On the table is are 10 jars, labeled with the number of assorted marbles contained within (say 1,000 and 1,000,000).
The goal is to estimate how many blue marbles are in each jar. Unfortunately the marbles are not evenly distributed and non-blue marbles may be clumped together (at an unpredictable rate).
You can pull as many samples of any size from the jar as you'd like (from any place in the jar). But each marble included in a sample costs $1 to check.
What are the most cost-effective strategies to estimate how many blue marbles are in each jar (say, ±20% with 95% probability)? Does the strategy differ based on the size of the jar?
For example, the first jar contains 1000 marbles, some of which are blue. I can pay $1000 to learn the exact amount of marbles.
What is the best strategy to reach 95% confidence that my estimate would be within 20% of the actual?
Example strategies for this first jar of 1k marbles:
50 Samples of 2 marbles (costs $100)
20 Samples of 10 marbles each (costs $200)
100 Samples of 5 marbles each (costs $500)