Is there a way to reduce a sample size such that I gain roughly the same amount of information out of it as I would get from a bigger one? I was thinking of Cochran's formula for determining the ideal sample size under maximum variability, however, I couldn't find a way to formulate my problem such that I end up with some sort of relevant attribute.
Nonetheless, suppose I want to have 99.9% accuracy for a given task –– this means 1 fail out of 1000 experiments. However, performing 1000 experiments is quite expensive, so I want to reduce that number. Is there a way, for example, to perform 100 experiments and use that data to talk "as if" I would have done 1000?
Any hints would be appreciated!

I'm assuming you want to confirm your 99.9% accuracy based on experiments. But actually, running 1000 experiments won't confirm a 99.9% succes rate. You may be unlucky, and have 2 fails, even if the "true" success rate is 99.9%. There are various ways to arrive at a confidence interval for the succes rate, based on data, and there are also ways to figure out how many experiments you need for getting a small enough (for your purposes) confidence interval. As you can imagine, if you want to be sure the "true" success rate is between 99.45% and 99.55%, you're going to need to run many more than 1000 experiments.
