a better approximation for these "rare event" scenarios. Fix the sample size at, say, 100. If we're assuming 1% of balls are black, then that's 1 ball in our sample. A Poisson(1) distribution has a 37% chance of seeing 0 blacks, 37% seeing 1 black, and 18% seeing 2 blacks. The cumulative probability of those 3 cases is 92%. So if we observe 3 or more balls being black in a sample of 100, we could reject our starting assumption that k=1 with 92% confidence.
The probability of 3 blacks in that distribution is 6%, and the CDF up to 3 blacks is 98%, so if we observe 4 or more balls we can reject the hypothesis of 1% black with 98% confidence.
Bottom line - this all depends on my interpretation of your question being correct, and there are multiple approximations for these "rare event" scenarios. (see the Wikipedia article)
