Say that you are given a coin that you are informed has a slight unfairness, somewhere between 1% and 10% and that you know which side the coin is biased in favor of (that is you know heads will show somewhere between 51 and 60 percent of the time).
How many times must this single coin be flipped in order to narrow your believed unfairness to the nearest integral percent with 95% confidence? (That is, you are to say "With 95% confidence this coin is no more than _% unfair")
I know from questions like How many tosses for 95% centainty that coin is not fair that having that level of confidence that a particular coin is 100% unfair is quick (8 heads in a row will give you that level of confidence) but an only slightly biased coin is a very different matter.
My thinking is that you approach this problem saying first "if this coin is 10% unfair then X trials will be needed" and if those trials are that unfair you can stop but if not then you say "9% unfair requires an additional __ trials" and so on.
My suspicion is that this is going to require an enormous number of flips and the closer to fair the quicker "enormous" grows (simply because even an actually perfectly fair coin will very frequently produce numbers approaching these levels). But narrowing that rough suspicion to actual numbers is proving elusive.
